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Trapped modes in finite quantum waveguides

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Abstract

The eigenstates of an electron in an infinite quantum waveguide (e.g., a bent strip or a twisted tube) are often trapped or localized in a bounded region that prohibits the electron transmission through the waveguide at the corresponding energies. We revisit this statement for resonators with long but finite branches that we call “finite waveguides”. Although the Laplace operator in bounded domains has no continuous spectrum and all eigenfunctions have finite L 2 norm, the trapping of an eigenfunction can be understood as its exponential decay inside the branches. We describe a general variational formalism for detecting trapped modes in such resonators. For finite waveguides with general cylindrical branches, we obtain a sufficient condition which determines the minimal length of branches for getting a trapped eigenmode. Varying the branch lengths may switch certain eigenmodes from non-trapped to trapped or, equivalently, the waveguide state from conducting to insulating. These concepts are illustrated for several typical waveguides (L-shape, bent strip, crossing of two strips, etc.). We conclude that the well-established theory of trapping in infinite waveguides may be incomplete and require further development for applications to finite-size microscopic quantum devices.

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References

  1. G. Timp, H.U. Baranger, P. deVegvar, J.E. Cunningham, R.E. Howard, R. Behringer, P.M. Mankiewich, Phys. Rev. Lett. 60, 2081 (1988)

    Article  ADS  Google Scholar 

  2. R.L. Schult, D.G. Ravenhall, H.W. Wyld, Phys. Rev. B 39, 5476 (1989)

    Article  ADS  Google Scholar 

  3. J.P. Carini, J.T. Londergan, K. Mullen, D.P. Murdock, Phys. Rev. B 46, 15538 (1992)

    Article  ADS  Google Scholar 

  4. J.P. Carini, J.T. Londergan, K. Mullen, D.P. Murdock, Phys. Rev. B 48, 4503 (1993)

    Article  ADS  Google Scholar 

  5. J.P. Carini, J.T. Londergan, D.P. Murdock, D. Trinkle, C.S. Yung, Phys. Rev. B 55, 9842 (1997)

    Article  ADS  Google Scholar 

  6. J.T. Londergan, J.P. Carini, D.P. Murdock, Binding and Scattering in Two-Dimensional Systems: Applications to Quantum Wires, Waveguides and Photonic Crystals (Springer, Berlin, 1999)

  7. P. Duclos, P. Exner, Rev. Math. Phys. 7, 73 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  8. J.D. Jackson, Classical Electrodynamics, 3rd edn. (Wiley & Sons, New York, 1999)

  9. C.M. Linton, P. McIver, Wave Motion 45, 16 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. O. Olendski, L. Mikhailovska, Phys. Rev. E 81, 036606 (2010)

    Article  ADS  Google Scholar 

  11. F. Rellich, Das Eigenwertproblem von in Halbrohren, (Studies and Essays Presented to R. Courant, New York, 1948), pp. 329–344

  12. D.S. Jones, Math. Proc. Camb. Phil. Soc. 49, 668 (1953)

    Article  ADS  MATH  Google Scholar 

  13. F. Ursell, Math. Proc. Camb. Phil. Soc. 47, 347 (1951)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. F. Ursell, J. Fluid Mech. 183, 421 (1987)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. F. Ursell, Proc. R. Soc. Lond. A 435, 575 (1991)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. R. Parker, J. Sound Vib. 4, 62 (1966)

    Article  ADS  Google Scholar 

  17. R. Parker, J. Sound Vib. 5, 330 (1967)

    Article  ADS  Google Scholar 

  18. P. Exner, P. Seba, J. Math. Phys. 30, 2574 (1989)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. J. Goldstone, R.L. Jaffe, Phys. Rev. B 45, 14100 (1992)

    Article  ADS  Google Scholar 

  20. B. Chenaud, P. Duclos, P. Freitas, D. Krejcirík, Diff. Geom. Appl. 23, 95 (2005)

    Article  MATH  Google Scholar 

  21. D.V. Evans, IMA J. Appl. Math. 49, 45 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  22. D.V. Evans, M. Levitin, D. Vassiliev, J. Fluid Mech. 261, 21 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. E.B. Davies, L. Parnovski, Q.J. Mech. Appl. Math. 51, 477 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  24. W. Bulla, F. Gesztesy, W. Renger, B. Simon, Proc. Amer. Math. Soc. 125, 1487 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  25. J. Dittrich, J. Kríz, J. Phys. A: Math. Gen. 35, L269 (2002)

    Article  ADS  MATH  Google Scholar 

  26. P. Freitas, D. Krejčiřík, Math. Phys. Anal. Geom. 9, 335 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. A.S. Bonnet-Ben Dhia, P. Joly, SIAM J. Appl. Math. 53, 1507 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  28. M.S. Ashbaugh, P. Exner, Phys. Lett. A 150, 183 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  29. P. Exner, P. Freitas, D. Krejčiřík, Proc. R. Soc. Lond. A 460, 3457 (2004)

    Article  ADS  MATH  Google Scholar 

  30. B. Sapoval, T. Gobron, A. Margolina, Phys. Rev. Lett. 67, 2974 (1991)

    Article  ADS  Google Scholar 

  31. C. Even, S. Russ, V. Repain, P. Pieranski, B. Sapoval, Phys. Rev. Lett. 83, 726 (1999)

    Article  ADS  Google Scholar 

  32. S. Felix, M. Asch, M. Filoche, B. Sapoval, J. Sound. Vib. 299, 965 (2007)

    Article  ADS  Google Scholar 

  33. J.L. Lions, E. Magenes, Non-homogeneous Boundary value Problems and Applications (Springer-Verlag, Berlin, New York, 1972)

  34. M.S. Birman, Mat. Sbornik 55, 125 (1961) [in Russian]

    MathSciNet  Google Scholar 

  35. J. Schwinger, Proc. Natl. Acad. Sci. 47, 122 (1961)

    Article  ADS  MathSciNet  Google Scholar 

  36. A.L. Delitsyn, Diff. Equ. 40, 207 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  37. Y. Avishai, D. Bessis, B.G. Giraud, G. Mantica, Phys. Rev. B 44, 8028 (1991)

    Article  ADS  Google Scholar 

  38. E.N. Bulgakov, P. Exner, K.N. Pichugin, A.F. Sadreev, Phys. Rev. B 66, 155109 (2002)

    Article  ADS  Google Scholar 

  39. P. Freitas, D. Krejčiřík, Proc. Amer. Math. Soc. 136, 2997 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  40. P. Exner, P. Seba, M. Tater, D. Vanek, J. Math. Phys. 37, 4867 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  41. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Products (Academic Press, New York, 1980)

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Authors and Affiliations

  1. Mathematical Department of the Faculty of Physics, Moscow State University, 119991, Moscow, Russia

    A. L. Delitsyn

  2. Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS — Ecole Polytechnique, 91128, Palaiseau, France

    B. T. Nguyen & D. S. Grebenkov

  3. Laboratoire Poncelet (UMI 2615), CNRS — Independent University of Moscow, Bolshoy Vlasyevskiy Pereulok 11, 119002, Moscow, Russia

    D. S. Grebenkov

  4. Chebyshev Laboratory, Saint Petersburg State University, 14th line of Vasil’evskiy Ostrov 29, Saint Petersburg, Russia

    D. S. Grebenkov

Authors
  1. A. L. Delitsyn
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  2. B. T. Nguyen
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  3. D. S. Grebenkov
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Correspondence to D. S. Grebenkov.

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Delitsyn, A.L., Nguyen, B.T. & Grebenkov, D.S. Trapped modes in finite quantum waveguides. Eur. Phys. J. B 85, 176 (2012). https://doi.org/10.1140/epjb/e2012-21038-y

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  • DOI: https://doi.org/10.1140/epjb/e2012-21038-y

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