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