# American Institute of Mathematical Sciences

February  2019, 12(1): 217-242. doi: 10.3934/krm.2019010

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

 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

1 Institute of Engineering Univ. Grenoble Alpes

Received  January 2017 Revised  March 2018 Published  July 2018

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.

Citation: Clément Jourdana, Paola Pietra. A quantum Drift-Diffusion model and its use into a hybrid strategy for strongly confined nanostructures. Kinetic & Related Models, 2019, 12 (1) : 217-242. doi: 10.3934/krm.2019010
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##### References:
Schematic longitudinal section of the CNTFET
3D (left) and 2D (right) representation of atom positions in a (10, 0) 'zig-zag" CNT
Mobility influence on the current-voltage characteristics obtained with the S-QDD approach
Current-Voltage characteristics obtained with the five different models
2D slice of the potential energy (eV) at thermal equilibrium (left: DD model, right: QDD model)
2D slice of the density in logarithm scale at thermal equilibrium (left: DD model, right: QDD model)
2D slice of the density in logarithm scale for $V_{DS} = 0.2$ V (left: DD model, right: QDD model)
Comparison of the potential energy (left) and the density (right) at thermal equilibrium. Curves obtained with S (dashed), DD (dotted) and QDD (solid)
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)
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)
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)
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)
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)
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)
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)
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)
Current-Voltage characteristics obtained with S-QDD in the isotropic case (solid curves) and in the anisotropic case (dashed curves)
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