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 & 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).  doi: 10.1103/PhysRevB.74.165409.  Google Scholar

[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.  doi: 10.1007/s10955-010-9940-9.  Google Scholar

[3]

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

[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.   Google Scholar

[5]

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

[6]

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

[7]

G. Folland, "Harmonic Analysis in Phase Space,'', Annals of Mathematics Studies, 122 (1989).   Google Scholar

[8]

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

[9]

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

[10]

A. Jüngel, "Transport Equations for Semiconductors,'', Annals of Mathematics Studies, 122 (2009).   Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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 (2005).   Google Scholar

[17]

Nicola Zamponi, "Trasporto Quantistico Degli Elettroni nel Grafene: Un Approccio Cinetico e Fluidodinamico,'', Tesi di Laurea Specialistica, (2009).   Google Scholar

[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.  doi: 10.1002/mma.1403.  Google Scholar

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).  doi: 10.1103/PhysRevB.74.165409.  Google Scholar

[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.  doi: 10.1007/s10955-010-9940-9.  Google Scholar

[3]

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

[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.   Google Scholar

[5]

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

[6]

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

[7]

G. Folland, "Harmonic Analysis in Phase Space,'', Annals of Mathematics Studies, 122 (1989).   Google Scholar

[8]

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

[9]

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

[10]

A. Jüngel, "Transport Equations for Semiconductors,'', Annals of Mathematics Studies, 122 (2009).   Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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 (2005).   Google Scholar

[17]

Nicola Zamponi, "Trasporto Quantistico Degli Elettroni nel Grafene: Un Approccio Cinetico e Fluidodinamico,'', Tesi di Laurea Specialistica, (2009).   Google Scholar

[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.  doi: 10.1002/mma.1403.  Google Scholar

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