A quantum Drift-Diffusion model and its use into a hybrid strategy for strongly confined nanostructures

Clément Jourdana; Paola Pietra

doi:10.3934/krm.2019010

Article Contents

2019, Volume 12, Issue 1: 217-242. Doi: 10.3934/krm.2019010

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A quantum Drift-Diffusion model and its use into a hybrid strategy for strongly confined nanostructures

Clément Jourdana^1, , and
Paola Pietra^2,

1.
Univ. Grenoble Alpes, CNRS, Grenoble INP¹, LJK, 38000 Grenoble, France
2.
Istituto di Matematica Applicata e Tecnologie Informatiche "E. Magenes" - CNR, Via Ferrata 5a, 27100 Pavia, Italy

* Corresponding author: C. Jourdana
* Corresponding author: C. Jourdana

¹ Institute of Engineering Univ. Grenoble Alpes

Received: December 31, 2016

Revised: February 28, 2018

Published: July 2018

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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 Schrödinger 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 Schrödinger 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.

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Mathematics Subject Classification: 35Q40, 65Z05, 82D37, 82D80.

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Figure 1. Schematic longitudinal section of the CNTFET

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Figure 2. 3D (left) and 2D (right) representation of atom positions in a (10, 0) 'zig-zag" CNT

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Figure 3. Mobility influence on the current-voltage characteristics obtained with the S-QDD approach

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Figure 4. Current-Voltage characteristics obtained with the five different models

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Figure 5. 2D slice of the potential energy (eV) at thermal equilibrium (left: DD model, right: QDD model)

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Figure 6. 2D slice of the density in logarithm scale at thermal equilibrium (left: DD model, right: QDD model)

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Figure 7. 2D slice of the density in logarithm scale for $V_{DS} = 0.2$ V (left: DD model, right: QDD model)

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Figure 8. Comparison of the potential energy (left) and the density (right) at thermal equilibrium. Curves obtained with S (dashed), DD (dotted) and QDD (solid)

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Figure 9. Comparison of the potential energy (left) and the density (right) for $V_{DS} = 0.2$ V. Curves obtained with S (dashed), DD (dotted) and QDD (solid)

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Figure 10. Comparison of the inverse of the density at thermal equilibrium (left) and for $V_{DS} = 0.2$ V (right). Curves obtained with S (dashed), DD (dotted) and QDD (solid)

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Figure 11. Comparison of the potential energy (left) and the inverse of the density (right) at thermal equilibrium. Curves obtained with S (dashed), S-DD (dotted) and S-QDD (solid)

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Figure 12. Comparison of the potential energy (left) and the inverse of the density (right) for $V_{DS} = 0.2$ V. Curves obtained with S (dashed), S-DD (dotted) and S-QDD (solid)

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Figure 13. Potential energy at thermal equilibrium obtained with S-QDD, moving the left interface position $x_{I_{1}}$ (left) and the right interface position $x_{I_{2}}$ (right)

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Figure 14. Inverse of the density at thermal equilibrium obtained with S-QDD, moving the left interface position $x_{I_{1}}$ (left) and the right interface position $x_{I_{2}}$ (right)

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Figure 15. Current-Voltage characteristics obtained with S-QDD, moving the left interface position $x_{I_{1}}$ (left) and the right interface position $x_{I_{2}}$ (right)

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Figure 16. Potential energy (left) and density (right) at thermal equilibrium obtained with S-QDD in the isotropic case (solid curves) and in the anisotropic case (dashed curves)

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Figure 17. Current-Voltage characteristics obtained with S-QDD in the isotropic case (solid curves) and in the anisotropic case (dashed curves)

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