March  2012, 5(1): 203-221. doi: 10.3934/krm.2012.5.203

Some fluid-dynamic models for quantum electron transport in graphene via entropy minimization

1. 

Dipartimento di matematica Ulisse Dini, Viale Morgagni 67/A, Firenze, Italy

Received  May 2011 Revised  August 2011 Published  January 2012

We derive some fluid-dynamic models for electron transport near a Dirac point in graphene. We start from a kinetic model constituted by a set of spinorial Wigner equations, we make suitable scalings (hydrodynamic or diffusive) of the model and we build moment equations, which we close through a minimum entropy principle. In order to do this we make some assumptions: the usual semiclassical approximation (ħ $\ll 1$), and two further hypothesis, namely Low Scaled Fermi Speed (LSFS) and Strongly Mixed State (SMS), which allow us to explicitly compute the closure.
Citation: Nicola Zamponi. Some fluid-dynamic models for quantum electron transport in graphene via entropy minimization. Kinetic and Related Models, 2012, 5 (1) : 203-221. doi: 10.3934/krm.2012.5.203
References:
[1]

C. Auer, F. Schürrer and C. Ertler, Hot phonon effects on the high-field transport in metallic carbon nanotubes, Physical Review B, 74 (2006), 165409. doi: 10.1103/PhysRevB.74.165409.

[2]

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

[3]

C. Beenakker, Colloquium: Andreev reflection and Klein tunneling in graphene, Reviews of Modern Physics, 80 (2008), 1337-1354. doi: 10.1103/RevModPhys.80.1337.

[4]

P. Degond and C. Ringhofer, A note on quantum moment hydrodynamics and the entropy principle, C. R. Math. Acad. Sci. Paris, 335 (2002), 967-972.

[5]

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.

[6]

P. Degond, F. 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.

[7]

G. Folland, "Harmonic Analysis in Phase Space,'' Annals of Mathematics Studies, 122, Princeton University Press, Princeton, NJ, 1989.

[8]

M. Freitag, Graphene: Nanoelectronics goes flat out, Nature Nanotechnology, 3 (2008), 455-457. doi: 10.1038/nnano.2008.219.

[9]

A. Jüngel, D. Matthes and J. Milšić, Derivation of new quantum hydrodynamic equations using entropy minimization, SIAM J. Appl. Math., 67 (2006), 46-68. doi: 10.1137/050644823.

[10]

A. Jüngel, "Transport Equations for Semiconductors,'' Annals of Mathematics Studies, 122, Springer-Verlag, Berlin, 2009.

[11]

A. Jüngel, Dissipative quantum fluid models, to appear in Riv. Mat. Univ. Parma, 2011.

[12]

M. Katsnelson, K. Novoselov and A. Geim, Chiral tunneling and the Klein paradox in graphene, Nature Physics 2, 9 (2006), 620-625. doi: 10.1038/nphys384.

[13]

C. D. Levermore, Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996), 1021-1065. doi: 10.1007/BF02179552.

[14]

E. Madelung, Quantentheorie in hydrodynamischer form, Zeitschrift für Physik A Hadrons and Nuclei, 40 (1927), 322-326.

[15]

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

[16]

C. Zachos, D. Fairlie and T. Curtright, eds., "Quantum Mechanics in Phase Space. An Overview with Selected Papers,'' Worl Scientific Series in 20th Century Physics, 34, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.

[17]

Nicola Zamponi, "Trasporto Quantistico Degli Elettroni nel Grafene: Un Approccio Cinetico e Fluidodinamico,'' Tesi di Laurea Specialistica, Università di Firenze, Facoltà di Scienze Mat. Fis. Nat., 2009.

[18]

N. Zamponi and L. Barletti, Quantum electronic transport in graphene: A kinetic and fluid-dynamic approach, Mathematical Methods in the Applied Sciences, 34 (2011), 807-818. doi: 10.1002/mma.1403.

show all references

References:
[1]

C. Auer, F. Schürrer and C. Ertler, Hot phonon effects on the high-field transport in metallic carbon nanotubes, Physical Review B, 74 (2006), 165409. doi: 10.1103/PhysRevB.74.165409.

[2]

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

[3]

C. Beenakker, Colloquium: Andreev reflection and Klein tunneling in graphene, Reviews of Modern Physics, 80 (2008), 1337-1354. doi: 10.1103/RevModPhys.80.1337.

[4]

P. Degond and C. Ringhofer, A note on quantum moment hydrodynamics and the entropy principle, C. R. Math. Acad. Sci. Paris, 335 (2002), 967-972.

[5]

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.

[6]

P. Degond, F. 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.

[7]

G. Folland, "Harmonic Analysis in Phase Space,'' Annals of Mathematics Studies, 122, Princeton University Press, Princeton, NJ, 1989.

[8]

M. Freitag, Graphene: Nanoelectronics goes flat out, Nature Nanotechnology, 3 (2008), 455-457. doi: 10.1038/nnano.2008.219.

[9]

A. Jüngel, D. Matthes and J. Milšić, Derivation of new quantum hydrodynamic equations using entropy minimization, SIAM J. Appl. Math., 67 (2006), 46-68. doi: 10.1137/050644823.

[10]

A. Jüngel, "Transport Equations for Semiconductors,'' Annals of Mathematics Studies, 122, Springer-Verlag, Berlin, 2009.

[11]

A. Jüngel, Dissipative quantum fluid models, to appear in Riv. Mat. Univ. Parma, 2011.

[12]

M. Katsnelson, K. Novoselov and A. Geim, Chiral tunneling and the Klein paradox in graphene, Nature Physics 2, 9 (2006), 620-625. doi: 10.1038/nphys384.

[13]

C. D. Levermore, Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996), 1021-1065. doi: 10.1007/BF02179552.

[14]

E. Madelung, Quantentheorie in hydrodynamischer form, Zeitschrift für Physik A Hadrons and Nuclei, 40 (1927), 322-326.

[15]

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

[16]

C. Zachos, D. Fairlie and T. Curtright, eds., "Quantum Mechanics in Phase Space. An Overview with Selected Papers,'' Worl Scientific Series in 20th Century Physics, 34, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.

[17]

Nicola Zamponi, "Trasporto Quantistico Degli Elettroni nel Grafene: Un Approccio Cinetico e Fluidodinamico,'' Tesi di Laurea Specialistica, Università di Firenze, Facoltà di Scienze Mat. Fis. Nat., 2009.

[18]

N. Zamponi and L. Barletti, Quantum electronic transport in graphene: A kinetic and fluid-dynamic approach, Mathematical Methods in the Applied Sciences, 34 (2011), 807-818. doi: 10.1002/mma.1403.

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