# A numerical method to efficiently calculate the transport properties of large systems: an algorithm optimized for sparse linear solvers

## TL;DR

This paper introduces an efficient numerical method for calculating electronic transport in large nanostructures by leveraging sparse linear algebra, significantly reducing computational complexity.

## Contribution

The paper develops a wave-function matching method optimized for sparse solvers, enabling scalable transport calculations for large systems with improved efficiency.

## Key findings

- Operation count scales with N_S x N_P for large systems
- Method effectively handles large nanostructures
- Sparse linear solver accelerates calculations

## Abstract

We present a self-contained description of the wave-function matching (WFM) method to calculate electronic quantum transport properties of nanostructures using the Landauer-B\"uttiker approach. The method is based on a partition of the system between a central region ("conductor") containing $N_S$ sites and an asymptotic region ("leads") characterized by $N_P$ open channels. The two subsystems are linearly coupled and solved simultaneously using an efficient sparse linear solver. Invoking the sparsity of the Hamiltonian matrix representation of the central region, we show that the number of operations required by the WFM method in conductance calculations scales with $\sim N_S\times N_P$ for large $N_S$.