Abstract
In this paper we derive by an entropy minimization technique a local Quantum Drift-Diffusion (QDD) model that allows to describe with accuracy the transport of electrons in confined nanostructures. The starting point is an effective mass model, obtained by considering the crystal lattice as periodic only in the one dimensional longitudinal direction and keeping an atomistic description of the entire two dimensional cross-section. It consists of a sequence of one dimensional device dependent Schrodinger equations, one for each energy band, in which quantities retaining the effects of the confinement and of the transversal crystal structure are inserted. These quantities are incorporated into the definition of the entropy and consequently the QDD model that we obtain has a peculiar quantum correction that includes the contributions of the different energy bands. Next, in order to simulate the electron transport in a gate-all-around Carbon Nanotube Field Effect Transistor, we propose a spatial hybrid strategy coupling the QDD model in the Source/Drain regions and the Schrodinger equations in the channel. Self-consistent computations are performed coupling the hybrid transport equations with the resolution of a Poisson equation in the whole three dimensional domain.
Highlights
The extreme miniaturization reached in nanoelectronics brings the necessity of developing new models to describe the electron transport
In order to complete the picture of possible models for Carbon Nanotube Field-Effect Transistor (CNTFET)’s, we investigate in a second part the use of this Quantum Drift-Diffusion (QDD) model in a hybrid strategy, spatially coupling it with the effective mass Schrodinger model proposed in [8]
We compare the numerical results obtained with five different approaches: - approach S: the Schrodinger model proposed in [9] is used in the entire domain, - approach DD: the Drift-Diffusion model proposed in [20] is used in the entire domain, - approach QDD: the Quantum Drift-Diffusion proposed in Section 3 is used in the entire domain, - approach S-DD: the hybrid approach detailed in [19] couples spatially the Schrodinger system with the Drift-Diffusion model, - approach S-QDD: the hybrid approach described in Section 4 couples spatially the Schrodinger system with the Quantum Drift-Diffusion model
Summary
The extreme miniaturization reached in nanoelectronics brings the necessity of developing new models to describe the electron transport. We propose a formal derivation of a Quantum Drift-Diffusion (QDD) model in this context of strongly confined nanostructures. We assume that adiabatic decoupling occurs and, associating to each wave function a “densitymatrix” function, we obtain a sequence of Wigner equations After a description of the obtained QDD model (Proposition 3.2), we detail the three main steps of the derivation: the entropy minimization (Section 3.1), the diffusive limit using a ChapmanEnskog method (Section 3.2) and the semiclassical expansion up to the second order (Section 3.3). We describe the physical device and compare the numerical results obtained with the different models
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