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.
|Publication status||Published - Jul 2008|