Fys 208, Solid State

Notes about the lectures, Spring Semester, 2003

Updated 02.06. 2003

Please, return to this page regularly for news, reports, tasks, electronic material.....


Note: On this page we shall list what is to be done/has been done
      in the lectures and discussions.
      It is planned to put here links to all electronic versions of notes
      You can also obtain electronic materials from the previous years.
      See the links below


In the first lecture: 20.01.03 Introduction. Walking through the topics, about molecular physics Interesting phenomena Quantum Mechanics, Statistical Physics 22.01.03 Some photos of the blackboard Quantum Mechanics Highlights Probability density - square modulus of wave function Expectation values in classical probability Quant.Mech. expectation values To every quantity - an operator Hamilton operator and total energy Schrödinger equation with time; stationary Schrödinger equation H Psi = E Psi (seperation of time dependence) Eigenvalues - discrete solutions (sometimes continuous) Commutator Heisenberg uncertainety relations (Heisenberg in acoustics) Harmonic Oscillator - Newton Harmonic Oscillator - Quant. Ladder spectrum (equidistant) hbar omega (circular frequency vs. frequency) Schrödinger equation vs. Wave equation Plane waves (free particles in QM; simplest waves in Class. Wave Th.) 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) Splitting of levels, origins of bands This might be seen as origin of the bonding The lower level goes down (the united nuclei) Why harmonic oscillators? Close to any minimum V(x)=V_0+1/2 x^2 + ... 27.01.03 Some photos of the blackboard The linear chain of point masses with springs (1 dim. crystal) Lagrange equation - Newton equations Waves; Wave Ansatz Alternative? Eigenmodes (matrix diagonalization) Dispersion relation Ashcroft-Mermin chapter 22 pages 430-433 Some older notes (gif files) on linear chain Matlab files for chains etc: (download click by right mouse):CHAINS.tar or go through the files online: matlab files Used lchain3.m (needs also lchaigo3.m) to look at k>pi/a ADDED 04.02.03: lchain3.m has now variable delay slider 29.01.03 Some photos of the blackboard Linear chain with 2 different point masses Relation for the density of modes -NOTE: postscript or acrobat PDF Densities (for lattice vibrations; explanation in the NOTE) 2001: modedens.gif 2003: modedens.gif modedens.m Here are all the files (as well as matlab files) for the NOTE 3. dimensional density of modes: Density of modes (frequency.gif) Matlab files for chains etc: (download click by right mouse):CHAINS.tar or go through the files online: matlab files ADDED 04.02.03: lchain3.m has now variable delay slider 03.02.03 Some photos of the blackboard The linear chain of two different masses (or 2 different strings) Handwritten notes from earlier years (see also the blackboard above) 1. Two-atomic chain. 2. Two-atomic chain - with Omega. Blackboard photos: Phonons: quanta of normal modes (travelling waves) Neutron scattering Algebraic method for harmonic oscillators 05.02.03 Some photos of the blackboard Boltzmann factor (simple derivation) Discussing the Boltzmann Factor Work with Einstein Model Work with Debye Model Einstein vs. Debye model: Ashcroft-Mermin page 462 distributed notes: scanned copies (copied from 04.02.02 ) 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 10.02.03 Some photos of the blackboard 1. Debye; 2. Heat Conductivity Work with Debye Model: High temperature limit (see the note) The low temperature world: calorimetry with C ~ T3 Combined Einstein and Debye Model Einstein vs. Debye model: Ashcroft-Mermin page 462 Heat conductivity Thermal Conductivity: Ashcroft-Mermin page 499-500 Thermal Conductivity: see also electrons, Ashcroft-Mermin page 22 mean free path 12.02.03 Some photos of the blackboard 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'' Thermal Conductivity Umklapp process - see "chains" simulation earlier phonon scattering 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 Anharmonic effects: Thermal expansion 17.02.03 Photos of the blackboard with comments 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) 19.02.03 Photos of the blackboard with comments FERMI GAS MODELS Pauli principle Independent particles-independent probabilities Indistinguishable particles -- new form of probability P(r1,r2)=P(r2,r1) Slater Determinant Boltzmann and Fermi distributions Fermi Gas Available states (density of states in k-space) NOTE (scanned) fermi-density.jpg Fermi momentum Fermi Temperature, Fermi energy=Fermi Level Dependence on electron density 24.02.03 Photos of the blackboard with comments 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) Heat capacity etc Note: Due to Fermi statistics vanishingly small contribution to heat capacity from electrons 26.02.03 Photos of the blackboard with comments Wiedemann-Franz law in Fermi-gas version (Table 4, section THERMAL CONDUCTIVITY OF METALS, Kittel, 7.ed.p. 166) (A scan of the above will be added) (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)) Started: Bloch theorem via Fourier NOTE: Bloch.ps (postscript file) and Bloch.pdf (acrobat PDF file) Expansions in function sets (Fourier series, other sets) (also Ashcroft-Mermin p. 135-139) This is referred to as "second proof of Bloch theorem" 03.03.03 Photos of the blackboard with comments Bloch theorem via Fourier NOTE: Bloch.ps (postscript file) and Bloch.pdf (acrobat PDF 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" matlab file and drawings: Chain with two masses diagonalized (see 03.02.03 above) 05.03.03 Photos of the blackboard (comments later) Bloch theorem via Fourier - more 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 10.03.03 Notes on 3-dim and 2-dim bands 3dim.pdf: The same notes as a PDF file (set resolution 100% for the last 2 pages in 3dim.pdf) 3-dim crystals electronic bands the shape of bands 2-dim (see pictures above) Notes from 2001 (nearly as above): 3dim-1.jpg 3dim-2.jpg 3dim-3.jpg Ashcroft and Mermin (page 181) (chapter 10) Applications to an s-band arising from a single atomic s-level 12.03.03 Photos of the blackboard (comments later) GROUP VELOCITY 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) 17.03.03 Photos of the blackboard (comments later) Continued Semiclassical motion 19.03.03 Photos of the blackboard (with detailed comments) Bands, gaps, periodic bands Band gaps: Metals, Insulators, Semiconductors Semiconductors Energy Gap: diagram of resistivity, photon excitation Evaluation of number of electrons in bands (textbook) Semiconductor equation (textbook) Population of states in the conduction band Electrons and holes 'Number of available states' in a band Population of states in the valence band Semiconductor equation first part derived (Number of missing electrons, i.e. holes) 24.03.03 Photos of the blackboard (no comments yet) Semiconductor equation second part (Number of electrons in conduction band) Impurity states (Hydrogen-atom like) Electron states (States just below the lower edge of the conduction band) Hole states above the valence band upper edge (Lower edge for hole states is the inverse of valence band upper edge) 26.03.03 Photos of the blackboard with comments. Reciprocal Lattices (belongs to Crystals) Impurity states, position of mu Position of mu-level in doped crystals 31.03.03 Photos of the blackboard with comments. p-n-junction following the distributed notes. Drawings to p-n junction from L.G. Johansens thesis 02.04.03 Photos of the blackboard p-n-junction following the distributed notes. p-n-junction: Scanned notes for last and this lecture Discussed Drawings to p-n junction from L.G. Johansens thesis Powerpoint presentation of solid-state sensors The powerpoint file contains some nice visualizations of processes in semiconductors (Jan Kocbach, dr.scient. given subject nov. 2000) 07.04.03 Photos of the blackboard with detailed comments Metals, Fermi surfaces NOTE: distributed at lecture (cyclotron frequency) 09.04.03 Photos of the blackboard with detailed comments Hall effect. Magnetism. NOTE: distributed at lecture (Hall effect) NOTE: distributed at lecture ('all' on magnetism) 23.04.03 Photos of the blackboard (comments later) Magnetism; diamagnetism NOTE: distributed at lecture (Magnetic Properties of materials dia-, para-) 28.04.03 Photos of the blackboard (with comments) Magnetism; paramagnetism 30.04.03 Photos of the blackboard (comments later) Magnetism; Ferromagnetism 05.05.03 Superconductivity Crystal structure 07.05.03 NO LECTURE 12.05.03 NO LECTURE 14.05.03 NO LECTURE 19.05.03 Colloquium - topics review Some blackboard notes with comments PLAN for the rest 21.05.03 Colloquium - topics review 26.05.03 Presentations 28.05.03 Presentations 02.06.03 Additional texts, prepared 02.06.03 New (preliminary) version of Bloch-Fourier method The text below might help to work with the p-n junction From cand.scient. thesis of Lars Gimmestad Johansen The text is only meant to assist in the work with other sources. 06.06.2003 Friday: Oral exams. Compare with this file for year 2002 Compare with this file for year 2001