Quantum Phase-Space Transport and Applications to the Solid State Physics

Abstract

Quantum modeling is becoming a crucial aspect in nanoelectronics research in perspective of analog and digital applications. Devices like resonant tunneling diodes or graphene sheets are examples of solid state structures that are receiving great importance in the modern nanotechnology for high-speed and miniaturized systems. Differing from the usual transport where the electronic current flows in a single band, the remarkable feature of this new solid state structures is the possibility to achieve a sharp coupling among states belonging to different bands. Under some conditions, a non negligible contribution to the particle transport induced by interband tunneling can be observed and, consequently, the single band transport or the classical phase-space description of the charge motion based on the Boltzmann equation are no longer accurate. Different approaches have been proposed for the full quantum description of the electron transport with the inclusion of the interband processes. Among them, the phase-space formulation of quantum mechanics offers a framework in which the quantum phenomena can be described with a classical language and the question of the quantum-classical correspondence can be directly investigated. In particular, the visual representation of the quantummechanical motion by quantum-corrected phase-plane trajectories is a valuable instrument for the investigation of the particle-particle quantum coherence. However, due to the non-commutativity of quantum mechanical operators, there is no unique way to describe a quantum system by a phase-space distribution function. Among all the possible definitions of quantum phase-space distribution functions, the Wigner function, the Glauber-Sudarshan P and Q functions, the Kirkwood and the Husimi distribution have attained a considerable interest (Lee, 1995). The Glauber-Sudarshan distribution function has turned out to be particularly useful in quantum optics and in the field of solid state physics and the Wigner formalism represents a natural choice for including quantum corrections in the classical phase-space motion (see, for example (Jungel, 2009)). This Chapter is intended to present different approaches for modeling the quantum transport in nano-structures based on the Wigner, or more generally, on the quantum phase-space formalism. Our discussion will be focused on the application of the Weyl quantization procedure to various problems. In particular, we show the existence of a quite general multiband formalism and we discuss its application to some relevant cases. In accordance with the Schrodinger representation, where a physical system can be characterized by a set of projectors, we extend the original Wigner approach by considering a wider class of representations. The applications of this formalism span among different subjects: the 1