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Nanowaveguides. Bloch Waves

  • Sergey LebleEmail author
Chapter
Part of the Springer Series on Atomic, Optical, and Plasma Physics book series (SSAOPP, volume 109)

Abstract

The quantum description of transport properties in nanostructures are directly connected with the geometry of the object and the corresponding electron states in a field of atomic systems [1, 2]. Waveguide properties in optics and microwaves demonstrate the very rich set of possibilities for solitonic behavior [3], and also constructive possibilities [4]. The conventional quantum mechanics of pure electron states originated from the mathematical results of Floquet [5], which lead to the fundamental notion of the Bloch  function. The quantum state is here defined as the common eigenfunction of commuting translational symmetric Hamiltonian and shift operators [6]

References

  1. 1.
    D. Christodoulides, F. Lederer, Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature 424, 817–823 (2003)CrossRefADSGoogle Scholar
  2. 2.
    A. Blanco-Redondo, I. Andonegui, M.J. Collins, G. Harari, Y. Lumer, M.C. Rechtsman, B.J. Eggleton, M. Segev, Topological Optical Waveguiding in Silicon and the Transition between Topological and Trivial Defect States. Phys. Rev. Lett. 116, 163901 (2016)CrossRefADSGoogle Scholar
  3. 3.
    A.A. Sukhorukov, Y.S. Kivshar, H.S. Eisenberg, Y. Silberberg, Spatial optical solitons in waveguide arrays. IEEE J. Quantum Electron. 39, 31–50 (2003)CrossRefADSGoogle Scholar
  4. 4.
    R.S. Savelev, A.P. Slobozhanyuk, A.E. Miroshnichenko, Y.S. Kivshar, P.A. Belov, Subwavelength waveguides composed of dielectric nanoparticles. Phys. Rev. B 89, 035435 (2014)CrossRefADSGoogle Scholar
  5. 5.
    G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques. Ann. de l’Ecole Norm. Supérieure 12, 47–88 (1883)MathSciNetCrossRefGoogle Scholar
  6. 6.
    V.A. Geyler, IYu. Popov, Group-theoretical analysis of lattice Hamiltonians with a magnetic field. Phys. Lett. A 201, 359–364 (1995)MathSciNetCrossRefADSGoogle Scholar
  7. 7.
    S.B. Leble, Kolmogorov equation for Bloch electrons and electrical resistivity models for nanowires. Nanosyst. Phys. Chem. Math. 8(2), 247–259 (2017)Google Scholar
  8. 8.
    E. Fermi, Sopra lo spostamento per pressione delle righe elevate delle serie spettrali. Il Nuovo Cim. 11, 157–166 (1934)CrossRefADSGoogle Scholar
  9. 9.
    G. Breit, The scattering of slow neutrons by bound protons: methods of calculation. Phys. Rev. 71, 215 (1947)CrossRefADSGoogle Scholar
  10. 10.
    YuN Demkov, V.N. Ostrovskii, Zero-Range Potentials and Their Applications in Atomic Physics (Plenum, New York, 1988)CrossRefGoogle Scholar
  11. 11.
    B.S. Pavlov, The theory of extensions and explicitly-solvable models. Uspekhi Mat. Nauk. 258, 99–131 (1987); Russ. Math. Surv. 42 (6), 127–168 (1987)CrossRefADSGoogle Scholar
  12. 12.
    S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics (Springer, New York, 1988)CrossRefGoogle Scholar
  13. 13.
    S.B. Leble, S. Yalunin, Multiple scattering and electron-uracil collisions at low energies. EPJ 144, 115–122 (2007)ADSGoogle Scholar
  14. 14.
    K. Huang, C.N. Yang, Quantum-mechanical many-body problem with hard-sphere interaction. Phys. Rev. 105, 767–775 (1957)MathSciNetCrossRefADSGoogle Scholar
  15. 15.
    A. Derevianko, Revised Huang-Yang multipolar pseudopotential. Phys. Rev. A 72, 044701 (2005)CrossRefADSGoogle Scholar
  16. 16.
    S.B. Leble, S. Yalunin, A dressing of zero-range potentials and electron-molecule scattering problem at low energies. Phys. Lett. A 339, 83–88 (2005)CrossRefADSGoogle Scholar
  17. 17.
    S.B. Leble, S. Yalunin, Generalized zero range potentials and multi-channel electron-molecule scattering. Rad. Phys. Chem. 68, 181–186 (2003)CrossRefADSGoogle Scholar
  18. 18.
    S. Leble, D.V. Ponomarev, Dressing of zero-range potentials into realistic molecular potentials of finite range. Task Q. 14(1–2), 29–34 (2010)Google Scholar
  19. 19.
    V.M. Adamyan, I.V. Blinova, A.I. Popov, I.Yu. Popov: Waveguide bands for a system of macromolecules. Nanosyst. Phys. Chem. Math. 6(5), 611–617 (2015)Google Scholar
  20. 20.
    E.N. Grishanov, I.Y. Popov, Electron spectrum for aligned SWNT array in a magnetic field. Superlattices Microstruct. 100, 1276–1282 (2016)CrossRefADSGoogle Scholar
  21. 21.
    E.N. Grishanov, I.Y. Popov, Computer simulation of periodic nanostructures. Nanosyst. Phys. Chem. Math. 7(5), 865–868 (2016)Google Scholar
  22. 22.
    S. Leble, Cyclic periodic ZRP structures. Scattering problem for generalized Bloch functions and conductivity. Nanosyst. Phys. Chem. Math. 9 (2), 225–243 (2018)MathSciNetGoogle Scholar
  23. 23.
    D.V. Ponomarev, Electronic states in zero-range potential models of nanostructures with a cyclic symmetry. M.Sc. Thesis. Supervisor: S. Leble, Gdansk University of Technology (2010)Google Scholar
  24. 24.
    E.V. Doktorov, S.B. Leble, Dressing Method in Mathematical Physics (Springer, Dordrecht, 2007)CrossRefGoogle Scholar
  25. 25.
    IYu. Popov, S.L. Popova, Eigenvalues and bands imbedded in the continuous spectrum for a system of resonators and a waveguide: solvable model. Phys. Lett. A 222, 286–290 (1996)MathSciNetCrossRefADSGoogle Scholar
  26. 26.
    V. Fock, Fundamentals of Quantum Mechanics. (Mir publishers, Moscow, 1978, 1982)Google Scholar
  27. 27.
    R. Szmytkowski, Zero-range potentials for Dirac particles: scattering and related continuum problems. Phys. Rev. A 71(052708), 1–19 (2005)Google Scholar
  28. 28.
    S. Botman, S. Leble, Bloch wave scattering on pseudopotential impurity in 1D Dirac comb model. arXiv:1511.04758v1 [cond-mat.mes-hall]
  29. 29.
    M. Le Bellac, Quantum Physics (Cambridge University Press, Cambridge, 2006)Google Scholar
  30. 30.
    C.A. Coulson, B. O’Leary, R.B. Mallion, Hückel Theory for Organic Chemists (Academic Press, London, 1978)Google Scholar
  31. 31.
    M. Nakahara, C. Wakai, N. Matubayasi, Jump in the rotational mobility of Benzene induced by the Clathrate Hydrate formation. J. Phys. Chem. 99(5), 1377–1379 (1995)CrossRefGoogle Scholar
  32. 32.
    B.S. Pavlov, A.V. Strepetov, Exactly solvable model of electron scattering by an inhomogeneity in a thin conductor. TMF 90(2), 226–232 (1992); Theoret. Math. Phys. 90 (2), 152–156 (1992)MathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Immanuel Kant Baltic Federal UniversityKaliningradRussia

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