**4. Discussion**

The investigation of energy and electrical transport in nanoscale devices has recently attracted a consistent amount of interest for its wide range of potential applications in technology, biomedicine and energy materials to name just a few. A comprehensive look at the field would, therefore, require more than a reasonably short review. We have preferred a bird-eye introduction to what we believe are the actual more promising theoretical methods, the Boltzmann's equation and the Landauer-Büttiker formalism for quantum transport. These methods are somewhat complementary since they apply to different regimes, namely a diffusive regime where scattering events are important, or a ballistic regime where the quantum-mechanical effects are dominant. While these methods might not be the most accurate for any situation, most of the recent progress in our understanding of thermal, energy, and electronic transport at the nanoscale is based upon them or some of their extensions. Both methods benefit from an accurate description of the quantum-mechanical properties of the material under investigation. In this respect, the method of choice is Density Functional Theory, since its practical implementation allow for an almost straightforward interfacing with both the Boltzmann or Landauer theories (the last one through the Non-Equilibrium Green's functions formalism). Therefore, the use of DFT methods for electronic structure calculations (including phonon and vibration properties) combined with Boltzmann's or Landauer's theories is to some extent the state of the art of modern calculations.

We have briefly discussed the need to go beyond this *state of the art* both in terms of the limits of DFT or the limits of the approximations used in the methods themselves. Clearly, this is a very fertile field of investigation and in most cases it requires novel paradigms. We have exposed a few here: the introduction of methods beyond the constant relaxation time approximations ameliorates the description of electron and phonon transport by introducing energy-dependent relaxation times; The static DFT cannot, in principle, be the theory to study transport, an inherent out-of-equilibrium

problem. We therefore introduce extension of DFT able to properly take into account the dynamics of the electrons and show how this theory naturally allows for the description of strongly correlated effect in nanoscale molecular transport, in particular how it corrects the calculation of the Seebeck coefficient. We finally introduced the theory of open-quantum system, where the system under investigation is "open" to an external environment with which it could exchange energy, momentum, and particles. The external environment might have a passive role of driving an otherwise out-of-equilibrium system toward a steady or ground state, or an active role in which it drives the system itself allowing for novel quantum-mechanical effects to be exploited. As the OQS approach describes the dynamics of the density operator, this theory can be also used to investigate the influence of coherence in quantum transport devices. This might pave the way for novel strategies to construct quantum devices for thermoelectric energy conversion, photonics, spintronic injection, and sensing, to just name a few.

Clearly, the previous description cannot be complete. For example, we are aware of other DFT methods to include thermoelectric phenomena [64,65] or strong correlations [52,112–114] in the Kohn-Sham system, or an accurate description of vibrational modes and energy transport with classical methods by solving the Newton's equations [115,116], or the inclusion of non-harmonic effects in the non-equilibrium Green's function for the phonons [117]. Each of these topics deserves a complete introduction per se.

We hope this review serves both to the young researcher to have an outlook of the actual state of the art and potential ways beyond it, without striving through the immense amount of literature currently available and in continuous development, and to the experienced scientist as a reference where some of the most fundamental results and future outlooks are collected.

**Author Contributions:** R.B. and R.D. contributed equally in writing this review.

**Funding:** This research was funded by the Spanish Ministerio de Economia y Competitividad (MINECO) gran<sup>t</sup> number FIS2016-79464-P (SElecT-DFT) and MINECOG17/A01 (TOWTherm), by the Basque Government (Eusko Jaurlaritza) through the Grupos Consolidados (IT578-13 and IT1249-19). R.B. acknowledges funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie gran<sup>t</sup> agreemen<sup>t</sup> No. 793318. The APC was funded by Dresden University of Technology (TU Dresden).

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
