Density Functional Theory and Block Tridiagonal Matrix Invertion
Ph.D. - Forsvar, Dan Erik Petersen
Simulation software for the analysis of electronic structure at the atomic level is a problem of fundamental interest in today's world of ever increasing use of nano-devices designed at the atomic level. At these length scales, the usual methods of classical physics to predict device characteristics fails in the face of quantum mechanical effects, and new methods rooted in quantum mechanics are the norm.
Since the seminal work of Kohn, Sham, Mermin and others, the method of Density Functional Theory (DFT) has become one of the most studied and employed methods, of theoreticians and experimentalists alike, in the modelling of properties of materials where quantum mechanical effects matter. The leaps and bounds of computing power and availability has enabled the useful modelling of larger and more complex systems, and as computing resources improve, simulation will likely become the norm for assessing and designing most, if not all, nanoscopic devices, materials and molecules in the future.
One of the most important and costly calculation steps that arises in DFT is that of the generating the Green's function matrix, used both in the process of determining ground state properties of materials or devices, as well as the transmission, conductance, and other non-equilibrium properties of these modelled systems. The Green's function matrix is essentially the inverse of a large matrix with block tridiagonal structure. However, as only the same block tridiagonal structure of the original matrix is desired from the inverse, algorithms can be designed to take advantage of both the sparsity and structure of the input matrix as well as the desired output matrix.
This talk will give a short introduction to DFT, the origin and properties of the block tridiagonal matrix for which we need to calculate the Green's function matrix, some algorithms - in serial and parallel - for this calculation, as well as a much improved method for the calculation of transmission of nanodevices over the usual method found in DFT literature.
Mads Brandbyge, Assoc. Prof., PhD (DTU Nanotech,
Technical University of Denmark)
Martin Bastiaan van Gijzen, Assist. Prof., PhD (Delft University of Technology)
Jørgen Bonde Sand, Assoc. Prof., PhD (DIKU, University of Copenhagen)
Vejleder: Stig Skelboe (DIKU, University of Copenhagen)
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