Abstract
Interface conditions between a classical transport model described by the Boltzmann equation and a quantum model described by a set of Schrödinger equations are presented in the one-dimensional stationary setting. These interface conditions, derived thanks to an asymptotic analysis of the Wigner transform, are shown to be flux-preserving and are used to build a hybrid model consisting of a quantum zone surrounded by two classical ones. The hybrid model is shown to be well posed when the potential is either prescribed or computed self-consistently, and the semiclassical limit of the problem is shown to give the right interface conditions between two kinetic regions (the electrostatic potential being fixed). This model can be used to describe far-from-equilibrium electron transport in a resonant tunneling diode.
Similar content being viewed by others
REFERENCES
P. N. Argyres, Quantum Kinetic equations for electrons in high electric and phonon fields, Phys. Lett. A. 171:373-379 (1992).
A. M. Anile and O. Muscato, Extended thermodynamics tested beyond the linear regime: The case of electron transport in silicon semiconductors, Contin. Mech. Thermodyn. 8(3):131-142 (1996).
A. Arnold, Numerically Absorbing Boundary Conditions for Quantum Evolution Equations, to appear 1996.
N. Ben Abdallah, On a multi-dimensional Schrödinger Poisson Scattering model for semiconductors, submitted.
N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, Journal of Mathematical Physics 37(7):3306-3333 (1996).
N. Ben Abdallah, P. Degond, and S. Genieys, An Energy-transport model derived from the Boltzmann equation of semiconductors, J. Stat. Phys. 84:1-2, 205–231 (1996).
N. Ben Abdallah, P. Degond and P. A. Markowich, On a one-dimensional Schrödinger Poisson Scattering model, ZAMP 48:135-155 (1997).
N. Ben Abdallah, P. Degond, and P. A. Markowich, The Quantum Child Langmuir Problem, Nonlinear Analysis TMA 31(5–6):629-648 (1998).
F. Brezzi and P. A. Markowich, A mathematical analysis of quantum transport in three dimensional crystals, Anna. di Matematica Pura Applicata 160:171-191 (1991).
R. Brunetti, C. Jacoboni and F. Rossi, Quantum Theory of Transient Transport in Semiconductors: A Monte Carlo Approach, Phys. Rev. B 39(15):10781-10790 (1989).
F. Chevoir, Effet tunnel résonant assisté par diffusion dans les diodes double-barrière, PHD thesis, (Univ. Orsay 1992).
F. Chevoir and B. Vinter, Scattering-assisted tunneling in double-barrier diodes: scattering rates and valley current, Phys. Rev B. 47(12):7260-7274 (1993).
P. Degond and P. A. Markowich, A quantum transport model for semiconductors: The Wigner-Poisson model on a bounded Brillouin zone, M2AN 24:697-710 (1990).
P. Degond and C. Schmeiser, Kinetic boundary layers and fluid-kinetic coupling in semiconductors, in preparation.
P. Degond and C. Schmeiser, Macroscopic models for semiconductor heterostructures, in preparation.
L. Di Menza, Transparent and Absorbing Boundary Conditions for the Schrödinger Equation in a Bounded Domain, Internal report, (Univ. Bordeaux 1, 1995).
W. R. Frensley, Boundary conditions for open quantum systems driven far from equilibrium, Reviews of Modern Physics 62(3):745-791 (1990).
W. R. Frensley, Wigner-function model of a resonant-tunneling semiconductor device, Phys. Rev. B. 36(3):1570-1580 (1987).
P. Gérard, Mesures semiclassiques et ondes de bloch, Sém. Ecole Polytechnique XVI:1-19 (1990–1991).
P. Gérard, P. A. Markowich, N. J. Mauser, and F. Poupaud, Homogenization Limits and Wigner Transforms, submitted, 1996.
C. Greengard and P. A. Raviart, A boundary value problem for the stationary Vlasov-Poisson equations: The plane diode, Comm. Pure Appl. Math. 43:473-507 (1990).
J. R. Hellums and W. R. Frensley, Non-Markovian open-system boundary conditions for the time-dependent Schrödinger equation, Phys. Rev. B. 49:2904-2906 (1994).
A. Klar, Asymptotic induced domain decomposition methods for kinetid and drift-diffusion semiconductor equation, preprint Univ. Kaiserslautern.
N. C. Kluksdahl, A. M. Kriman, D.K. Ferry, and C. Ringhofer, Self-consistent study of the resonant tunneling diode, Phys. Rev. B. 39(11):7720-7735 (1989).
C. Lent and D. Kirkner, The Quantum Transmitting Boundary Method, J. Appl. Phys. 67(10):6353-6359 (1990).
P. L. Lions and T. Paul, Sur les mesures de Wigner, Revista Mathematica Iberoamericana 9:553-618 (1993).
P. A. Markowich and N. J. Mauser, The classical limit of a selfconsistent quantum-Vlasov equation in 3-D, Math. Meth. Mod. 16(6):409-442 (1993).
P. A. Markowich, N. J. Mauser and F. Poupaud, A Wigner function approach to semiclassical limits: electrons in a periodic potential, J. Math. Phys. 35:1066-1094 (1994).
A. Messiah, Mécanique Quantique, Tome 1, 2, (Dunod, Paris, 1995).
P. Mounaix, O. Vanbésien and D. Lippens, Effect of cathode space layer on the current-voltage characteristics of resonant tunneling diodes, Appl. Phys. Lett. 57:8, 1517–1519 (1990).
F. Nier, A stationary Schrödinger-Poisson system arising from the modeling of electronic devices, 2(5):489-510 (1990).
F. Nier, A variational formulation of Schrödinger-Poisson systems in dimension d ⩽ 3, Comm. Part. Diff. Equations 18(7–8):1125-1147 (1993).
A. Nouri and F. Poupaud, Stationary solutions of boundary value problems for Maxwell Boltzmann system modeling degenerate semiconductors, SIAM J. Math. Anal. 26(5):1143-1156 (1995).
O. Vanbésien and D. Lippens, Theoretical analysis of a branch line quantum directional coupler, Appl. Phys. Lett. 65(19):2439-2441 (1994).
F. Poupaud, Boundary value problems for the stationary Vlasov-Maxwell system, Forum Mathematicum 4:499-527 (1992).
F. Poupaud, Diffusion approximation of the linear semiconductor equation: analysis of boundary layers, Asymp. Anal. 4:293-317 (1991).
F. Poupaud and C. Ringhofer, Quantum hydrodynamic models in a crystal, Appl. Math. Lett. 8(6):55-59 (1995).
C. Ringhofer, D. K. Ferry, and N. C. Kluksdahl, Absorbing boundary conditions for the simulation of quantum transport phenomena, Trans. Theo. Stat. Phys. 18:331-346 (1989).
C. Schmeiser and A. Zwirchmayr, Elastic and drift-diffusion limits of electron phonon interaction in semiconductors, M3AS to appear.
R. Stratton, Diffusion of hot and cold electrons in semiconductor barriers, Phys. Rev. 126:2002-2014 (1962).
D. Ventura, A. Gnudi, G. Baccarani, and F. Odeh, Multidimensional spherical harmonics expansion of Boltzmann equation for transport in semiconductors, Appl. Math. Lett. 5:85-90 (1992).
E. P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40:749-759 (1932).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Abdallah, N.B. A Hybrid Kinetic-Quantum Model for Stationary Electron Transport. Journal of Statistical Physics 90, 627–662 (1998). https://doi.org/10.1023/A:1023216701688
Issue Date:
DOI: https://doi.org/10.1023/A:1023216701688