1.- Motivation and Presentation
For over four decades, microelectronics industry has been characterized by an exponential growth of the performance of its products [1,2]. On one hand, the level of integration has increased ("Moore's law"), as well as the switching velocity and functionality of integrated circuits. On the other hand, power consumption and cost per operation has decreased. Most of these developments have arisen as a direct consequence of the ability of the electronics industry to further reduce the size of conventional MOSFET. However, there is nowadays a broad consensus that the geometric scaling of MOSFET is not enough to provide the expected performance gain of future electronic devices [1,2]. Given this, the scientific community has identified the "More Moore" domain, which looks for avenues to evolve traditional CMOS devices by means of a tradeoff between the traditional miniaturizing and the introduction of new technological solutions (high-κ dielectric, multiple gate transistors, stressed silicon, metal gates, etc.), known as "equivalent scaling" . There is a wide agreement that these are currently the best strategies for the electronics industry in the 2007-2022 period predicted by the latest ITRS . However, the scientific community is searching for completely different alternatives to CMOS, since the long-term scaling required by Moore's law (4 nm channel length transistors predicted for 2022 ) will be technologically and economically unattainable. In this sense, the "Beyond CMOS" domain explores emerging electronic devices whose operation is based on different physical principles than MOSFET, being able to improve at least some aspects of the MOSFET performance. For example, devices based on the spin orientation control rather than the electron charge dynamics are being pursued (“spintronics”), as well as devices based on tunnel transport such as “resonant tunneling diodes” (RTD) and ”single-electron devices”. Building block materials diffe rent from bulk Silicon are also under investigation, such as “silicon nanowires” or "Carbon-based Nanoelectronics" (i.e. carbon nanotubes and graphene). It is currently not clear which of these proposals may replace the MOSFET. In any case, it is believed that in the near future some of these emerging devices can coexist with nanometer CMOS structures by using technologies not necessarily based on electronics (MEMS, sensors, etc), combined with new architectures (quantum computing, bio-inspired, etc.) or new connections (3D, "Silicon-on-Package"), what has come to be known as the “More than Moore" domain [1,2].
This research project is basically devoted to the simulation of nanoelectronic devices (and their quantum correlations) in the transition between the "More Moore" and "Beyond CMOS" domains. The Monte Carlo (MC) technique, applied to the solution of the Boltzmann equation, has been the tool chosen for the international scientific community to simulate electronic devices for several and well-founded reasons: it provides a strictly accurate solution of the Boltzmann equation through an intuitive picture of the dynamics of electrons by using trajectories, it allows us to obtain the I-V characteristic of the devices and other information of relevance, such as the local velocity distribution or the local electric field, and it is a very versatile technique as a "simulated experiment" to save costs and efforts in the development of prototypes. At a national level, we highlight the Spanish groups at the University of Salamanca (D. Pardo, T. González, E. Velázquez), University of Granada (F. Gamiz, A. Godoy, J.A. López Villanueva, J.A. Jimenez Tejada) and University of Santiago de Compostela (A.J. Garcia-Loureiro) for their well-known experience in the use of MC simulations for nanoelectronics. In addition, international groups of relevance are the University of Illinois at Urbana-Champaign (K. Hess and U. Ra vaioli), the Arizona State University (D. Ferry and D. Vasileska), University of Modena (C. Jacoboni), University of Lecce (L. Reggiani) and the MC simulator called Damocles  by D. Fischetti, to provide predictions for prototypes of the IBM company. Silvaco  and Synopsys  also provide commercial software for MC device simulations.
From a theoretical point of view, the Boltzmann equation is not able to directly include the relevant quantum effects in structures with a characteristic size comparable to the de Broglie wavelength of an electron. Therefore, the scientific community has developed many simulators to delve into quantum and/or atomistic properties as required in the "More Moore" and "Beyond CMOS" domains. We mention the NEMO software , designed initially by S. Datta, R. Lake and G. Klimeck to study “high speed electronics” with RTDs for the Texas Instruments company, and subsequently evolved into a general-purpose nanoelectronic simulator used by Intel, Motorola, HP, Texas Instruments and many universities . At the European level, we highlight two similar projects: NEXTNANO  developed by the group of P. Vogl, from the Walter Schottky Institute, and TiberCad  by the group of Aldo Di Carlo, from the Department of Electronics Engineering of the “Università di Rome Tor Vergata ". These three simulators for emerging electronic devices share a tight-binding description  of the band structure in addition to a stationary treatment of electron transport based on the "Non-Equilibrium Green function formalism” (NEGF) . These are very good tools to study the I-V characteristics (DC) of a variety of "quantum-based" devices, but they face many problems obtaining results in the frequency regime (AC) or describing stochastic fluctuations in time (noise). The evaluation of the quantum correlations of electrons (due to the Coulomb int eraction and Pauli principle) was and is an extremely complicated issue  but very relevant for the rigorous estimation of the time-dependent current and its fluctuations . The current measured with an ammeter is not only related to the rate of electrons passing through a surface, but also related to the time variation of the electric field at that same surface (i.e. displacement current) . Therefore, better approximations to the "many-body" problem (i.e. to the quantum correlations) should be considered when quantum simulators are built to predict the frequency behavior of emerging devices . On the other hand, there are also several “first-principle” simulators [14-17] dealing with a much improved treatment of the quantum correlations using the Density Functional Theory technique (DFT) proposed by Dr. W. Kohn  (which received the Nobel Prize in Chemistry in 1998 "for his development of the density-functional theory”) to calculate equilibrium states (i.e. minimum energy). We mention the Transiesta project  (which has generated the Danish spin-off company Atomistix) using the DFT technique together with NEGF to study DC transport in nanostructures. Despite its undeniable success for qualitative DC predictions (quantitative comparison with experimental results may differ by orders of magnitude, stemming from the use of a ground state theory for electronic devices out of equilibrium and the treatment of transport with single-particle techniques), obtaining information about the device frequency behavior is not possible. The Time-Dependent DFT technique  is an excited state theory, enabling the rigorous analysis of out of equilibrium systems. The Octopus simulator , developed by the group of A. Rubio belonging to the "Nano-Bio Spectroscopy Group” of San Sebastián, is designed for a future implementation of methods to estimate the current in electronic systems, although their main activity is the study of temporal dynamics of more basic physical-chemical phenomena. Other simulat ors and other techniques  well assessed for the rigorous treatment of quantum correlations are designed solely for physical-chemical studies, but they are far from the possible application to estimate the time-dependent current in electronic devices.
Thus, we are at a moment where the scientific community has an acceptable capability to predict the I-V characteristics (DC) of quantum/atomistic devices in the "Beyond CMOS" domain (with simple approximations to quantum correlations), but no simulators are able to complement these studies with AC, transient or noise behavior. The latter predictions are equally (or even more) important than DC predictions in order to properly assess the final role that the several proposals in the “Beyond CMOS” domain will have in the near future. We should mention that the fundamental theory of AC transport in mesoscopic devices currently shows a great level of activity, highlighting the relevant work of Drs. M. Büttiker  and W. Frensley  among others, but this fundamental theory has been developed at a purely theoretical level, with simplified Hamiltonians where geometry and quantum correlations are modeled with adjustable parameters that do not allow the development of general purpose simulators based on "first principles techniques” for quantitative predictions.
In summary, the scientific community needs a versatile, rigorous and all-inclusive nanoelectronics simulation tool to study the quantum/atomistic devices in the "Beyond CMOS" domain with the capabilities (DC, AC, noise, transients) that the MC simulation technique has provided for traditional classical devices. Nowadays, there is no publicly available simulator that can adequately fill this gap. Our propuse is developing a quantum-trajectory simulator trying to achive this goal. The ideas behind the BITLLE S simulator are described at http://europe.uab.es/xoriols.
 Intl. Technology Roadmap for Semiconductors, 2007 Edition (and 2008 update); http://www.itrs.net/home.html
 ENIAC Strategic Research Agenda 2007, http://www.eniac.eu
"It would indeed be remarkable if Nature fortified herself against further advances in knowledge behind the analytical difficulties of the many-body problem." Max Born, 1960
 X.Oriols, Phys. Rev. Lett. 98, 066803 (2007)
 A.Alarcón and X.Oriols, J. Stat. Mech. 2009, 01051 (2009)
 Y.M.Blanter and M. Buttiker, Phys. Rep. 1 336 (2000)
 W.Kohn and L.J.Sham, Phys. Rev. A, 140 1163 (1965)
 E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52 997 (1984)
 C. Fernando and W. Frensley Phys. Rev. B 52, 5092 (1995)