Pergamon M athl. Comput. M odelling Vol. 25, No. 12, pp. 33- 53, 1997 Copyright@1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 08957177/97 $17.00 + 0.09 PII: SO8957177(97)00093-9 Using the Wigner F’unction for Quantum Transport in Device Simulation M. NEDJALKOV AND I. DIMOV Center for Inform&& and Computer Technology Acad. G. Bountchev Str. B125A, Sofia, Bulgaria P. BORDONE, R. BRUNETTI AND C. JACOBONI Istituto Nazionale per la Fisica della Materia Dipartimento di Fisica, Universitit di Modena Via Campi 213/A, 41100 Modena, Italy (Received March 1997; accepted April 1997) Abstract-The Wigner function was introduced as a generalization of the concept of distribution function for quantum statistics. The aim of this work is pushing further the formal analogy between quantum and classical approaches. The Wigner function is defined as an ensemble average, i.e., in terms of a mixture of pure states. From the point of view of basic physics, it would be very appealing to be able to define a Wigner function also for pure states and the associated expectation values for quantum observables, in strict analogy with the definition of mean value of a physical quantity in classical mechanics; then correct results for any quantum system should be recovered as appropriate superpositions of such “ pure-state” quantities. We will show that this is actually possible, at the co& of dealing with generalized functions in place of proper functions, Keywords-Wigner function, Quantum transport, Convergence procedure, Pure-state Wigner function, Generalized functions. 1. INTRODUCTION Transport in mesoscopic systems has been widely investigated in recent years [l-4]. This increas- ing interest is mainly related to the fundamental problems of the basic physics involved, as well as to possible utilization of such systems in the electronic device production. Since the dimensions of mesoscopic systems are comparable with typical electron coherence lengths, a correct analysis of transport phenomena in such systems requires a detailed quantum- mechanical treatment. The general case of interest is that of a mesoscopic semiconductor structure coupled with two leads considered as reservoirs at equilibrium. The potential can have an arbitrary profile inside the structure, while it takes constant and different values in the two leads. Thus, the system is open and, in the general case, far from equilibrium. The transport properties of the electron system are provided by quantum equations originating from the Liouville-von Neumann equation for the density matrix operator. In order to study such structures, the Wigner function [5-81 is a valid tool. In fact, it provides a rigorous quantum-mechanical approach and it constitutes a direct link between quantum and The authors would like to thank T. Gramchev (University of Cagliari) for helpful discussions and comments. This work has been partially funded by A.R.O. and O.N.R. through E.R.O. and 1501, MM449 of MSETB. 33