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 & 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), 5042-5044.  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), 567-607. Google Scholar

[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.  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), 173-206.  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), 580-614. 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), 513-549. 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), 1129-1142. 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), 829-857. Google Scholar

[11]

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

[12]

A. Jüngel, "Transport Equations for Semiconductors,'' Lecture Notes in Physics, 773, Springer-Verlag, Berlin, 2009.  Google Scholar

[13]

J.-L. Lions, "Équations Différentielles Opérationnelles et Problèmes aux Limites,'' Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.  Google Scholar

[14]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations,'' Springer-Verlag, Vienna, 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), 1788-1807. doi: 10.1137/050630763.  Google Scholar

[16]

M. S. Mock, "Analysis of Mathematical Models of Semiconductor Devices,'' Advances in Numerical Computation Series, 3, Boole Press, Dún Laoghaire, 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), 52-65. 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), 293-317.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

T. Wenckebach, "Essential of Semiconductor Physics,'' Wiley, Chichester, 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), 5042-5044.  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), 567-607. Google Scholar

[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.  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), 173-206.  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), 580-614. 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), 513-549. 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), 1129-1142. 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), 829-857. Google Scholar

[11]

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

[12]

A. Jüngel, "Transport Equations for Semiconductors,'' Lecture Notes in Physics, 773, Springer-Verlag, Berlin, 2009.  Google Scholar

[13]

J.-L. Lions, "Équations Différentielles Opérationnelles et Problèmes aux Limites,'' Die Grundlehren der mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.  Google Scholar

[14]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations,'' Springer-Verlag, Vienna, 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), 1788-1807. doi: 10.1137/050630763.  Google Scholar

[16]

M. S. Mock, "Analysis of Mathematical Models of Semiconductor Devices,'' Advances in Numerical Computation Series, 3, Boole Press, Dún Laoghaire, 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), 52-65. 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), 293-317.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[1]

Takayoshi Ogawa, Hiroshi Wakui. Stability and instability of solutions to the drift-diffusion system. Evolution Equations & Control Theory, 2017, 6 (4) : 587-597. doi: 10.3934/eect.2017029

[2]

T. Ogawa. The degenerate drift-diffusion system with the Sobolev critical exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 875-886. doi: 10.3934/dcdss.2011.4.875

[3]

Ronald E. Mickens. A nonstandard finite difference scheme for the drift-diffusion system. Conference Publications, 2009, 2009 (Special) : 558-563. doi: 10.3934/proc.2009.2009.558

[4]

Claire Chainais-Hillairet, Ingrid Lacroix-Violet. On the existence of solutions for a drift-diffusion system arising in corrosion modeling. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 77-92. doi: 10.3934/dcdsb.2015.20.77

[5]

José Antonio Carrillo, Yingping Peng, Aneta Wróblewska-Kamińska. Relative entropy method for the relaxation limit of hydrodynamic models. Networks & Heterogeneous Media, 2020, 15 (3) : 369-387. doi: 10.3934/nhm.2020023

[6]

Masaki Kurokiba, Toshitaka Nagai, T. Ogawa. The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system. Communications on Pure & Applied Analysis, 2006, 5 (1) : 97-106. doi: 10.3934/cpaa.2006.5.97

[7]

Elio E. Espejo, Masaki Kurokiba, Takashi Suzuki. Blowup threshold and collapse mass separation for a drift-diffusion system in space-dimension two. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2627-2644. doi: 10.3934/cpaa.2013.12.2627

[8]

Corrado Lattanzio, Pierangelo Marcati. The relaxation to the drift-diffusion system for the 3-$D$ isentropic Euler-Poisson model for semiconductors. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 449-455. doi: 10.3934/dcds.1999.5.449

[9]

H.J. Hwang, K. Kang, A. Stevens. Drift-diffusion limits of kinetic models for chemotaxis: A generalization. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 319-334. doi: 10.3934/dcdsb.2005.5.319

[10]

Dietmar Oelz, Alex Mogilner. A drift-diffusion model for molecular motor transport in anisotropic filament bundles. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4553-4567. doi: 10.3934/dcds.2016.36.4553

[11]

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

[12]

Denis Serre, Alexis F. Vasseur. The relative entropy method for the stability of intermediate shock waves; the rich case. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4569-4577. doi: 10.3934/dcds.2016.36.4569

[13]

Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic & Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701

[14]

Luiza H. F. Andrade, Rui F. Vigelis, Charles C. Cavalcante. A generalized quantum relative entropy. Advances in Mathematics of Communications, 2020, 14 (3) : 413-422. doi: 10.3934/amc.2020063

[15]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935

[16]

Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods. Kinetic & Related Models, 2017, 10 (1) : 61-91. doi: 10.3934/krm.2017003

[17]

Junmin Yang, Maoan Han. On the number of limit cycles of a cubic Near-Hamiltonian system. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 827-840. doi: 10.3934/dcds.2009.24.827

[18]

David Damanik, Anton Gorodetski. The spectrum of the weakly coupled Fibonacci Hamiltonian. Electronic Research Announcements, 2009, 16: 23-29. doi: 10.3934/era.2009.16.23

[19]

Patrick Henning, Mario Ohlberger. The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Networks & Heterogeneous Media, 2010, 5 (4) : 711-744. doi: 10.3934/nhm.2010.5.711

[20]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003

2020 Impact Factor: 1.432

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]