## Nanoelectronic Modeling Lecture 21: Recursive Green Function Algorithm

Electrical and Computer Engineering, Purdue University, West Lafayette, IN

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#### Abstract

The Recursive Green Function (RGF) algorithms is the primary workhorse for the numerical solution of NEGF equations in quasi-1D systems. It is particularly efficient in cases where the device is partitioned into reservoirs which may be characterized by a non-Hermitian Hamiltonian and a central device region which is Hermitian. Until now (2009) it also appears to be the only scalable algorithm that enables the rapid computation of incoherent transport with NEGF.

The NEGF equations formally require the inversion of a sparse system Hamiltonian matrix to compute the full matrix Green function. However, in most practical computations one does not need the full inverse but only a few elements of the inverse. RGF provides the formidable feature that it allows for the targeted computation of a few blocks of the inverse. RGF is numerically efficient in that it scales as O(N B3) where N is the number of layers and B the number of coupled orbitals in that layer. RGF is therefore particularly efficient for long and thin devices, and in the limit of an effective mass model or a 1D heterostructure effectively scales as order N.

This lecture derives the RGF equations from standard Dyson equations and provides an intuitive overview of the computational flow. The lecture indicates that for a pure current calculation only a single fully connected green function element is needed, while the computation of quantum charge requires the computation of all diagonal elemets of the inverse and a few partial rows to off-diagonal elements.

**Learning Objectives:**

- Demonstrate the need for an efficient computational algorithm that targets only a few inverse matrix elements, rather than the full matrix inverse.
- Computational flow of the Recursive Green Function (RGF) algorithm for
- Pure current calculations – only a single matrix element is needed
- Quantum charge calculations – all diagonal matrix elements are needed and a few partial rows of off-diagonal elements

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Università di Pisa, Pisa, Italy