Modeling of quantum transport in open systems

Whereas classical transport physics is based on the concept of a probability distribution which is defined over the phase space of the system, the concept of a phase-space distribution function in the quantum formulation of transport is difficult, since the non-commutation of the position and momentum operators (the Heisenberg uncertainty principle) precludes the precise specification of a point in phase space. However, within the formulation of quantum mechanics, various formalisms based on density matrices, Wigner functions, Feynman path integrals and Green’s functions have been developed. These embrace the quantum nature of transport; moreover, in recent years, each technique has been utilized to address key aspects of quantum transport in semiconductors. At present, there is no unifying, user-friendly approach to quantum transport in semiconductors. Density matrices, and the associated Wigner function approach, Green’s functions, and Feynman path integrals all have their application and computational strength and weakness, and all their are equivalent representations of the quantum nature of transport. In the present work the density-matrix and Wigner-function formalisms will be employed. This choice is due to the fact that the “open system problem” that here is faced, is better managed using such an approach; Indeed the density-matrix formalism is extremely useful to show the degree of quantum coherence of the system under investigations while the Wigner-function picture is the ideal instrument to describe real-space quantum devices. Such a choice will be better understood looking in more detail to the problem of open systems, i.e., systems with open spatial boundaries. The most interesting products of micro- and nanoelectronics technology are systems that operate far from equilibrium. A closer inspection of a few examples of such systems reveals that they are generally open, in the sense that they exchange matter with their environment. The present work is aimed at developing a fully microscopic theory to describe open quantum systems starting from the so-called Semiconductor Bloch equations, namely the equations that describe the coherent versus incoherent dynamics of a closed quantum system. In the context of the present work, an open system is a system that exchange locally particles with its environment. Moreover, we wish to focus upon its far-from-equilibrium behaviour, and thus the specific definition of open system will be further restricted to describe a system coupled to at least two separate particle reservoirs, so that a non-equilibrium state may be created and maintained. To specify such a system we must regard it as occupying a finite region of space, and thus the exchange of particles must consist of a current flowing through the system surface which is taken to be the boundary of the system. Our aim is to analyse in detail the problem of openness in the present sense with also the possibility to consider energy-relaxation and dephasing processes within the device active region. More specifically, our analysis will allow us to point out and overcome some basic limitations of the conventional Wigner-function formalism; this will be accomplished by introducing a Generalized Weyl-Wigner approach, able to remove such anomalies, thus recovering typical results of partially phenomenological models. In this context we shall propose a theoretical scheme where the boundary conditions are described via a source term, i.e., a term representing the particles entering the simulated region from its spatial boundaries. In particular, we shall propose and demonstrate two fully equivalent theoretical models able to describe adequately an open quantum device: the first one is characterized by a non-diagonal source term (i.e., coherent source) while the second one is characterized by a diagonal source term.