Skip to main content
Springer Nature Link
Log in

Multi-Scale Modeling of Quantum Semiconductor Devices

  • Conference paper
Analysis, Modeling and Simulation of Multiscale Problems

  • 1707 Accesses

Summary

This review is concerned with three classes of quantum semiconductor equations: Schrödinger models, Wigner models, and fluid-type models. For each of these classes, some phenomena on various time and length scales are presented and the connections between micro-scale and macro-scale models are explained. We discuss Schrödinger-Poisson systems for the simulation of quantum waveguides and illustrate the importance of using open boundary conditions. We present Wigner-based semiconductor models and sketch their mathematical analysis. In particular we discuss the Wigner-Poisson-Focker-Planck system, which is the starting point of deriving subsequently the viscous quantum hydrodynamic model. Furthermore, a unified approach to derive macroscopic quantum equations is presented. Two classes of models are derived from a Wigner equation with elastic and inelastic collisions: quantum hydrodynamic equations and their variants, as well as quantum diffusion models.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. M. Ancona and G. Iafrate. Quantum correction to the equation of state of an electron gas in a semiconductor. Phys. Rev. B 39 (1989), 9536–9540.

    Article  Google Scholar 

  2. M. Ancona and H. Tiersten. Macroscopic physics of the silicon inversion layer. Phys. Rev. B 35 (1987), 7959–7965.

    Article  Google Scholar 

  3. A. Arnold. Self-consistent relaxation-time models in quantum mechanics. Comm. PDE 21(3 & 4) (1995), 473–506.

    Google Scholar 

  4. A. Arnold. The relaxation-time von Neumann-Poisson equation. in: Proceedings of ICIAM 95, Hamburg (1995), Oskar Mahrenholtz, Reinhard Mennicken (eds.), ZAMM 76–S2 (1996), 293–296.

    Google Scholar 

  5. A. Arnold. Numerically absorbing boundary conditions for quantum evolution equations. VLSI Design 6(1–4) (1998) 313–319.

    Google Scholar 

  6. A. Arnold, E. Dhamo, and C. Manzini. The Wigner-Poisson-Fokker-Planck system: global-in-time solution and dispersive effects, submitted (2005). Technical Report 10/04-N, Angewandte Mathematik, Universität Münster.

    Google Scholar 

  7. A. Arnold. Mathematical concepts of open quantum boundary conditions. Transp. Theory Stat. Phys. 30(4–6) (2001) 561–584.

    Article  MATH  Google Scholar 

  8. A. Arnold, E. Dhamo, and C. Manzini. Dispersive effects in quantum kinetic equations, submitted (2005). Technical Report 07/05-N, Angewandte Mathematik, Universität Münster.

    Google Scholar 

  9. A. Arnold, M. Ehrhardt, I. Sofronov. Approximation, stability and fast calculation of non-local boundary conditions for the Schrödinger equation. Commun. Mathematical Sciences 1–3 (2003) 501–556.

    MathSciNet  Google Scholar 

  10. A. Arnold, J.L. López, P.A. Markowich and J. Soler. An analysis of quantum Fokker-Planck models: A Wigner function approach. Rev. Mat. Iberoam. 20(3) (2004), 771–814.

    MATH  Google Scholar 

  11. A. Arnold, M. Schulte. Discrete transparent boundary conditions for the Schrödinger equatioon — a compact higher order scheme. in preparation 2006.

    Google Scholar 

  12. A. Arnold, C. Sparber. Conservative quantum dynamical semigroups for mean-field quantum diffusion models. Comm. Math. Phys. 251(1) (2004), 179–207.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Arnold, C. Ringhofer. Operator splitting methods applied to spectral discretizations of quantum transport equations. SIAM J. of Num. Anal. 32–6 (1995) 1876–1894.

    Article  MathSciNet  Google Scholar 

  14. N. Ben Abdallah and P. Degond. On a hierarchy of macroscopic models for semiconductors. J. Math. Phys. 37 (1996), 3308–3333.

    Article  MathSciNet  Google Scholar 

  15. N. Ben Abdallah, F. Méhats, O. Pinaud. On an open transient Schrödinger-Poisson system. Math. Models Methods Appl. Sci. 15–5 (2005), 667–688.

    Article  Google Scholar 

  16. N. Ben Abdallah, O. Pinaud. Multiscale simulation of transport in an open quantum system: resonances and WKB interpolation. J. Comput. Phys. 213–1 (2006) 288–310.

    Article  MathSciNet  Google Scholar 

  17. N. Ben Abdallah and A. Unterreiter. On the stationary quantum driftdiffusion model. Z. Angew. Math. Phys. 49 (1998), 251–275.

    Article  MATH  MathSciNet  Google Scholar 

  18. P. Bhatnagar, E. Gross, and M. Krook. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Review 94 (1954), 511–525.

    Article  MATH  Google Scholar 

  19. F. Brezzi, P.A. Markowich. The three-dimensional Wigner-Poisson problem: existence, uniqueness and approximation. Math. Methods Appl. Sci. 14(1) (1991), 35–61.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Caldeira and A. Leggett. Path integral approach to quantum Brownian motion. Phys. A 121A (1983), 587–616.

    Article  MathSciNet  Google Scholar 

  21. J.A. Cañizo, J.L. López, J. Nieto. Global L 1-theory and regularity for the 3D nonlinear Wigner-Poisson-Fokker-Planck system. J. Diff. Eq. 198 (2004), 356–373.

    Article  MATH  Google Scholar 

  22. F. Castella. The Vlasov-Poisson-Fokker-Planck System with Infinite Kinetic Energy. Indiana Univ. Math. J. 47(3) (1998), 939–964.

    Article  MATH  MathSciNet  Google Scholar 

  23. F. Castella, L. Erdös, F. Frommlet, and P. Markowich. Fokker-Planck equations as scaling limits of reversible quantum systems. J. Stat. Phys. 100 (2000), 543–601.

    Article  MATH  Google Scholar 

  24. P. Degond, S. Génieys, and A. Jüngel. A system of parabolic equations in nonequilibrium thermodynamics including thermal and electrical effects. J. Math. Pures Appl. 76 (1997), 991–1015.

    MATH  MathSciNet  Google Scholar 

  25. P. Degond, F. Méhats, and C. Ringhofer. Quantum hydrodynamic models derived from the entropy principle. Contemp. Math. 371 (2005), 107–131.

    MATH  Google Scholar 

  26. P. Degond, F. Méhats, and C. Ringhofer. Quantum energy-transport and drift-diffusion models. J. Stat. Phys. 118 (2005), 625–665.

    Article  MATH  MathSciNet  Google Scholar 

  27. P. Degond and C. Ringhofer. Quantum moment hydrodynamics and the entropy principle. J. Stat. Phys. 112 (2003), 587–628.

    Article  MATH  MathSciNet  Google Scholar 

  28. B. Derrida, J. L. Lebowitz, E. R. Speer, and H. Spohn. Fluctuations of a stationary nonequilibrium interface. Phys. Rev. Lett. 67 (1991), 165–168.

    Article  MATH  MathSciNet  Google Scholar 

  29. E. Dhamo, A. Arnold. An operator splitting method on the periodic Wigner-Poisson-Fokker-Planck system. in preparation, 2006.

    Google Scholar 

  30. J. Dolbeault, I. Gentil, and A. Jüngel. A nonlinear fourth-order parabolic equation and related logarithmic Sobolev inequalities. To appear in Commun. Math. Sci., 2006.

    Google Scholar 

  31. A. El Ayyadi and A. Jüngel. Semiconductor simulations using a coupled quantum drift-diffusion Schrödinger-Poisson model. SIAM J. Appl. Math. 66 (2005), 554–572.

    Article  MATH  MathSciNet  Google Scholar 

  32. D. Ferry and J.-R. Zhou. Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling. Phys. Rev. B 48 (1993), 7944–7950.

    Article  Google Scholar 

  33. W.R. Frensley. Wigner-function model of a resonant-tunneling semiconductor device Phys. Rev. B 36 (1987) 1570–1580.

    Article  Google Scholar 

  34. S. Gallego and F. Méhats. Entropic discretization of a quantum drift-diffusion model. SIAM J. Numer. Anal. 43 (2005), 1828–1849.

    Article  MATH  MathSciNet  Google Scholar 

  35. I. Gamba and A. Jüngel. Positive solutions of singular equations of second and third order for quantum fluids. Arch. Rat. Mech. Anal. 156 (2001), 183–203.

    Article  MATH  Google Scholar 

  36. I. Gamba and A. Jüngel. Asymptotic limits in quantum trajectory models. Commun. Part. Diff. Eqs. 27 (2002), 669–691.

    Article  MATH  Google Scholar 

  37. C. Gardner. The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54 (1994), 409–427.

    Article  MATH  MathSciNet  Google Scholar 

  38. U. Gianazza, G. Savaré, and G. Toscani. A fourth-order nonlinear PDE as gradient flow of the Fisher information in Wasserstein spaces. In preparation, 2006.

    Google Scholar 

  39. H.L. Gruvin, T.R. Govindan, J.P. Kreskovsky, M.A. Stroscio. Transport via the Liouville equation and moments of quantum distribution functions, Solid State Electr. 36 (1993), 1697–1709.

    Article  Google Scholar 

  40. M. Gualdani and A. Jüngel. Analysis of the viscous quantum hydrodynamic equations for semiconductors. Europ. J. Appl. Math. 15 (2004), 577–595.

    Article  Google Scholar 

  41. M. T. Gyi and A. Jüngel. A quantum regularization of the one-dimensional hydrodynamic model for semiconductors. Adv. Diff. Eqs. 5 (2000), 773–800.

    MATH  Google Scholar 

  42. F. Huang, H.-L. Li, A. Matsumura, and S. Odanaka. Well-posedness and stability of multi-dimensional quantum hydrodynamics in whole space. Preprint, Osaka University, Japan, 2004.

    Google Scholar 

  43. A. Jüngel. A steady-state quantum Euler-Poisson system for semiconductors. Commun. Math. Phys. 194 (1998), 463–479.

    Article  MATH  Google Scholar 

  44. A. Jüngel. Quasi-hydrodynamic Semiconductor Equations. Birkhäuser, Basel, 2001.

    Google Scholar 

  45. A. Jüngel and H.-L. Li. Quantum Euler-Poisson systems: global existence and exponential decay. Quart. Appl. Math. 62 (2004), 569–600.

    MATH  MathSciNet  Google Scholar 

  46. A. Jüngel, H.-L. Li, and A. Matsumura. The relaxation-time limit in the quantum hydrodynamic equations for semiconductors. To appear in J. Diff. Eqs., 2006.

    Google Scholar 

  47. A. Jüngel, M. C. Mariani and D. Rial. Local existence of solutions to the transient quantum hydrodynamic equations. Math. Models Meth. Appl. Sci. 12 (2002), 485–495.

    Article  MATH  Google Scholar 

  48. A. Jüngel and D. Matthes. A derivation of the isothermal quantum hydrodynamic equations using entropy minimization. Z. Angew. Math. Mech. 85 (2005), 806–814.

    Article  MATH  Google Scholar 

  49. A. Jüngel and D. Matthes. An algorithmic construction of entropies in higher-order nonlinear PDEs. Nonlinearity 19 (2006), 633–659.

    Article  MATH  MathSciNet  Google Scholar 

  50. A. Jüngel and D. Matthes. The multi-dimensional Derrida-Lebowitz-Speer-Spohn equation. In preparation, 2006.

    Google Scholar 

  51. A. Jüngel, D. Matthes, and J.-P. Milišić. Derivation of new quantum hydrodynamic equations using entropy minimization. Preprint, Universität Mainz, 2005.

    Google Scholar 

  52. A. Jüngel and J.-P. Milišić. Macroscopic quantum models with and without collisions. To appear in Proceedings of the Sixth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics, Kyoto, Japan. Transp. Theory Stat. Phys., 2006.

    Google Scholar 

  53. A. Jüngel and J.-P. Milišić. Numerical approximation of the nonisothermal quantum hydrodynamic equations for semiconductors with viscous terms. In preparation, 2006.

    Google Scholar 

  54. A. Jüngel and R. Pinnau. Global non-negative solutions of a nonlinear fourth-oder parabolic equation for quantum systems. SIAM J. Math. Anal. 32 (2000), 760–777.

    Article  MathSciNet  Google Scholar 

  55. A. Jüngel and R. Pinnau. A positivity-preserving numerical scheme for a nonlinear fourth-order parabolic equation. SIAM J. Num. Anal. 39 (2001), 385–406.

    Article  MATH  Google Scholar 

  56. A. Jüngel and S. Tang. Numerical approximation of the viscous quantum hydrodynamic model for semiconductors. To appear in Appl. Numer. Math., 2006.

    Google Scholar 

  57. A. Jüngel and I. Violet. The quasineutral limit in the quantum drift-diffusion equations. Preprint, Universität Mainz, Germany, 2005.

    Google Scholar 

  58. N. Kluksdahl, A. M. Kriman, D. K. Ferry, and C. Ringhofer. Self-consistent study of the resonant tunneling diode. Phys. Rev. B 39 (1989), 7720–7735.

    Article  Google Scholar 

  59. H. Kosina, M. Nedjalkov. Wigner function-based device modeling. in: Handbook of Theoretical and Computational Nanotechnology vol. 10 (eds: M. Rieth, W. Schommers), American Scientific Publishers, 2006

    Google Scholar 

  60. C. Levermore. Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83 (1996), 1021–1065.

    Article  MATH  MathSciNet  Google Scholar 

  61. I.B. Levinson. Translational invariance in uniform fields and the equation for the density matrix in the Wigner representation. Sov. Phys. JETP 30 (1970) 362–367.

    MathSciNet  Google Scholar 

  62. H.-L. Li and C.-K. Lin. Zero Debye length asymptotic of the quantum hydrodynamic model for semiconductors. Commun. Math. Phys. 256 (2005), 195–212.

    Article  MATH  MathSciNet  Google Scholar 

  63. G. Lindblad. On the generators of quantum mechanical semigroups. Comm. Math. Phys. 48 (1976), 119–130.

    Article  MATH  MathSciNet  Google Scholar 

  64. P.L. Lions. T. Paul. Sur les mesures de Wigner. Rev. Math. Iberoam., 9(3) (1993), 553–561.

    MATH  MathSciNet  Google Scholar 

  65. E. Madelung. Quantentheorie in hydrodynamischer Form. Z. Physik 40 (1927), 322–326.

    Article  Google Scholar 

  66. P. Markowich, C. Ringhofer, and C. Schmeiser. Semiconductor Equations. Springer, Vienna, 1990.

    MATH  Google Scholar 

  67. D.A. Rodrigues, A.D. Armour. Quantum master equation descriptions of a nanomechanical resonator coupled to a single-electron transistor. New J. Phys. 7 (2005) 251–272.

    Article  Google Scholar 

  68. B. Perthame. Time decay, propagation of low moments and dispersive effects for kinetic equations. Comm. P.D.E. 21(1 & 2) (1996), 659–686.

    MATH  MathSciNet  Google Scholar 

  69. L. Ramdas Ram-Mohan. Finite element and boundary emelent applications in quantum mechanics. Oxford Univ. Press, 2002.

    Google Scholar 

  70. C. Ringhofer. A spectral method for the numerical simulation of quantum tunneling phenomena SIAM J. Num. Anal. 27 (1990) 32–50.

    Article  MATH  MathSciNet  Google Scholar 

  71. H. Risken. The Fokker-Planck equation. Springer, 1984.

    Google Scholar 

  72. C. Sparber, J.A. Carrillo, J. Dolbeault, P.A. Markowich. On the long time behavior of the quantum Fokker-Planck equation. Monatsh. f. Math. 141(3) (2004), 237–257.

    Article  MATH  MathSciNet  Google Scholar 

  73. M.A. Stroscio. Moment-equation representation of the dissipative quantum Liouville equation. Superlattices and microstructures 2 (1986), 83–87.

    Article  Google Scholar 

  74. E. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40 (1932), 749–759.

    Article  MATH  Google Scholar 

  75. A. Zisowsky, A. Arnold, M. Ehrhardt, T. Koprucki. Discrete Transparent Boundary Conditions for transient kp-Schrödinger Equations with Application to Quantum-Heterostructures. ZAMM 85 11 (2005) 793–805.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8, A-1040, Wien, Austria

    Anton Arnold

  2. Institut für Mathematik, Universität Mainz, Staudingerweg 9, 55099, Mainz

    Ansgar Jüngel

Authors
  1. Anton Arnold
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ansgar Jüngel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Weierstraß-Institut für Angewandte, Analysis und Stochastik, Mohrenstraße 39, 10117, Berlin, Germany

    Alexander Mielke

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin, Germany

    Alexander Mielke

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Arnold, A., Jüngel, A. (2006). Multi-Scale Modeling of Quantum Semiconductor Devices. In: Mielke, A. (eds) Analysis, Modeling and Simulation of Multiscale Problems. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-35657-6_12

Download citation

Publish with us

Policies and ethics