Symbolic-Numeric Algorithms for Computer Analysis of Spheroidal Quantum Dot Models

  • A. A. Gusev
  • O. Chuluunbaatar
  • V. P. Gerdt
  • V. A. Rostovtsev
  • S. I. Vinitsky
  • V. L. Derbov
  • V. V. Serov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6244)


A computational scheme for solving elliptic boundary value problems with axially symmetric confining potentials using different sets of one-parameter basis functions is presented. The efficiency of the proposed symbolic-numerical algorithms implemented in Maple is shown by examples of spheroidal quantum dot models, for which energy spectra and eigenfunctions versus the spheroid aspect ratio were calculated within the conventional effective mass approximation. Critical values of the aspect ratio, at which the discrete spectrum of models with finite-wall potentials is transformed into a continuous one in strong dimensional quantization regime, were revealed using the exact and adiabatic classifications.


Wall Height Slow Subsystem Minor Semiaxis Interband Light Absorption Optical Absorption Cross Section 
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  1. 1.
    Harrison, P.: Quantum Well, Wires and Dots. In: Theoretical and Computational Physics of Semiconductor Nanostructures. Wiley, New York (2005)Google Scholar
  2. 2.
    Gambaryan, K.M.: Interaction and Cooperative Nucleation of InAsSbP Quantum Dots and Pits on InAs(100) Substrate. Nanoscale. Res. Lett. (2009), doi:10.1007/s11671-009-9510-8Google Scholar
  3. 3.
    Wojs, A., Hawrylak, P., Fafard, S., Jacak, L.: Electronic structure and magneto-optics of self-assembled quantum dots. Phys. Rev. B 54, 5604–5608 (1996)CrossRefGoogle Scholar
  4. 4.
    Juharyan, L.A., Kazaryan, E.M., Petrosyan, L.S.: Electronic states and interband light absorption in semi-spherical quantum dot under the influence of strong magnetic field. Solid State Comm. 139, 537–540 (2006)CrossRefGoogle Scholar
  5. 5.
    Dvoyan, K.G., Hayrapetyan, D.B., Kazaryan, E.M., Tshantshapanyan, A.A.: Electron States and Light Absorption in Strongly Oblate and Strongly Prolate Ellipsoidal Quantum Dots in Presence of Electrical and Magnetic Fields. Nanoscale Res. Lett. 2, 601–608 (2007)CrossRefGoogle Scholar
  6. 6.
    Cantele, G., Ninno, D., Iadonisi, G.: Confined states in ellipsoidal quantum dots. J. Phys. Condens. Matt. 12, 9019–9036 (2000)CrossRefGoogle Scholar
  7. 7.
    Trani, F., Cantele, G., Ninno, D., Iadonisi, G.: Tight-binding calculation of the optical absorption cross section of spherical and ellipsoidal silicon nanocrystals. Phys. Rev. B 72, 075423 (2005)CrossRefGoogle Scholar
  8. 8.
    Lepadatu, A.-M., Stavarache, I., Ciurea, M.L., Iancu, V.: The influence of shape and potential barrier on confinement energy levels in quantum dots. J. Appl. Phys. 107, 033721 (2010)CrossRefGoogle Scholar
  9. 9.
    Vinitsky, S.I., Gerdt, V.P., Gusev, A.A., Kaschiev, M.S., Rostovtsev, V.A., Samoilov, V.N., Tupikova, T.V., Chuluunbaatar, O.: A symbolic-numerical algorithm for the computation of matrix elements in the parametric eigenvalue problem. Programming and Computer Software 33, 105–116 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chuluunbaatar, O., Gusev, A., Gerdt, V., Kaschiev, M., Rostovtsev, V., Samoylov, V., Tupikova, T., Vinitsky, S.: A Symbolic-numerical algorithm for solving the eigenvalue problem for a hydrogen atom in the magnetic field: cylindrical coordinates. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 118–133. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Chuluunbaatar, O., Gusev, A.A., Abrashkevich, A.G., Amaya-Tapia, A., Kaschiev, M.S., Larsen, S.Y., Vinitsky, S.I.: KANTBP: A program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach. Comput. Phys. Commun. 177, 649–675 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chuluunbaatar, O., Gusev, A.A., Gerdt, V.P., Rostovtsev, V.A., Vinitsky, S.I., Abrashkevich, A.G., Kaschiev, M.S., Serov, V.V.: POTHMF: A program for computing potential curves and matrix elements of the coupled adiabatic radial equations for a hydrogen-like atom in a homogeneous magnetic field. Comput. Phys. Commun. 178, 301–330 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chuluunbaatar, O., Gusev, A.A., Vinitsky, S.I., Abrashkevich, A.G.: ODPEVP: A program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem. Comput. Phys. Commun. 180, 1358–1375 (2009)CrossRefzbMATHGoogle Scholar
  14. 14.
    Vinitsky, S.I., Chuluunbaatar, O., Gerdt, V.P., Gusev, A.A., Rostovtsev, V.A.: Symbolic-numerical algorithms for solving parabolic quantum well problem with hydrogen-like impurity. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2009. LNCS, vol. 5743, pp. 334–349. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  15. 15.
    Kantorovich, L.V., Krylov, V.I.: Approximate Methods of Higher Analysis. Wiley, New York (1964)zbMATHGoogle Scholar
  16. 16.
    Born, M., Huang, X.: Dynamical Theory of Crystal Lattices. The Clarendon Press, Oxford (1954)zbMATHGoogle Scholar
  17. 17.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965)zbMATHGoogle Scholar
  18. 18.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1. Wiley, Chichester (1989)CrossRefzbMATHGoogle Scholar
  19. 19.
    Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973)zbMATHGoogle Scholar
  20. 20.
    Schultz, M.H.: L 2 Error Bounds for the Rayleigh-Ritz-Galerkin Method. SIAM J. Numer. Anal. 8, 737–748 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Harper, P.G.: Single Band Motion of Conduction Electrons in a Uniform Magnetic Field. Proc. Phys. Soc. A 68, 874–878 (1955)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • A. A. Gusev
    • 1
    • 2
  • O. Chuluunbaatar
    • 1
  • V. P. Gerdt
    • 1
  • V. A. Rostovtsev
    • 1
    • 2
  • S. I. Vinitsky
    • 1
  • V. L. Derbov
    • 3
  • V. V. Serov
    • 3
  1. 1.Joint Institute for Nuclear ResearchDubnaRussia
  2. 2.Society & ManDubna International University of NatureDubnaRussia
  3. 3.Saratov State UniversitySaratovRussia

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