# Nested Dissection Solver for Transport in 3D Nano-Electronic Devices

## TL;DR

This paper extends the Hierarchical Schur Complement method to efficiently solve non-equilibrium Green's functions in three-dimensional nanoscale devices, achieving over 1000x speed-up compared to traditional algorithms.

## Contribution

The work applies the HSC-extension to 3D NEGF problems, reducing computational complexity from N^7 to N^6 and enabling practical simulations of large nanoscale systems.

## Key findings

- HSC-extension reduces operation count from N^7 to N^6.
- Runtime speed-ups exceed three orders of magnitude.
- Enables simulation of large realistic 3D nanoscale devices.

## Abstract

The Hierarchical Schur Complement method (HSC), and the HSC-extension, have significantly accelerated the evaluation of the retarded Green's function, particularly the lesser Green's function, for two-dimensional nanoscale devices. In this work, the HSC-extension is applied to determine the solution of non-equilibrium Green's functions (NEGF) on three-dimensional nanoscale devices. The operation count for the HSC-extension is analyzed for a cuboid device. When a cubic device is discretized with $N \times N \times N$ grid points, the state-of-the-art Recursive Green Function (RGF) algorithm takes $\mathcal{O}(N^7)$ operations, whereas the HSC-extension only requires $\mathcal{O}(N^6)$ operations. %Realistic operation counts also depend on the system dimensions in $xyz$-directions and the form of contact self-energy matrix. Operation counts and runtimes are also studied for three-dimensional nanoscale devices of practical interest: a graphene-boron\- nitride-graphene multilayer system, a silicon nanowire, and a DNA molecule. The numerical experiments indicate that the cost for the HSC-extension is proportional to the solution of one linear system (or one LU-factorization) and that the runtime speed-ups over RGF exceed three orders of magnitude when simulating realistic devices, such as a graphene-boron nitride-graphene multilayer system with 40,000 atoms.