No Power Point Slides for Lecture 8 - See Ch. 12 in text and manual for TBPW code linked below
Link to TBPW - simple pedagogical plane wave code.
The big picture: Fourier Transforms always work in periodic systems. Direct proof of the Bloch theorem;
alternative way to understand the Brillouin Zone, etc.
But are plane waves useful? How many Fourier components are required for a realistic calculation?
Easy to show that plane wave calculations are out of the question for heavy elements.
Plane waves are widely used only because of pseudopotentials!; MANY aspects become much easier, e.g., Hellmann-Feynman Forces, Car-Parrinello type simulations, ....
- The Schrodinger equation for a periodic crystal
- General Expressions for Eqs. in a Plane Wave Basis
- Express periodic potential in Fourier components: Veff(G)
- Express wave function in basis of plane waves
- Equations lead directly to Bloch theorem, definition of Brillouin Zone, etc.
- Form of equations if V(r) is a sum of spherical potentials around each atom
- Form factors and structure factors
- The nearly free electron approximation
- Free electrons
- Effect of a weak lattice potential
- Remarkable fact that this seems to describe bands in many crystals
- Empirical Pseudopotentials
- Approximate the potential due to the electrons as spherical atomic-like
- Phillips-Kleinman - 1959 - can describe bands by a few parameters
- More recent accurate work - Chelikowsky and Cohen Book
- Computer program for plane wave calculation
- "TBPW" F90 code available at http://www.mcc.uiuc.edu/software/ under "instructional software"
- Flow Chart
- Identify parts of program common to all band structure calculations
- Crystal structure information
- Diagonalization routines
- Identify parts of program specific to plane waves
- Form of hamiltonian: essential ingredient is the set of Fourier components of the potential
- Empirical Pseudopotential: specified by a parameterized analytic form, or by values of the Fourier components of at the reciprocal lattice vectors
- Example input file
- Examples of Results in crystals - Si, GaAs
- Beyond the local approximation for the potential - needed for accurate calculations
- Conclusions