Dynamics of electrons in gradient nanostructures (exactly solvable model)

  • A. Shvartsburg
  • V. Kuzmiak
  • G. Petite
Mesoscopic and Nanoscale Systems


A flexible multi-parameter exactly solvable model of potential profile, containing an arbitrary number of continuous smoothly shaped barriers and wells, both equal or unequal, characterized by finite values and continuous profiles of the potential and of its gradient, is presented. We demonstrate an influence of both gradient and curvature of these potentials on the electron transport and spectra of symmetric and asymmetric double-well (DW) potentials. The use of this model is simplified due to one to one correspondence between the algorithms of calculation of the transmittance of convex barriers and energy spectra of concave wells. We have shown that the resonant contrast between maximum and minimum in over-barrier reflectivity of curvilinear barrier exceeds significantly the analogous effect for rectangular barrier with the same height and width. Reflectionless tunneling of electrons below the bottom of gradient nanostructures forming concave potential barriers is considered. The analogy between dynamics of electrons in gradient fields and gradient optics of heterogeneous photonic barriers is illustrated.


03.65.Ge Solutions of wave equations: bound states 42.25.Bs Wave propagation, transmission and absorption 73.63.-b Electronic transport in nanoscale materials and structures 


  1. 1.
    D.K. Ferry, C. Jacobony, Quantum Transport in Semiconductors, edited by D.K. Ferry, C. Jacobony (Plenum, NY, 1992)Google Scholar
  2. 2.
    H.-H. Tung, C.-P. Lee, IEEE J. Quant. Electron. 32, 507 (1996)CrossRefADSGoogle Scholar
  3. 3.
    K. Nakamura, A. Shimizu, K. Fujee, IEEE J. Quant. Electron. 28, 1670 (1992)CrossRefADSGoogle Scholar
  4. 4.
    V.M. Kenkre, in Exciton Dynamics in Molecular Crystals, edited by G. Hohler (Springer, Berlin, 1982)Google Scholar
  5. 5.
    A. Wurger, From Coherent Tunneling to Relaxation (Springer, Berlin, 1997)Google Scholar
  6. 6.
    E.M. Chudnovsky, J. Tejada, Macroscopic Quantum Tunneling of Magnetic Moment (Cambridge University Press, 1998)Google Scholar
  7. 7.
    J. Esteve, T. Schumm, J.-B. Treblia, I. Bouchoule, A. Aspect, C.I. Westbrook, Eur. Phys. J. D 35 141(2005)Google Scholar
  8. 8.
    S.K. Dutta, B.K. Teo, G. Raithel, Phys. Rev. Lett. 83, 1994 (1999)CrossRefADSGoogle Scholar
  9. 9.
    I.H. Deutsch, P.S. Jessen, Phys. Rev. A 57, 1972 (1998)CrossRefADSGoogle Scholar
  10. 10.
    A.J. Leggett, Rev. Mod. Phys. 73, 307 (2001)CrossRefADSGoogle Scholar
  11. 11.
    I.H. Deutsch, P.M. Alsing, J. Grondalski, S. Chose, D.L. Haycock, P.S. Jessen, J. Opt. B: Quantum Semiclass. Opt. 2, 633 (2000)CrossRefADSGoogle Scholar
  12. 12.
    J. Esteve, C. Aussibal, T. Schumm, C. Figl, D. Mailly, I. Bouchoule, C.I. Westbrook, A. Aspect, Phys. Rev. A 70, 043629 (2004)CrossRefADSGoogle Scholar
  13. 13.
    P.L. Shkolnikov, A.E. Kaplan, S.F. Straub, Phys. Rev. A 59, 490 (1999)CrossRefADSGoogle Scholar
  14. 14.
    A.B. Shvartsburg, G. Petite, Eur. Phys. J. D 36, 111 (2005)CrossRefADSGoogle Scholar
  15. 15.
    L.D. Landau, E.M. Lifshitz, Quantum Mechanics (Pergamon Press, 1967)Google Scholar
  16. 16.
    L. Kronig, W.G. Penney, Proc. Roy. Soc. A 130, 499 (1931)CrossRefADSGoogle Scholar
  17. 17.
    A.O.E. Animalu, Intermediate Quantum Theory of Crystalline Solids (Prentice — Hall, Englewood Cliffs, 1977)Google Scholar
  18. 18.
    A. Iwamoto, V.M. Aquino, V.C. Aquilera-Nawarro, Int. J. Theor. Phys. 43, 483 (2004)CrossRefGoogle Scholar
  19. 19.
    S. Flugge, Practical Quantum Mechanics (Springer-Verlag, Berlin, 1971)Google Scholar
  20. 20.
    A.B. Shvartsburg, V. Kuzmiak, G. Petite, Phys. Rev. E 76, 016603 (2007)CrossRefADSGoogle Scholar
  21. 21.
    G. Poschl, E. Teller, Z. Physik 83, 143 (1933)CrossRefADSGoogle Scholar
  22. 22.
    S.-H. Dong, Factorization Method in Quantum Mechanics (Springer, 2007)Google Scholar
  23. 23.
    M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1968)Google Scholar
  24. 24.
    A.B. Shvartsburg, G. Petite, Opt. Lett. 31, 1127 (2006)CrossRefADSGoogle Scholar
  25. 25.
    I. Bloch, J. Phys. B: At. Mol. Opt. Phys. 38, 629 (2005)CrossRefADSGoogle Scholar
  26. 26.
    C. Zhang, S.L. Rolston, S. Das Sarma, Phys. Rev. A 74, 042316 (2006)CrossRefADSGoogle Scholar
  27. 27.
    D. Hayes, P.S. Julienne, L.H. Deutsch, Phys. Rev. Lett. 98, 070501 (2007)CrossRefADSGoogle Scholar
  28. 28.
    P.J. Lee, M. Anderlini, B.L. Brown, J. Sebby-Strabley, W.D. Phillips, J.V. Porto, Phys. Rev. Lett. (2007)Google Scholar
  29. 29.
    A.B. Shvartsburg, V. Kuzmiak, G. Petite, Phys. Rep. 452, 33 (2007)CrossRefADSGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Joint Institute of High TemperaturesRussian Acad. of SciencesMoscowRussia
  2. 2.Institute of Photonics and Electronics, Czech Acad. of SciencesPraha 8Czech Republic
  3. 3.Laboratoire des Solides Irradies, CEA – DSM, CNRS, École PolytechniquePalaiseauFrance

Personalised recommendations