Journal of Statistical Physics

, Volume 171, Issue 4, pp 696–726 | Cite as

Quantum Transmission Conditions for Diffusive Transport in Graphene with Steep Potentials

  • Luigi Barletti
  • Claudia Negulescu


We present a formal derivation of a drift-diffusion model for stationary electron transport in graphene, in presence of sharp potential profiles, such as barriers and steps. Assuming the electric potential to have steep variations within a strip of vanishing width on a macroscopic scale, such strip is viewed as a quantum interface that couples the classical regions at its left and right sides. In the two classical regions, where the potential is assumed to be smooth, electron and hole transport is described in terms of semiclassical kinetic equations. The diffusive limit of the kinetic model is derived by means of a Hilbert expansion and a boundary layer analysis, and consists of drift-diffusion equations in the classical regions, coupled by quantum diffusive transmission conditions through the interface. The boundary layer analysis leads to the discussion of a four-fold Milne (half-space, half-range) transport problem.


Transmission conditions Graphene Diffusion limit Boundary layer Milne problem 



Support is acknowledged from the Italian-French project Projet International de Coopration Scientifique (PICS) “MANUS—Modelling and Numerics for Spintronics and Graphene” (Ref. PICS07373).


  1. 1.
    Ashcroft, N.W., Mermin, N.D.: Solid State Physics. Saunders College Publishing, Philadelphia (1976)zbMATHGoogle Scholar
  2. 2.
    Bardos, C., Santos, R., Sentis, R.: Diffusion approximation and the computation of the critical size. T. Am. Math. Soc. 284, 617–649 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barletti, L.: Hydrodynamic equations for electrons in graphene obtained from the maximum entropy principle. J. Math. Phys. 55, 083303 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barletti, L.: Hydrodynamic equations for an electron gas in graphene. J. Math. Ind. 6, 7 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Barletti, L., Negulescu, C.: Hybrid classical-quantum models for charge transport in graphene with sharp potentials. J. Comput. Theor. Transp. 46, 159–175 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Barletti, L., Frosali, G., Morandi, O.: Kinetic and hydrodynamic models for multi-band quantum transport in crystals. In: Ehrhardt, M., Koprucki, T. (eds.) Multi-band Effective Mass Approximations: Advanced Mathematical Models and Numerical Techniques, pp. 3–56. Springer, Heidelberg (2014)Google Scholar
  7. 7.
    Ben Abdallah, N.: A hybrid kinetic-quantum model for stationary electron transport. J. Stat. Phys. 90, 627–662 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ben, Abdallah N., Degond, P., Gamba, I.: Coupling one-dimensional time-dependent classical and quantum transport models. J. Math. Phys. 43, 1–24 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Borysenko, K.M., Mullen, J.T., Barry, E.A., Paul, S., Semenov, Y.G., Zavada, J.M., Buongiorno Nardelli, M., Kim, K.W.: First-principles analysis of electron-phonon interactions in graphene. Phys. Rev. B 81, 121412(R) (2010)ADSCrossRefGoogle Scholar
  10. 10.
    Castro Neto, A.H., Guinea, F., Peres, N.M.R., Novoselov, K.S., Geim, A.K.: The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009)ADSCrossRefGoogle Scholar
  11. 11.
    Cheianov, V.V., Fal’ko, V., Altshuler, B.L.: The focusing of electron flow and a Veselago lens in graphene. Science 315, 1252–1255 (2007)ADSCrossRefGoogle Scholar
  12. 12.
    Degond, P.: Macroscopic limits of the Boltzmann equation: a review. In: Degond, P., Pareschi, L., Russo, G. (eds.) Modeling and Computational Methods for Kinetic Equations, pp. 3–57. Birkhäuser, Basel (2004)CrossRefGoogle Scholar
  13. 13.
    Degond, P., El Ayyadi, A.: A coupled Schrödinger drift-diffusion model for quantum semiconductor device simulations. J. Comput. Phys. 181, 222–259 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Degond, P., Schmeiser, C.: Macroscopic models for semiconductor heterostructures. J. Math. Phys. 39, 4634–4663 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Duderstadt, J.J., Martin, W.R.: Transport Theory. Wiley, New York (1979)zbMATHGoogle Scholar
  16. 16.
    Golse, F., Klar, A.: A numerical method for computing asymptotic states and outgoing distributions for kinetic linear half-space problems. J. Stat. Phys. 80, 1033–1061 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Katsnelson, M.I., Novoselov, K.S., Geim, A.K.: Chiral tunnelling and the Klein paradox in graphene. Nat. Phys. 2, 620–625 (2006)CrossRefGoogle Scholar
  18. 18.
    Lee, G.H., Park, G.H., Lee, H.J.: Observation of negative refraction of Dirac fermions in graphene. Nat. Phys. 11, 925–929 (2015)CrossRefGoogle Scholar
  19. 19.
    Lejarreta, J.D., Fuentevilla, C.H., Diez, E., Cerveró, J.M.: An exact transmission coefficient with one and two barriers in graphene. J. Phys. A 46, 155304 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Majorana, A., Mascali, G., Romano, V.: Charge transport and mobility in monolayer graphene. J. Math. Ind. 7, 4 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Morandi, O.: Wigner-function formalism applied to the Zener band transition in a semiconductor. Phys. Rev. B 80, 024301 (2009)ADSCrossRefGoogle Scholar
  22. 22.
    Poupaud, F.: Diffusion approximation of the linear semiconductor Boltzmann equation: analysis of boundary layers. Asymptot. Anal. 4, 293–317 (1991)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Slonczewski, J.C., Weiss, P.R.: Band structure of graphite. Phys. Rev. 109, 272–279 (1958)ADSCrossRefGoogle Scholar
  24. 24.
    Young, A.F., Kim, P.: Quantum interference and Klein tunnelling in graphene heterojunctions. Nat. Phys. 5, 222–226 (2009)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Dipartimento di Matematica e Informatica “U. Dini”FlorenceItalia
  2. 2.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouseFrance

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