American Institute of Mathematical Sciences

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December  2011, 4(4): 1121-1142. doi: 10.3934/krm.2011.4.1121

Analysis of a diffusive effective mass model for nanowires

Clément Jourdana 1, and  Nicolas Vauchelet 2,

1. 

Istituto di Matematica Applicata e Tecnologie Informatiche CNR, via Ferrata 1, 27100 Pavia
2. 

UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, INRIA Paris-Rocquencourt, EPI MAMBA, F-75005, Paris, France

Received  May 2011 Revised  August 2011 Published  November 2011

We propose in this paper to derive and analyze a self-consistent model describing the diffusive transport in a nanowire. From a physical point of view, it describes the electron transport in an ultra-scaled confined structure, taking into account the interactions of charged particles with phonons. The transport direction is assumed to be large compared to the wire section and is described by a drift-diffusion equation including effective quantities computed from a Bloch problem in the crystal lattice. The electrostatic potential solves a Poisson equation where the particle density couples on each energy band a two dimensional confinement density with the monodimensional transport density given by the Boltzmann statistics. On the one hand, we study the derivation of this Nanowire Drift-Diffusion Poisson model from a kinetic level description. On the other hand, we present an existence result for this model in a bounded domain.
Keywords: Hamiltonian's spectrum., relative entropy method, diffusive limit, Drift-diffusion system.
Mathematics Subject Classification: Primary: 35Q40, 76R99, 49K20, 82D80; Secondary: 81Q1.
Citation: Clément Jourdana, Nicolas Vauchelet. Analysis of a diffusive effective mass model for nanowires. Kinetic & Related Models, 2011, 4 (4) : 1121-1142. doi: 10.3934/krm.2011.4.1121
References:
[1]

N. W. Ashcroft and N. D. Mermin, "Solid State Physics,'', Saunders College Publishing, (1976). Google Scholar

[2]

J.-P. Aubin, Un théor\`eme de compacité,, C. R. Acad. Sci. Paris, 256 (1963), 5042. Google Scholar

[3]

N. Ben Abdallah and L. Barletti, Quantum transport in crystals : Effective-mass theorem and k.p Hamiltonians,, Comm. Math. Phys., 307 (2011), 567. Google Scholar

[4]

N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), 3306. doi: 10.1063/1.531567. Google Scholar

[5]

N. Ben Abdallah, C. Jourdana and P. Pietra, An effective mass model for the simulation of ultra-scaled confined devices,, preprint, (). Google Scholar

[6]

N. Ben Abdallah and F. Méhats, On a Vlasov-Schrödinger-Poisson model,, Comm. Partial Differential Equations, 29 (2004), 173. Google Scholar

[7]

N. Ben Abdallah and F. Méhats, Semiclassical analysis of the Schrödinger equation with a partially confining potential,, J. Math. Pures Appl. (9), 84 (2005), 580. doi: 10.1016/j.matpur.2004.10.004. Google Scholar

[8]

N. Ben Abdallah, F. Méhats and N. Vauchelet, Diffusive transport of partially quantized particles: Existence, uniqueness and long-time behaviour,, Proc. Edinb. Math. Soc. (2), 49 (2006), 513. doi: 10.1017/S0013091504000987. Google Scholar

[9]

N. Ben Abdallah and M. L. Tayeb, Diffusion approximation for the one dimensional Boltzmann-Poisson system,, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1129. doi: 10.3934/dcdsb.2004.4.1129. Google Scholar

[10]

C. Heitzinger and C. Ringhofer, A transport equation for confined structures derived from the Boltzmann equation,, Comm. Math. Sci., 9 (2011), 829. Google Scholar

[11]

A. Henrot, "Extremum Problems for Eigenvalues of Elliptic Operators,'', Frontiers in Mathematics, (2006). Google Scholar

[12]

A. Jüngel, "Transport Equations for Semiconductors,'', Lecture Notes in Physics, 773 (2009). Google Scholar

[13]

J.-L. Lions, "Équations Différentielles Opérationnelles et Problèmes aux Limites,'', Die Grundlehren der mathematischen Wissenschaften, (1961). Google Scholar

[14]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations,'', Springer-Verlag, (1990). doi: 10.1007/978-3-7091-6961-2. Google Scholar

[15]

N. Masmoudi and M. L. Tayeb, Diffusion limit of a semiconductor Boltzmann-Poisson system,, SIAM J. Math. Anal., 38 (2007), 1788. doi: 10.1137/050630763. Google Scholar

[16]

M. S. Mock, "Analysis of Mathematical Models of Semiconductor Devices,'', Advances in Numerical Computation Series, 3 (1983). Google Scholar

[17]

P. Pietra and N. Vauchelet, Modeling and simulation of the diffusive transport in a nanoscale Double-Gate MOSFET,, J. Comp. Elec., 7 (2008), 52. doi: 10.1007/s10825-008-0253-z. Google Scholar

[18]

F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers,, Asymptotic Anal., 4 (1991), 293. Google Scholar

[19]

N. Vauchelet, Diffusive limit of a two dimensional kinetic system of partially quantized particles,, J. Stat. Phys., 139 (2010), 882. doi: 10.1007/s10955-010-9970-3. Google Scholar

[20]

N. Vauchelet, Diffusive transport of partially quantized particles: $L\log L$ solutions,, Math. Models Methods Appl. Sci., 18 (2008), 489. doi: 10.1142/S0218202508002759. Google Scholar

[21]

T. Wenckebach, "Essential of Semiconductor Physics,'', Wiley, (1999). Google Scholar

show all references

References:
[1]

N. W. Ashcroft and N. D. Mermin, "Solid State Physics,'', Saunders College Publishing, (1976). Google Scholar

[2]

J.-P. Aubin, Un théor\`eme de compacité,, C. R. Acad. Sci. Paris, 256 (1963), 5042. Google Scholar

[3]

N. Ben Abdallah and L. Barletti, Quantum transport in crystals : Effective-mass theorem and k.p Hamiltonians,, Comm. Math. Phys., 307 (2011), 567. Google Scholar

[4]

N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), 3306. doi: 10.1063/1.531567. Google Scholar

[5]

N. Ben Abdallah, C. Jourdana and P. Pietra, An effective mass model for the simulation of ultra-scaled confined devices,, preprint, (). Google Scholar

[6]

N. Ben Abdallah and F. Méhats, On a Vlasov-Schrödinger-Poisson model,, Comm. Partial Differential Equations, 29 (2004), 173. Google Scholar

[7]

N. Ben Abdallah and F. Méhats, Semiclassical analysis of the Schrödinger equation with a partially confining potential,, J. Math. Pures Appl. (9), 84 (2005), 580. doi: 10.1016/j.matpur.2004.10.004. Google Scholar

[8]

N. Ben Abdallah, F. Méhats and N. Vauchelet, Diffusive transport of partially quantized particles: Existence, uniqueness and long-time behaviour,, Proc. Edinb. Math. Soc. (2), 49 (2006), 513. doi: 10.1017/S0013091504000987. Google Scholar

[9]

N. Ben Abdallah and M. L. Tayeb, Diffusion approximation for the one dimensional Boltzmann-Poisson system,, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1129. doi: 10.3934/dcdsb.2004.4.1129. Google Scholar

[10]

C. Heitzinger and C. Ringhofer, A transport equation for confined structures derived from the Boltzmann equation,, Comm. Math. Sci., 9 (2011), 829. Google Scholar

[11]

A. Henrot, "Extremum Problems for Eigenvalues of Elliptic Operators,'', Frontiers in Mathematics, (2006). Google Scholar

[12]

A. Jüngel, "Transport Equations for Semiconductors,'', Lecture Notes in Physics, 773 (2009). Google Scholar

[13]

J.-L. Lions, "Équations Différentielles Opérationnelles et Problèmes aux Limites,'', Die Grundlehren der mathematischen Wissenschaften, (1961). Google Scholar

[14]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations,'', Springer-Verlag, (1990). doi: 10.1007/978-3-7091-6961-2. Google Scholar

[15]

N. Masmoudi and M. L. Tayeb, Diffusion limit of a semiconductor Boltzmann-Poisson system,, SIAM J. Math. Anal., 38 (2007), 1788. doi: 10.1137/050630763. Google Scholar

[16]

M. S. Mock, "Analysis of Mathematical Models of Semiconductor Devices,'', Advances in Numerical Computation Series, 3 (1983). Google Scholar

[17]

P. Pietra and N. Vauchelet, Modeling and simulation of the diffusive transport in a nanoscale Double-Gate MOSFET,, J. Comp. Elec., 7 (2008), 52. doi: 10.1007/s10825-008-0253-z. Google Scholar

[18]

F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers,, Asymptotic Anal., 4 (1991), 293. Google Scholar

[19]

N. Vauchelet, Diffusive limit of a two dimensional kinetic system of partially quantized particles,, J. Stat. Phys., 139 (2010), 882. doi: 10.1007/s10955-010-9970-3. Google Scholar

[20]

N. Vauchelet, Diffusive transport of partially quantized particles: $L\log L$ solutions,, Math. Models Methods Appl. Sci., 18 (2008), 489. doi: 10.1142/S0218202508002759. Google Scholar

[21]

T. Wenckebach, "Essential of Semiconductor Physics,'', Wiley, (1999). Google Scholar

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