Fudan University - April 2010 - Instructor: Richard M. Martin
Schedule of Lectures
(Last modified April 30, 2010)
Every student must make a project report. Here is an extensive list possible topics for projects
|
|
|
|
|
|
||||
|
[For motivation and information only - students are not responsible for this material] |
||||||||
|
|
|
Electronic Structure
and the Properties of Matter: Overview of status and challenges A. Examples of extended band-like (weakly-correlated) and localized atomic-like (strongly-correlated) behavior: Na, Si, C - diamond, graphene, Cu, Ni, NiO, La2CuO4, Rare earths ... B. The Fundamental Hamiltonian for electrons and nuclei in materials given to emphasize the difficulty of the problem and the importance of the work that makes it possible to calculate properties of materials. |
|
|
||||
|
The needed material can be found in standard texts on solid state physics or will be given in the lectures. Extensive additional description is given in the text; this is not required. |
||||||||
|
|
|
Theory of electrons in materials treated as independent
as particles that obey the exclusion principle. This is the basis of all the methods used in this course A. Typical types of binding and structures of crystals. B. The Bloch theorem, the Brillouin zone, and electron bands in crystals. C. Two alternative descriptions of the bands provide insight in to the nature of electronic states: formation of bands as atoms are brought together, and evolution of bands starting from free electrons. |
|
|
||||
|
|
|
Density Functional Theory: the Hohenberg-Kohn Theorem,
and the Kohn-Sham Equations A. An amazingly simple idea that has made possible accurate calculations of many properties of materials - including the many-body electron-electron interactions - using independent-electron methods. B. The force (Hellman-Feynman) theorem: another amazing simple expressions that allows one to calculate forces on the nuclei - the basis for molecular dynamics calculations (see lecture 6). |
|
|
||||
|
|
|
Solving the Kohn-Shams Eqs. - a self-consistent set of independent electron
Schrodinger-like equations. A. Choices of the functionals - examples of the widely-used simple expressions that that have proved to be very useful: the local density approximation (LDA) and improved forms. B. Tests on simple systems (atoms, Hydrogen molecule) and examples of successes and failures in real systems |
|
|
||||
|
These cannot be treated in detail. All the methods can be found in solid state texts; we will emphasize relations between the methods and typical applications |
||||||||
|
|
|
A. Tight-binding methods that express the wavefunctions in
terms of atomic-like orbitals; this
provides the simplest interpretation of bands B. Plane waves that are Fourier transforms appropriate for periodic crystals. This is the most direct method that is most easily extended to large calculations; very efficient due to the fast Fourier transform. C. Augmented methods that combine plane waves with spherical harmonics in regions around each nucleus. This is most complete approach but more difficult to use. |
|
|
||||
|
|
|
Continuation of previous lecture and efficient iterative methods described using place waves. The iterative methods are described in courses on computational physics. |
|
|
||||
|
The physical quantities are found in a standard solid state text, and the calculations described here use methods defined in previous lectures. |
||||||||
|
|
|
A. Examples of calculations using plane waves: structures
and elastic constants of crystals; phase transitions under pressure, calculations of surface structures. B. Molecular dynamics the describe thermal motion of the atoms using Newton;s laws and forces from the electrons. Car-Parrinello (and related) simulations. |
|
|
||||
|
|
|
Continuation of the previous lecture and conclusions. |
|
|
||||
|
|
||||||||
|
|
||||||||
|
|
||||||||
|
|
|
Overview of many-body methods (without mathematical formulation!) Quasiparticles, Excitations and the "GW" approximation |
|
|
||||
|
|
|
Overview of Strongly Correlated Systems: Local moments, Metal-Insulator Transitions, Dynamical Mean Field Theory (DMFT) |
|
|
||||
|
|
|
If there is time for a third special lecture: Berry phases and Electric Polarization: Berry phases are a beautiful idea that generalizes the Bohm-Aharonov effect (the phase of the wavefunction for a charge circling a magnetic solenoid) to create a new approach to many problems in physics. Why polarization is difficult, and why it is related to a Berry phase. |
|
|
||||
Send reports of problems to:
Prof. Martin, rmartin@illinois.edu
Click here to go the Course Home
Page