# American Institute of Mathematical Sciences

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

## Analysis of a diffusive effective mass model for nanowires

 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.
Citation: Clément Jourdana, Nicolas Vauchelet. Analysis of a diffusive effective mass model for nanowires. Kinetic and 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. [2] J.-P. Aubin, Un théor\eme de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044. [3] N. Ben Abdallah and L. Barletti, Quantum transport in crystals : Effective-mass theorem and k.p Hamiltonians, Comm. Math. Phys., 307 (2011), 567-607. [4] N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors J. Math. Phys., 37 (1996), 3306-3333. doi: 10.1063/1.531567. [5] N. Ben Abdallah, C. Jourdana and P. Pietra, An effective mass model for the simulation of ultra-scaled confined devices,, preprint, (). [6] N. Ben Abdallah and F. Méhats, On a Vlasov-Schrödinger-Poisson model, Comm. Partial Differential Equations, 29 (2004), 173-206. [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), 580-614. doi: 10.1016/j.matpur.2004.10.004. [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), 513-549. doi: 10.1017/S0013091504000987. [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), 1129-1142. doi: 10.3934/dcdsb.2004.4.1129. [10] C. Heitzinger and C. Ringhofer, A transport equation for confined structures derived from the Boltzmann equation, Comm. Math. Sci., 9 (2011), 829-857. [11] A. Henrot, "Extremum Problems for Eigenvalues of Elliptic Operators,'' Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. [12] A. Jüngel, "Transport Equations for Semiconductors,'' Lecture Notes in Physics, 773, Springer-Verlag, Berlin, 2009. [13] J.-L. Lions, "Équations Différentielles Opérationnelles et Problèmes aux Limites,'' Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. [14] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations,'' Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2. [15] N. Masmoudi and M. L. Tayeb, Diffusion limit of a semiconductor Boltzmann-Poisson system, SIAM J. Math. Anal., 38 (2007), 1788-1807. doi: 10.1137/050630763. [16] M. S. Mock, "Analysis of Mathematical Models of Semiconductor Devices,'' Advances in Numerical Computation Series, 3, Boole Press, Dún Laoghaire, 1983. [17] P. Pietra and N. Vauchelet, Modeling and simulation of the diffusive transport in a nanoscale Double-Gate MOSFET, J. Comp. Elec., 7 (2008), 52-65. doi: 10.1007/s10825-008-0253-z. [18] F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers, Asymptotic Anal., 4 (1991), 293-317. [19] N. Vauchelet, Diffusive limit of a two dimensional kinetic system of partially quantized particles, J. Stat. Phys., 139 (2010), 882-914. doi: 10.1007/s10955-010-9970-3. [20] N. Vauchelet, Diffusive transport of partially quantized particles: $L\log L$ solutions, Math. Models Methods Appl. Sci., 18 (2008), 489-510. doi: 10.1142/S0218202508002759. [21] T. Wenckebach, "Essential of Semiconductor Physics,'' Wiley, Chichester, 1999.

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##### References:
 [1] N. W. Ashcroft and N. D. Mermin, "Solid State Physics,'' Saunders College Publishing, 1976. [2] J.-P. Aubin, Un théor\eme de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044. [3] N. Ben Abdallah and L. Barletti, Quantum transport in crystals : Effective-mass theorem and k.p Hamiltonians, Comm. Math. Phys., 307 (2011), 567-607. [4] N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors J. Math. Phys., 37 (1996), 3306-3333. doi: 10.1063/1.531567. [5] N. Ben Abdallah, C. Jourdana and P. Pietra, An effective mass model for the simulation of ultra-scaled confined devices,, preprint, (). [6] N. Ben Abdallah and F. Méhats, On a Vlasov-Schrödinger-Poisson model, Comm. Partial Differential Equations, 29 (2004), 173-206. [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), 580-614. doi: 10.1016/j.matpur.2004.10.004. [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), 513-549. doi: 10.1017/S0013091504000987. [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), 1129-1142. doi: 10.3934/dcdsb.2004.4.1129. [10] C. Heitzinger and C. Ringhofer, A transport equation for confined structures derived from the Boltzmann equation, Comm. Math. Sci., 9 (2011), 829-857. [11] A. Henrot, "Extremum Problems for Eigenvalues of Elliptic Operators,'' Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. [12] A. Jüngel, "Transport Equations for Semiconductors,'' Lecture Notes in Physics, 773, Springer-Verlag, Berlin, 2009. [13] J.-L. Lions, "Équations Différentielles Opérationnelles et Problèmes aux Limites,'' Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. [14] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations,'' Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2. [15] N. Masmoudi and M. L. Tayeb, Diffusion limit of a semiconductor Boltzmann-Poisson system, SIAM J. Math. Anal., 38 (2007), 1788-1807. doi: 10.1137/050630763. [16] M. S. Mock, "Analysis of Mathematical Models of Semiconductor Devices,'' Advances in Numerical Computation Series, 3, Boole Press, Dún Laoghaire, 1983. [17] P. Pietra and N. Vauchelet, Modeling and simulation of the diffusive transport in a nanoscale Double-Gate MOSFET, J. Comp. Elec., 7 (2008), 52-65. doi: 10.1007/s10825-008-0253-z. [18] F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers, Asymptotic Anal., 4 (1991), 293-317. [19] N. Vauchelet, Diffusive limit of a two dimensional kinetic system of partially quantized particles, J. Stat. Phys., 139 (2010), 882-914. doi: 10.1007/s10955-010-9970-3. [20] N. Vauchelet, Diffusive transport of partially quantized particles: $L\log L$ solutions, Math. Models Methods Appl. Sci., 18 (2008), 489-510. doi: 10.1142/S0218202508002759. [21] T. Wenckebach, "Essential of Semiconductor Physics,'' Wiley, Chichester, 1999.
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