1 Scientific Computing, Department of Informatics and Mathematical Modeling, Technical University of Denmark2 Department of Informatics and Mathematical Modeling, Technical University of Denmark3 Department of Micro- and Nanotechnology, Technical University of Denmark4 Department of Applied Mathematics and Computer Science, Technical University of Denmark
This thesis is concerned with the modeling of electronic properties of nano-scale devices. In particular the computational aspects of calculating the transmission and current-voltage characteristics of Landauer-Büttiker two-probe systems are in focus. To begin with, the main existing methods are described in detail and benchmarked. These are the Green’s function method and the wave function matching method. The methods are subsequently combined in a hybrid scheme in order to benefit from a common formalism. The most time demanding stages of common electronic transport calculations are identified. For systems of more than about a hundred atoms, two specific tasks stand out; the evaluation of self-energy matrices to describe the coupling between the electrodes and the device, and the solution of the central region Schrödinger equation either by matrix inverse of by solving a system of linear equations. In this work the objective is to develop new efficient algorithms for these tasks in order to model nano-scale systems of larger size in the future. The starting point of the new methods is the combined formalism of the Green’s function and wave function matching methods. The first new algorithm described is for the calculation of the block tridiagonal matrix inverse of a block tridiagonal matrix in O(N) operations. This algorithm also leads to an optimal evaluation of the frequently used Caroli transmission formula. A modified wave function matching scheme is then developed which allows for a significant reduction in the cost of the self-energy matrix calculations when combined with an iterative eigensolver. Finally, such an iterative eigensolver is developed and implemented based of a shift-and-invert Krylov subspace approach. The method is applied to a selection of nano-scale systems and speed-ups of up to an order of magnitude are achieved.