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
References:
[1]

A. AkturkG. Pennington and N. Goldsman, Quantum modeling and proposed designs of cnt-embedded nanoscale mosfets, IEEE Transactions on Electron Devices, 52 (2005), 577-584. doi: 10.1109/TED.2005.845148.

[2]

A. M. Anile and S. Pennisi, Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors, Phys. Rev. B, 46 (1992), 13186-13193. doi: 10.1103/PhysRevB.46.13186.

[3]

L. Barletti and N. Ben Abdallah, Quantum transport in crystals: Effective mass theorem and k·P hamiltonians, Comm. Math. Phys., 307 (2011), 567-607. doi: 10.1007/s00220-011-1344-4.

[4]

L. Barletti and G. Frosali, Diffusive limit of the two-band k.p model for semiconductors, Journal of Statistical Physics, 139 (2010), 280-306. doi: 10.1007/s10955-010-9940-9.

[5]

L. Barletti and F. Méhats, Quantum drift-diffusion modeling of spin transport in nanostructures, Journal of Mathematical Physics, 51 (2010), 053304, 20 pp. doi: 10.1063/1.3380530.

[6]

M. BaroN. Ben AbdallahP. Degond and A. El Ayyadi, A 1d coupled Schrödinger drift-diffusion model including collisions, J. Comput. Phys., 203 (2005), 129-153. doi: 10.1016/j.jcp.2004.08.009.

[7]

N. Ben Abdallah, A hybrid kinetic-quantum model for stationary electron transport, J. Statist. Phys., 90 (1998), 627-662. doi: 10.1023/A:1023216701688.

[8]

N. Ben AbdallahP. Degond and P. A. Markowich, On a one-dimensional Schrödinger-Poisson scattering model, Z. Angew. Math. Phys., 48 (1997), 135-155. doi: 10.1007/PL00001463.

[9]

N. Ben Abdallah, C. Jourdana and P. Pietra, An effective mass model for the simulation of ultra-scaled confined devices, Math. Models Methods Appl. Sci., 22 (2012), 1250039, 40pp. doi: 10.1142/S021820251250039X.

[10]

Ph. CaussignacB. Zimmermann and R. Ferro, Finite element approximation of electrostatic potential in one-dimensional multilayer structures with quantized electronic charge, Computing, 45 (1990), 251-264. doi: 10.1007/BF02250636.

[11]

P. Degond and A. El Ayyadi, A coupled Schrödinger drift-diffusion model for quantum semiconductor device simulations, J. Comput. Phys., 181 (2002), 222-259. doi: 10.1006/jcph.2002.7122.

[12]

P. DegondF. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, Journal of Statistical Physics, 118 (2005), 625-667. doi: 10.1007/s10955-004-8823-3.

[13]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, Journal of Statistical Physics, 112 (2003), 587-628. doi: 10.1023/A:1023824008525.

[14]

A. El Ayyadi and A. Jüngel, Semiconductor simulations using a coupled quantum drift-diffusion Schrödinger-Poisson model, SIAM J. Appl. Math., 66 (2005), 554-572. doi: 10.1137/040610805.

[15]

E. GnaniA. GnudiS. ReggianiM. Luisier and G. Baccarani, Band effects on the transport characteristics of ultrascaled snw-fets, IEEE Trans. Nanotechnol., 7 (2008), 700-709. doi: 10.1109/TNANO.2008.2005777.

[16]

E. GnaniA. MarchiS. ReggianiM. Rudan and G. Baccarani, Quantum-mechanical analysis of the electrostatics in silicon-nanowire and carbon-nanotube FETs, Solid-state Electronics, 50 (2006), 709-715. doi: 10.1109/ESSDER.2005.1546610.

[17]

H. K. Gummel, A self-consistent iterative scheme for one-dimensional steady state transistor calculations, IEEE Trans. Electron Devices, 11 (1964), 455-465. doi: 10.1109/T-ED.1964.15364.

[18]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006.

[19]

C. Jourdana and P. Pietra, A hybrid classical-quantum transport model for the simulation of carbon nanotube transistors, SIAM J. Sci. Comp., 36 (2014), B486-B507. doi: 10.1137/130926353.

[20]

C. Jourdana and N. Vauchelet, Analysis of a diffusive effective mass model for nanowires, Kinet. Relat. Models, 4 (2011), 1121-1142. doi: 10.3934/krm.2011.4.1121.

[21]

A. Jüngel, Transport Equations for Semiconductors, Lecture Notes in Physics No. 773. Springer, Berlin, 2009. doi: 10.1007/978-3-540-89526-8.

[22]

A. Jüngel and R. Pinnau, A positivity-preserving numerical scheme for a nonlinear fourth order parabolic system, SIAM J. Num. Anal., 39 (2001), 385-406. doi: 10.1137/S0036142900369362.

[23]

B. Kozinsky and N. Marzari, Static dielectric properties of carbon nanotubes from first principles, Phys. Rev. Lett., 96 (2006), 166801. doi: 10.1103/PhysRevLett.96.166801.

[24]

C. S. Lent and D. J. Kirkner, The quantum transmitting boundary method, J. Appl. Phys., 67 (1990), 6353-6359. doi: 10.1063/1.345156.

[25]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.

[26]

A. MarchiE. GnaniS. ReggianiM. Rudan and G. Baccarani, Investigating the performance limits of silicon-nanowire and carbon-nanotube FETs, Solid-state Electronics, 50 (2006), 78-85. doi: 10.1016/j.sse.2005.10.039.

[27]

F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics, Journal of Statistical Physics, 140 (2010), 565-602. doi: 10.1007/s10955-010-0003-z.

[28]

F. Méhats and O Pinaud, A problem of moment realizability in quantum statistical physics, Kinetic and Related Models, 4 (2011), 1143-1158. doi: 10.3934/krm.2011.4.1143.

[29]

M. PourfathH. Kosina and S. Selberherr, Numerical study of quantum transport in carbon nanotube transistors, Math. Comput. Simul., 79 (2008), 1051-1059. doi: 10.1016/j.matcom.2007.09.004.

[30]

C. Ringhofer, Subband diffusion models for quantum transport in a strong force regime, SIAM Journal on Applied Mathematics, 71 (2011), 1871-1895. doi: 10.1137/100804164.

[31]

E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759. doi: 10.1007/978-3-642-59033-7_9.

show all references

References:
[1]

A. AkturkG. Pennington and N. Goldsman, Quantum modeling and proposed designs of cnt-embedded nanoscale mosfets, IEEE Transactions on Electron Devices, 52 (2005), 577-584. doi: 10.1109/TED.2005.845148.

[2]

A. M. Anile and S. Pennisi, Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors, Phys. Rev. B, 46 (1992), 13186-13193. doi: 10.1103/PhysRevB.46.13186.

[3]

L. Barletti and N. Ben Abdallah, Quantum transport in crystals: Effective mass theorem and k·P hamiltonians, Comm. Math. Phys., 307 (2011), 567-607. doi: 10.1007/s00220-011-1344-4.

[4]

L. Barletti and G. Frosali, Diffusive limit of the two-band k.p model for semiconductors, Journal of Statistical Physics, 139 (2010), 280-306. doi: 10.1007/s10955-010-9940-9.

[5]

L. Barletti and F. Méhats, Quantum drift-diffusion modeling of spin transport in nanostructures, Journal of Mathematical Physics, 51 (2010), 053304, 20 pp. doi: 10.1063/1.3380530.

[6]

M. BaroN. Ben AbdallahP. Degond and A. El Ayyadi, A 1d coupled Schrödinger drift-diffusion model including collisions, J. Comput. Phys., 203 (2005), 129-153. doi: 10.1016/j.jcp.2004.08.009.

[7]

N. Ben Abdallah, A hybrid kinetic-quantum model for stationary electron transport, J. Statist. Phys., 90 (1998), 627-662. doi: 10.1023/A:1023216701688.

[8]

N. Ben AbdallahP. Degond and P. A. Markowich, On a one-dimensional Schrödinger-Poisson scattering model, Z. Angew. Math. Phys., 48 (1997), 135-155. doi: 10.1007/PL00001463.

[9]

N. Ben Abdallah, C. Jourdana and P. Pietra, An effective mass model for the simulation of ultra-scaled confined devices, Math. Models Methods Appl. Sci., 22 (2012), 1250039, 40pp. doi: 10.1142/S021820251250039X.

[10]

Ph. CaussignacB. Zimmermann and R. Ferro, Finite element approximation of electrostatic potential in one-dimensional multilayer structures with quantized electronic charge, Computing, 45 (1990), 251-264. doi: 10.1007/BF02250636.

[11]

P. Degond and A. El Ayyadi, A coupled Schrödinger drift-diffusion model for quantum semiconductor device simulations, J. Comput. Phys., 181 (2002), 222-259. doi: 10.1006/jcph.2002.7122.

[12]

P. DegondF. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, Journal of Statistical Physics, 118 (2005), 625-667. doi: 10.1007/s10955-004-8823-3.

[13]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, Journal of Statistical Physics, 112 (2003), 587-628. doi: 10.1023/A:1023824008525.

[14]

A. El Ayyadi and A. Jüngel, Semiconductor simulations using a coupled quantum drift-diffusion Schrödinger-Poisson model, SIAM J. Appl. Math., 66 (2005), 554-572. doi: 10.1137/040610805.

[15]

E. GnaniA. GnudiS. ReggianiM. Luisier and G. Baccarani, Band effects on the transport characteristics of ultrascaled snw-fets, IEEE Trans. Nanotechnol., 7 (2008), 700-709. doi: 10.1109/TNANO.2008.2005777.

[16]

E. GnaniA. MarchiS. ReggianiM. Rudan and G. Baccarani, Quantum-mechanical analysis of the electrostatics in silicon-nanowire and carbon-nanotube FETs, Solid-state Electronics, 50 (2006), 709-715. doi: 10.1109/ESSDER.2005.1546610.

[17]

H. K. Gummel, A self-consistent iterative scheme for one-dimensional steady state transistor calculations, IEEE Trans. Electron Devices, 11 (1964), 455-465. doi: 10.1109/T-ED.1964.15364.

[18]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006.

[19]

C. Jourdana and P. Pietra, A hybrid classical-quantum transport model for the simulation of carbon nanotube transistors, SIAM J. Sci. Comp., 36 (2014), B486-B507. doi: 10.1137/130926353.

[20]

C. Jourdana and N. Vauchelet, Analysis of a diffusive effective mass model for nanowires, Kinet. Relat. Models, 4 (2011), 1121-1142. doi: 10.3934/krm.2011.4.1121.

[21]

A. Jüngel, Transport Equations for Semiconductors, Lecture Notes in Physics No. 773. Springer, Berlin, 2009. doi: 10.1007/978-3-540-89526-8.

[22]

A. Jüngel and R. Pinnau, A positivity-preserving numerical scheme for a nonlinear fourth order parabolic system, SIAM J. Num. Anal., 39 (2001), 385-406. doi: 10.1137/S0036142900369362.

[23]

B. Kozinsky and N. Marzari, Static dielectric properties of carbon nanotubes from first principles, Phys. Rev. Lett., 96 (2006), 166801. doi: 10.1103/PhysRevLett.96.166801.

[24]

C. S. Lent and D. J. Kirkner, The quantum transmitting boundary method, J. Appl. Phys., 67 (1990), 6353-6359. doi: 10.1063/1.345156.

[25]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.

[26]

A. MarchiE. GnaniS. ReggianiM. Rudan and G. Baccarani, Investigating the performance limits of silicon-nanowire and carbon-nanotube FETs, Solid-state Electronics, 50 (2006), 78-85. doi: 10.1016/j.sse.2005.10.039.

[27]

F. Méhats and O. Pinaud, An inverse problem in quantum statistical physics, Journal of Statistical Physics, 140 (2010), 565-602. doi: 10.1007/s10955-010-0003-z.

[28]

F. Méhats and O Pinaud, A problem of moment realizability in quantum statistical physics, Kinetic and Related Models, 4 (2011), 1143-1158. doi: 10.3934/krm.2011.4.1143.

[29]

M. PourfathH. Kosina and S. Selberherr, Numerical study of quantum transport in carbon nanotube transistors, Math. Comput. Simul., 79 (2008), 1051-1059. doi: 10.1016/j.matcom.2007.09.004.

[30]

C. Ringhofer, Subband diffusion models for quantum transport in a strong force regime, SIAM Journal on Applied Mathematics, 71 (2011), 1871-1895. doi: 10.1137/100804164.

[31]

E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759. doi: 10.1007/978-3-642-59033-7_9.

Figure 1.  Schematic longitudinal section of the CNTFET
Figure 2.  3D (left) and 2D (right) representation of atom positions in a (10, 0) 'zig-zag" CNT
Figure 3.  Mobility influence on the current-voltage characteristics obtained with the S-QDD approach
Figure 4.  Current-Voltage characteristics obtained with the five different models
Figure 5.  2D slice of the potential energy (eV) at thermal equilibrium (left: DD model, right: QDD model)
Figure 6.  2D slice of the density in logarithm scale at thermal equilibrium (left: DD model, right: QDD model)
Figure 7.  2D slice of the density in logarithm scale for $V_{DS} = 0.2$ V (left: DD model, right: QDD model)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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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