J Comput Electron (2009) 8: 267–286 DOI 10.1007/s10825-009-0293-z Ballistic quantum transport using the contact block reduction (CBR) method An introduction Stefan Birner · Christoph Schindler · Peter Greck · Matthias Sabathil · Peter Vogl Published online: 3 October 2009 © Springer Science+Business Media LLC 2009 Abstract The contact block reduction (CBR) method is a variant of the nonequilibrium Green’s function formalism and can be used to describe quantum transport in the bal- listic limit very efficiently. We present a numerical imple- mentation of a charge self-consistent version of the CBR al- gorithm. We show in detail how to calculate the electronic properties of open quantum systems such as the transmis- sion function, the local density of states and the carrier den- sity. Several 1D and 2D examples are provided to illustrate the key points. The CBR method is a very powerful tool to tackle the challenge of calculating transport in the ballistic limit for 3D devices of arbitrary shape and with an arbitrary number of contacts. Keywords Ballistic quantum transport · Nonequilibrium Green’s function formalism · NEGF · Transmission function · Landauer-Büttiker formalism · Device simulation 1 Introduction Since electronic devices have been shrinking steadily to nanometer dimensions, quantum transport is increasingly becoming a topic of interest not only to physicists but S. Birner ( ) · C. Schindler · P. Greck · M. Sabathil · P. Vogl Walter Schottky Institut and Physics Department, Technische Universität München, Am Coulombwall 3, 85478 Garching, Germany e-mail: stefan.birner@nextnano.de Present address: M. Sabathil OSRAM Opto Semiconductors GmbH, Leibnizstr. 4, 93055 Regensburg, Germany also to the electrical engineering community [1]. The non- equilibrium Green’s function (NEGF) formalism (e.g. [2]) provides a rigorous framework for the development of quantum device models. Here, we describe one of its implementations—the contact block reduction (CBR) me- thod [3]. It can be used to describe quantum transport in the ballistic limit very efficiently. Our aim in this article is to make the Green’s function formalism in the limit of ballistic quantum transport accessible to a more general audience. Thus, a detailed description of the underlying algorithm is given and numerical examples are provided as concrete il- lustrations. As it is very important to perform charge self- consistent calculations, we also give details on how to solve the nonlinear system of coupled Schrödinger and Poisson equations. Interested readers should be able to reproduce these results by setting up their own computer program. All results presented in the figures of this article can be repro- duced with the software that is provided as an Online Re- source [4]. 2 Ballistic quantum transport A conductor shows nonohmic behavior if its dimensions are smaller than certain characteristic lengths: The mean free path and the phase-relaxation length of the electron [5]. If the length of a conductor becomes shorter than the mean free path, the conductance approaches a limiting value. This classical ballistic limit has still nothing to do with quantum mechanics. Quantum mechanics does not become impor- tant until the dimensions of the conductor are smaller than the phase-relaxation length and interference-related effects come into play. In present day high-mobility semiconductor heterostructures such as modulation doped GaAs/AlGaAs heterojunctions or quantum wells, mean free paths and