Optimal block-tridiagonalization of matrices for coherent charge transport

TL;DR

This paper introduces an optimal matrix reordering algorithm based on graph partitioning that transforms tight-binding Hamiltonians into block-tridiagonal form, enhancing quantum transport calculations for complex geometries.

Contribution

The authors develop a novel graph partitioning-based algorithm for optimal block-tridiagonalization of Hamiltonians, enabling more efficient and versatile quantum transport simulations.

Findings

Significant performance improvements in transport calculations.

Applicable to complex multi-terminal geometries.

Demonstrated on semiconductor and graphene systems.

Abstract

Numerical quantum transport calculations are commonly based on a tight-binding formulation. A wide class of quantum transport algorithms requires the tight-binding Hamiltonian to be in the form of a block-tridiagonal matrix. Here, we develop a matrix reordering algorithm based on graph partitioning techniques that yields the optimal block-tridiagonal form for quantum transport. The reordered Hamiltonian can lead to significant performance gains in transport calculations, and allows to apply conventional two-terminal algorithms to arbitrary complex geometries, including multi-terminal structures. The block-tridiagonalization algorithm can thus be the foundation for a generic quantum transport code, applicable to arbitrary tight-binding systems. We demonstrate the power of this approach by applying the block-tridiagonalization algorithm together with the recursive Green's function algorithm to various examples of mesoscopic transport in two-dimensional electron gases in semiconductors and graphene.