Abstract
We derive semiclassical diffusive equations for the densities of electrons in the two energy bands of a semiconductor, as described by a k⋅p Hamiltonian. The derivation starts from a quantum kinetic (Wigner) description and resorts to the Chapman-Enskog method as well as to the quantum version of the minimum entropy principle. Four different regimes are investigated, according to different scalings of the k⋅p band-coupling and band-gap parameters with respect to the scaled Planck constant.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arnold, A.: Self-consistent relaxation-time models in quantum mechanics. Commun. Partial Differ. Equ. 21(3–4), 473–506 (1996)
Barletti, L., Demeio, L., Frosali, G.: Multiband quantum transport models for semiconductor devices. In: Cercignani, C., Gabetta, E. (eds.) Transport Phenomena and Kinetic Theory. Model. Simul. Sci. Eng. Technol., pp. 55–89. Birkhäuser, Boston (2007)
Barletti, L., Méhats, F.: Quantum drift-diffusion modeling of spin transport in nanostructures. J. Math. Phys. (in press)
Ben Abdallah, N., Méhats, F., Negulescu, C.: Adiabatic quantum-fluid transport models. Commun. Math. Sci. 4(3), 621–650 (2006)
Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden variables” I. Phys. Rev. 85, 166–179 (1952)
Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden variables” II. Phys. Rev. 85, 180–193 (1952)
Bonilla, L.L., Barletti, L., Alvaro, M.: Nonlinear electron and spin transport in semiconductor superlattices. SIAM J. Appl. Math. 69(2), 494–513 (2008)
Bourgade, J.P., Degond, P., Méhats, F., Ringhofer, C.: On quantum extensions to classical spherical harmonics expansion/Fokker-Planck models. J. Math. Phys. 47(4), 043302 (2006), 26 pp.
Degond, P., Gallego, S., Méhats, F.: An entropic quantum drift-diffusion model for electron transport in resonant tunneling diodes. J. Comput. Phys. 221(1), 226–249 (2007)
Degond, P., Gallego, S., Méhats, F.: Isothermal quantum hydrodynamics: derivation, asymptotic analysis, and simulation. Multiscale Model. Simul. 6(1), 246–272 (2007) (electronic)
Degond, P., Gallego, S., Méhats, F.: On quantum hydrodynamic and quantum energy transport models. Commun. Math. Sci. 5(4), 887–908 (2007)
Degond, P., Méhats, F., Ringhofer, C.: Quantum energy-transport and drift-diffusion models. J. Stat. Phys. 118(3–4), 625–667 (2005)
Degond, P., Ringhofer, C.: Quantum moment hydrodynamics and the entropy principle. J. Stat. Phys. 112(3–4), 587–628 (2003)
Folland, G.B.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton (1989)
Freitag, M.: Graphene: Nanoelectronics goes flat out. Nat. Nanotechnol. 3, 455–457 (2008)
Gardner, C.L.: The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54(2), 409–427 (1994)
Gasser, I., Markowich, P.A.: Quantum hydrodynamics, Wigner transforms and the classical limit. Asymptot. Anal. 14(2), 97–116 (1997)
Gasser, I., Markowich, P.A., Unterreiter, A.: Quantum hydrodynamics. In: Raviart, P.A. (ed.) Modeling of Collisions, pp. 179–216. Gauthier-Villars, Paris (1997)
Jüngel, A.: Quasi-Hydrodynamic Semiconductor Equations. Birkhäuser, Basel (2001)
Jüngel, A., Matthes, D.: A derivation of the isothermal quantum hydrodynamic equations using entropy minimization. ZAMM Z. Angew. Math. Mech. 85(11), 806–814 (2005)
Jüngel, A., Matthes, D., Milišić, J.P.: Derivation of new quantum hydrodynamic equations using entropy minimization. SIAM J. Appl. Math. 67(1), 46–68 (2006) (electronic)
Kane, E.O.: Zener tunneling in semiconductors. J. Phys. Chem. Solids 12, 181–188 (1959)
Kane, E.O.: The k⋅p method. In: Willardson, R.K., Beer, A.C. (eds.) Physics of III–V Compounds, Semiconductors and Semimetals, vol. 1. Academic Press, New York (1966). Chap. 3
Katsnelson, M.I., Novoselov, K.S., Geim, A.K.: Chiral tunnelling and the Klein paradox in graphene. Nat. Phys. 2(9), 620–625 (2006)
Levermore, C.D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83(5–6), 1021–1065 (1996)
Madelung, E.: Quantentheorie in hydrodynamischer Form. Z. Phys. 40, 322–326 (1926)
Mascali, G., Romano, V.: Hydrodynamic subband model for semiconductors based on the maximum entropy principle. Preprint (2009)
Nier, F.: A variational formulation of Schrödinger-Poisson systems in dimension d≤3. Commun. Partial Differ. Equ. 18(7–8), 1125–1147 (1993)
Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, London (1992)
Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs (1967)
Sweeney, M., Xu, J.M.: Resonant interband tunnel diodes. Appl. Phys. Lett. 54(6), 546–548 (1989)
Tatarskiĭ, V.I.: The Wigner representation of quantum mechanics. Sov. Phys. Usp. 26(4), 311–327 (1983)
Thaller, B.: The Dirac Equation. Springer, Berlin (1992)
Vasko, F.T., Raichev, O.E.: Quantum Kinetic Theory and Applications. Electrons, Photons, Phonons. Springer, New York (2005)
Wenckebach, T.: Essentials of Semiconductor Physics. Wiley, Chichester (1999)
Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)
Yang, R.Q., Sweeny, M., Day, D., Xu, J.M.: Interband tunneling in heterostructure tunnel diodes. IEEE Trans. Electron Devices 38(3), 442–446 (1991)
Young, A.F., Kim, P.: Quantum interference and Klein tunnelling in graphene heterojunctions. Nat. Phys. 5(3), 222–226 (2009)
Zachos, C.K., Fairlie, D.B., Curtright, T.L. (eds.): Quantum Mechanics in Phase Space. World Scientific Series in 20th Century Physics, vol. 34. World Scientific, Hackensack (2005). An overview with selected papers
Žutić, I., Fabian, J., Das Sarma, S.: Spin-polarized transport in inhomogeneous magnetic semiconductors: Theory of magnetic/nonmagnetic p-n junctions. Phys. Rev. Lett. 88(6), 066603 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barletti, L., Frosali, G. Diffusive Limit of the Two-Band k⋅p Model for Semiconductors. J Stat Phys 139, 280–306 (2010). https://doi.org/10.1007/s10955-010-9940-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-010-9940-9