Gæsteforelæsning - David Gontier, Paris 7 (Dauphine) og Antoine Levitt, INRIA Paris


27.06.2017 kl. 13.00 - 17.00


Title: Numerical methods in solid state physics

In this series of lessons, we focus on the electronic properties of
crystals, from a numerical point of view. Mathematically speaking, the
goal is to study the spectral properties of a self-consistent periodic
Hamiltonian. The lesson is divided in four parts:
1. First, we explain how the different current mathematical models are
designed from first principles (ab initio models). We focus on models
based on Density Functional Theory (DFT). The notion of self-consistent,
or mean-field, Hamiltonian is introduced, and the basic structure of a
DFT code is presented.
2. In a second part, we review the basic mathematical tools to study
periodic Hamiltonians, and in particular the Bloch-Floquet transform.
The theory of supercell models is then introduced. We prove that, for
insulator systems, such models are exponentially accurate for the
calculation of the energy, with respect to the size of the supercell.
3. For the third part, we focus on metallic systems. We review the
co-area formula, and explain its use for a priori numerical estimators.
We then introduce smearing methods, and provide numerical bounds for the
resulting algorithms.
4. For the last part (and if time allows), we present a non-linear
method for the calculation of defect energies in semi-conductors. We
introduce the notion of Wannier functions, and explain their use for
such models.


Institut for Matematiske Fag


Niels Jernes Vej 14, lokale 4.111