Molecular electronics – the use of single molecules as functional entities in electronic devices – is often heralded as a replacement for the conventional CMOS approach which faces physical limits in further miniaturization. Even if consumer products based on this emerging technology will not be available in the near future, single molecule devices offer a unique test bed to explore fundamental aspects of both physics and chemistry already today .
Molecular electronics – the use of single molecules as functional entities in electronic devices – is often heralded as a replacement for the conventional CMOS approach which faces physical limits in further miniaturization. Even if consumer products based on this emerging technology will not be available in the near future, single molecule devices offer a unique test bed to explore fundamental aspects of both physics and chemistry already today . Experiments can now directly probe the properties of a finite molecule with discrete level structure coupled to macroscopic metallic leads in and strongly out of equilibrium. Electronic transport mediated by charge and spin degrees of freedom is measured and found to be strongly influenced by electron-electron, electron-phonon as well as electron-photon interactions. These interactions give rise to effects like Spin- and Frank-Condon blockade of the current [2, 3], device heating  or light induced transport , to name just a few. Over the last six years, 29 groups in Germany, together with collaboration partners worldwide, interlinked their efforts to advance the field of quantum transport at the molecular scale under the auspices of the German Science Foundation in the Priority Program SPP 1243. The goal of this meeting is to reflect what has been achieved and frankly disclose open problems. Besides members of the SPP, leading international experts are invited to discuss about the future of the field and formulate the great challenges: Molecular electronics: Quo vadis?
In the past it turned out that the details of the geometry including the leads is important for the device performance . As this can vary from junction to junction one often obtains in experiment a wide range of different characteristics for the same molecule/lead combinations. To improve reproducibility, a control of the atomic-scale geometry is tried to be achieved by chemical methods. In another approach, the conduction between the leads and the molecule is enhanced, such that the overall conduction of the junction is not dominated by the molecule-to-lead coupling . In addition, in break-junction setups, a gating of the junction has become important, which enables three-terminal device geometry.
In recent year great progress has been achieved in exploiting also the spin-degree of freedom in molecular electronics [8, 9]. This holds true for different experimental setups such as break-junctions as well as various different STM geometries .
Very recently, in scanning tunneling spectroscopy temporal resolution down to the sub-nanosecond scale has been obtained . This has been enabled by the application of pulsed bias voltages, in analogy to optical pump-probe spectroscopy. Whereas so far this technique had been applied to metallic systems only, we expect that this interesting technique will also be applicable to molecular junctions and thereby open new research perspectives in the near future.
On the theoretical side, efforts may be divided into model based approaches and first-principles treatments. In the latter, non-equilibrium Greens function theory on the basis of Density Functional Theory (DFT) has emerged as the standard approach over the last years. These kinds of simulations provide important atomistic information on charge transport in realistic devices. For example, such calculations contributed to the search for optimal anchor groups that bind the molecule effectively to the electrodes while maintaining high transparency for electron flow .
In order to improve also the quantitative description of the current, several approaches based on many body perturbation theory in the GW approximation have been proposed [13, 14]. They overcome the overestimation of conductances in the standard approach and lead to excellent agreement with experimental observations. However, even cases which have been considered previously to be to difficult for DFT, were successfully described recently in this theoretical framework. To this class belong strongly correlated systems that exhibit Coulomb blockade  or Kondo physics [16, 17].
Another recent line of research in the field of first principles simulations is the description of dynamical effects in quantum transport in the context of time-dependent DFT. Here different groups work on transient effects and light assisted induced and suppressed currents [18, 19].
Work employing model Hamiltonians is based on the observation that frequently, only few electronic orbitals of a molecule contribute to charge transport through single molecules. The model-based approach has been instrumental in understanding the interaction between the electronic degrees of freedom and vibrational as well as spin modes of the molecule. These interactions are the source of numerous new quantum transport effects many of which have been observed in the laboratory.
Model based approaches are particularly important in the case when a mean-field (e.g. DFT based Landauer-Büttiker) approximation cannot be used. This is the case for strongly correlated systems, and in particular in the nonequilibrium current-carrying state, when inelastic processes are important. Recently several theoretical models and methods were developed and used intensively. They are based mainly on the electron-vibron model, the Anderson (multi-)impurity model, describing a discrete-level system with Coulomb interaction coupled to the leads, and the (multi-level) Anderson-Holstein model, which takes into account both Coulomb and electron-vibron interactions. In addition, to describe the Kondo effect special Kondo models are considered. All these models are solved roughly within two main approaches: the nonequilibrium Green function (NGF) technique, usually in the case of strong coupling to the leads, and the quantum master equation (QME) in the weak-coupling limit.
The Green function method can be used starting from a mean-field approximation and allows to include systematically interactions beyond the mean-field level, both perturbatively and nonperturbatively . In the case of weak to intermediate electron-vibron interaction the NGF method was developed recently and used to describe experiments with molecular junctions. It is especially applicable to inelastic electron tunneling spectroscopy and nonequilibrium vibrons [21-24]. Also progress was made in the case of strong electron-vibron interaction, leading to local polaron physics and important single-charge memory effects [25-26].
The quantum master equation is formulated in the basis of many-body eigenstates of a molecule, weakly coupled to leads. It is widely used in the case of intermediate to strong electro-electron and/or electron-vibron interactions. It gives a complete description of sequential tunneling and the main features of Coulomb blockade [27, 28], as well as Frank-Condon blockade , and was extended recently to investigate cotunneling effects in systems with Coulomb and electron-vibron interactions [30, 31].
Model-based approaches also intimately link molecular electronics to related fields such as quantum transport through quantum dots or nanoelectromechanical systems.