AMaSiS 2018 Workshop: Abstracts

Contour integral techniques for large-scale quantum transport simulations from first-principles

Mathieu Luisier and Sascha Brück

ETH Zurich, Integrated Systems Laboratory

The dimensions of transistors, the active components of all integrated circuits, and of memory cells have kept decreasing over the last 50 years so that they nowadays do not exceed a few nanometers. The presence or absence of one single atom at a critical location can, for example, profoundly affect the performance of these nano-devices, making their design more and more complex. To simplify this process, better understand the behavior of already manufactured structures, and predict the characteristics of not-yet-fabricated ones, it is therefore very useful to have access to a reliable modeling tool. The latter should be able to capture the atomic granularity of the simulated systems as well as the quantum mechanical effects that might affect them.

This can be ideally achieved by combining density-functional theory (DFT) and quantum transport (QT). After constructing a Hamiltonian matrix representing the device of interest with a localized basis, two tasks must be performed: (i) computing the open boundary conditions (OBCs) that connect the simulation domain with its environment and (ii) solving the resulting Schrödinger equation with the Non-equilibrium Green’s Function (NEGF) or Wave Function (WF) formalism. Here, the focus is set on the determination of the OBCs since these calculations can easily become the limiting factor of ab initio quantum transport simulations if not handled properly.

Decimation techniques [1] have long been applied to compute the OBCs, but they scale cubically with the number of atoms and require several iterations to converge. Transformations into standard eigenvalue problems [2] are usually faster, but also suffer from the O($N^3$) scaling. Recently developed methods based on contour integrals, e.g. the FEAST [3] and Beyn [4] algorithms, have paved the way for much faster evaluations of the OBCs. They allow to only account for the boundary states that matter, not the fast decaying ones, and they can be efficiently parallelized. Since the size of the numerical problems depends on the number of states that are retained, not on the device dimensions, the obtained computational complexity outperforms the O($N^3$) scaling limit of other methods, especially in structures composed of several thousands atoms.

After presenting the adaptation of the FEAST and Beyn algorithms to OBCs calculations, few QT examples will be introduced to demonstrate the advantages of these methods over competing approaches [5].

Acknowledgments: This work was supported by the Hartmann Müller-Fonds on ETH-Research Grant No. ETH-34 12-1 and by a grant from the Swiss National Supercomputing Centre under Project No. s662.