Quantum mechanical momentum and energy balance equations for transport of charge carriers

Abstract

In this paper some basic features of electrical transport are discussed in terms of quantum mechanical energy and momentum balance equations. The latter are presented as an appropriate tool for the investigation of carrier transport in electric circuits containing mesoscopic active areas in the presence of inhomogeneous electromagnetic fields.1 Investigating electrical transport It goes without saying that transport of electrical charges through conducting media has been a topic of utmost importance ever since the discovery of electricity and magnetism. However, realistic models and theories predicting both microscopic and macroscopic transport features have to cope with a fundamental complication: not only do they have to comply with Maxwell's equations governing the electromagnetic fields that are compatible with a given charge and current distribution, they also need to provide the tools to go the opposite way and to find out how charge and current distributions respond to a given electromagnetic field. In other words, all reliable transport theories have to overcome in one way or another a major problem of selfconsistency to arrive at a workable calculation scheme. Macroscopic models relating the electric currents to the voltage drops arising in electric circuits have been established and used long before sound microscopic theories relying merely on first principles were constructed. Ohm's law and Kirchoff's laws addressing the voltage drops in arbitrary network loops and the currents flowing through a network nodes are prominent successful examples of such models. Although practical calculations for complicated circuits can be very cumbersome, they are based on relatively simple recipes emerging from a circuit theory that is partially phenomenological. On one hand, the theory accounts correctly for the basic electric and magnetic induction phenomena by exploiting the integral form of Maxwell's equations and lumping together all microscopic details into global quantities such as capacitance and inductance, characterizing ideally behaving components of the circuit. No phenomenological approach has been invoked at this level, since characteristic quantities like inductance and capacitance can be expressed in a straightforward way in terms of the magnetic permeability, the electric permittivity and a set of geometry dependent form factors [1]. On the other hand, irreversible processes underlying the dissipation of both energy and momentum of the charge carriers due to various microscopic scattering mechanisms, are treated in a purely phenomenological fashion by postulating a frictional force that is

Tunneling-lifetime model for metal-oxide-semiconductor structures

“…G͑r , z , t͒ and R͑r , z , t͒, respectively, describe the local generation and recombination rates which, in general, are functionals of ͑r , z , t͒ and the hole density. As such, the continuity equation does not merely represent a classical conservation law but may also be derived from rigorous quantum dynamics 14 and can thus be used in the present context. Since we explicitly assume that all generation-recombination events are exclusively localized in the semiconductor part of the structure, we integrate Eq.…”