Diffusive Transport of Partially Quantized Particles: Existence, Uniqueness and Long-Time Behaviour

Abstract

A self-consistent model for charged particles, accounting for quantum confinement, diffusive transport and electrostatic interaction is considered. The electrostatic potential is a solution of a threedimensional Poisson equation with the particle density as the source term. This density is the product of a two-dimensional surface density and that of a one-dimensional mixed quantum state. The surface density is the solution of a drift-diffusion equation with an effective surface potential deduced from the fully three-dimensional one and which involves the diagonalization of a one-dimensional Schrödinger operator. The overall problem is viewed as a two-dimensional drift-diffusion equation coupled to a Schrödinger-Poisson system. The latter is proven to be well posed by a convex minimization technique. A relative entropy and an a priori L 2 estimate provide sufficient bounds to prove existence and uniqueness of a global-in-time solution. In the case of thermodynamic equilibrium boundary data, a unique stationary solution is proven to exist. The relative entropy allows us to prove the convergence of the transient solution towards it as time grows to infinity. Finally, the low-order approximation of the relative entropy is used to prove that this convergence is exponential in time.

Modeling and simulation of the diffusive transport in a nanoscale Double-Gate MOSFET

“…This fix point map is used in [27] to establish existence of solutions. More details about its use as numerical algorithm are given in Sect.…”

“…Indeed, if k is given by the subband model, i.e. it is the k-th eigenvalue of the stationary Schrödinger operator for an electrostatic potential energy U , then all these estimates hold true with constants μ and C 0 depending on U (see [29] and appendix of [27]). It remains to obtain some regularity on the potential energy U .…”

A Hybrid Classical-Quantum Transport Model for the Simulation of Carbon Nanotube Transistors

“…We also mention [12] where a hybrid strategy is studied with a quantum drift-diffusion equation. We point out that all these hybrid approaches are different from a "dimensional hybrid coupling" (see [8] e.g.) where electrons are described by a quantum model in the confined direction and a classical drift-diffusion equation along the transport direction.…”

A deterministic solver for a hybrid quantum-classical transport model in nanoMOSFETs

“…One approach for the modelling of such a device, sketched in Fig. 1, is the use of a different description of the electrons following the dimension [6][7][8]35]: they behave as waves along the confinement direction (z-direction) and as particles along the transport direction (x-direction). This kind of coupling goes by the name of dimensional coupling, to differentiate it by the geometrical coupling, where different zones of the device are described by different models put together by interface conditions [4,5,21].…”