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