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Electron transport is a ubiquitous phenomenon in modern technology, which underpins the operation of transistors, read heads for hard-disk drives and a multitude of chemical and biological sensors. The rapid approach of the typical device size to the nanoscale makes standard device simulators, based on effective Hamiltonian, inadequate and novel atomistic and materials specific transport theories are needed. Density functional theory combined with the non-equilibrium Green's functions scheme (NEGF+DFT) for electron transport provides today the most accurate and still numerically practical approach to the problem. In particular it is the only fully ab initio method which is currently capable of tackling systems with a large number of degrees of freedom. How this is achieved will be the subject of this Chapter, where I will also mention the pitfalls of such a method and the proposed remedies. Furthermore I will provide a glance over possible applications where transport theory has been demonstrated useful and efficient.

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