Fibonacci Superlattices

  • A. H. MacDonald
Chapter
Part of the NATO ASI Series book series (NATO ASI, volume 179)

Abstract

Many of the interesting properties of semiconductor superlattices are consequences of the artificially imposed one-dimensional (1D) periodicity along the growth direction. Fibonacci semiconductor superlattices have a 1D structure along the growth direction which is quasiperiodic, i.e. it is characterized by two different fundamental periods whose ratio is irrational. Both the structural properties and the various spectral properties (e.g. plasmon, phonon and electron) are very different for quasiperiodic structures when compared to those of periodic structures. These materials provide yet another example of the enormous variety of physical phenomena which can occur in semiconductor multi-layer systems.

Keywords

Form Factor Transfer Matrix Acoustic Phonon Fibonacci Number Fibonacci Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. 1.
    For a review of work up to 1982 see B. Simon, Adv. Appl. Math. 3, 463 (1982).CrossRefMATHGoogle Scholar
  2. 2.
    D. Schectman, I. Bloch, D. Gratlas and J.W. Calm, Phys. Rev. Lett. 53, 1951 (1984).ADSCrossRefGoogle Scholar
  3. 3.
    R. Merlin, K. Bajema, Roy Clarke, F.-Y. Juany and P.K. Battacharya, Phys. Rev. Lett. 55, 1768 (1985).ADSCrossRefGoogle Scholar
  4. 4.
    D. Levine and P. Steinhardt, Phys. Rev. Lett. 53, 2477 (1984).ADSCrossRefGoogle Scholar
  5. 5.
    M.G. Karkut, J.-M. Triscone, D. Ariosa and O. Fischer, Phys. Rev. B 34, 4390 (1986).ADSCrossRefGoogle Scholar
  6. 6.
    A. Behrooz, M.J. Burns, H. Deckman, D. Levine, B. Whitehead, and P.M. Chaikin, Phys. Rev. Lett. 57, 368 (1986),ADSCrossRefGoogle Scholar
  7. 7.
    P. Kramer and R. Neri, Acta Crystallogr. Sect. A 40, 580 (1984).CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    P.A. Kalugin, A. Yu Kitaeu and L.C. Lerutov, Pis’ma Zhu Eksp. Teor. Fiz. 41, 119 (1985) [JETP Lett. 4l, 119 (1985)].Google Scholar
  9. 9.
    V. Elser, Phys. Rev. B 32, 4892 (1985).ADSCrossRefGoogle Scholar
  10. 10.
    M. Duneau and A. Katz, Phys. Rev. Lett. 54, 2688 (1985).ADSCrossRefMathSciNetGoogle Scholar
  11. 11.
    R.K. Zia and W.J. Dallas, J. Phys. A 18, L341(1985).ADSCrossRefGoogle Scholar
  12. 12.
    The case of the Fibonacci lattice was first worked out in detail in M.W.C. Dharmawardana, A.H. MacDonald, D.J. Lockwood, J.-M. Baribeau and D.C. Houghton, Phys. Rev. Lett. 58, 1761 (1987).ADSCrossRefGoogle Scholar
  13. 13.
    We take the z direction to be the direction along the growth axis and write \(\mathop r\limits^\to = (\mathop {{r_\bot }}\limits^\to, z)\) and \(\mathop q\limits^\to = (\mathop {{q_\bot }}\limits^\to, {q_z})\). \(F(\mathop r\limits^\to )\) is, at this point \(F(\mathop {{r_\bot }}\limits^\to, z + d) = F(\mathop {{r_\bot }}\limits^\to, z)\). Frequently the case of interest for a superlattice is \(\mathop {{q_\bot }}\limits^\to = 0\).Google Scholar
  14. 14.
    For this calculation it is more convenient to define the form factor in terms of the charge density of the N layers centered between 0 and dx = Nxax, rather than in terms of all charge density between 0 and dx Google Scholar
  15. 15.
    J. Todd, R. Merlin, R. Clarke, K.M. Mohanty and J.D. Axe, Phys. Rev. Lett. 57, 1158 (1986).ADSCrossRefGoogle Scholar
  16. 16.
    C. Colvard, T.A. Gant, M.V. Klein, R. Merlin, R. Fischer, H. Morkoc and A.C. Gossard, Phys. Rev. B 31, 2080 (1985).ADSCrossRefGoogle Scholar
  17. 17.
    B. Jusserand and D. Paquet, in Semiconductor Heterojunctions and Superlattices, edited by N. Boccara, C. Allan, G. Bastard, M. Lannoo and M. Voss (Springer, Berlin, 1986).Google Scholar
  18. 18.
    The experiments are done in a backscattering configuration which allows scattering only by longitudinal acoustic phonons with wavevectors along the growth direction.Google Scholar
  19. 19.
    R. Merlin, K. Bajema, R. Clarke and J. Todd, in Proceedings of the 18th International Conference on the Physics of Semiconductors, ed. By O. Engström, (World Scientific, Singapore, 1987) p. 675.Google Scholar
  20. 20.
    K. Bajema and R. Merlin, Phys. Rev. B 36, 4555 (1987).ADSCrossRefGoogle Scholar
  21. 21.
    R. Merlin, K. Bajema, J. Nagle and K. Ploog, Proceedings of the 3rd International Conference on Modulated Semiconductor Structures, to be published (1988).Google Scholar
  22. 22.
    D.J. Lockwood, A.H. MacDonald, G.C. Aers, M.W.C. Dharmawardana, R.L.S. Devine and W.T. Moore, Phys. Rev. B 36, 9286 (1987).ADSCrossRefGoogle Scholar
  23. 23.
    B. Jusserand, F. Alexander, D. Paquet and G. LeRoux, Appl. Phys. Lett. 47, 301 (1985).ADSCrossRefGoogle Scholar
  24. 24.
    V. Matijasevic and M.R. Beasley, Phys. Rev. B 25, 3175 (1987).ADSCrossRefGoogle Scholar
  25. 25.
    P.G. Harper, Proc. Roy. Soc. London, Sect. A 68, 874 (1955).ADSCrossRefMATHGoogle Scholar
  26. 26.
    D.R. Hofstadter, Phys. Rev. B 14, 2239 (1976).ADSCrossRefGoogle Scholar
  27. 27.
    G. Andre and S. Aubrey, Ann. Isr. Phys. Soc. 3, 133 (1980).Google Scholar
  28. 28.
    D.J. Thouless, M. Kohmoto, P. Nightingale and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).ADSCrossRefGoogle Scholar
  29. 29.
    M. Kohmoto, Phys. Rev. Lett. 51, 1198 (1983).ADSCrossRefMathSciNetGoogle Scholar
  30. 30.
    C. Tang and M. Kohmoto, Phys. Rev. Lett. 34, 2041 (1986).ADSGoogle Scholar
  31. 31.
    J.B. Sokoloff, Phys. Rep. 126, 189 (1985).ADSCrossRefGoogle Scholar
  32. 32a.
    M. Kohmoto, L.P. Kadanoff and C. Tang, Phys. Rev. Lett. 50, 1870 (1983)ADSCrossRefMathSciNetGoogle Scholar
  33. 32b.
    M. Kohmoto and Y. Oono, Phys. Lett. 102A, 145 (1984)ADSCrossRefMathSciNetGoogle Scholar
  34. 32c.
    L.P. Kadanoff and C. Tang, Proc. Natl. Acad. Sci. USA 81, 1276 (1984).ADSCrossRefMATHMathSciNetGoogle Scholar
  35. 33.
    S. Ostlund, R. Pandit, D. Rand, H.J. Sehnellnhuber and E. Siggia, Phys. Rev. Lett. 50, 1973 (1983).ADSCrossRefGoogle Scholar
  36. 34.
    M. Kohmoto and J.R. Banavar, Phys. Rev. B 34, 563, (1986).ADSCrossRefGoogle Scholar
  37. 35.
    J.P. Lu, T. Odagaki and J.L. Birman, Phys. Rev. B 33, 4809 (1986).ADSCrossRefGoogle Scholar
  38. 36.
    F. Nori and J.P. Rodriquez, Phys. Rev. B 34, 2207 (1986).ADSCrossRefGoogle Scholar
  39. 37.
    T. Odagaki and L. Freidman, Solid State Commun. 57, 915 (1986).ADSCrossRefGoogle Scholar
  40. 38.
    K. Machida and M. Fujita, Phys. Rev. B 34, 7376 (1986).ADSCrossRefGoogle Scholar
  41. 39.
    A. Mookerjee and V.A. Singh, Phys. Rev. B 34, 7433 (1986).ADSCrossRefMathSciNetGoogle Scholar
  42. 40.
    J.M. Luck and D. Petretis, J. Stat. Phys. 42, 289 (1986).ADSCrossRefGoogle Scholar
  43. 41.
    M. Kohmoto, B. Sutherland and C. Tang, Phys. Rev. B 35, 1020 (1987).ADSCrossRefMathSciNetGoogle Scholar
  44. 42.
    T. Schneider, A. Politi and D. Wurtz (preprint).Google Scholar
  45. 43.
    I. Procaccia, in Proceedings of the Nobel Symposium on Chaos and Related Problems, Phys. Scr. T 9, 40 (1985).ADSCrossRefMathSciNetGoogle Scholar
  46. 44.
    B.B. Mandelbrot, Ann. Isr. Phys. Soc. 225 (1977).Google Scholar
  47. 45.
    T.A. Witten, Jr. and L.M. Sander, Phys. Rev. Lett. 47, 1400 (1981).ADSCrossRefGoogle Scholar
  48. 46.
    K.G. Wilson, Sci. Am. 241, 158 (1979).CrossRefGoogle Scholar
  49. 47.
    D. Katzen and I. Procaccia, Phys. Rev. Lett. 58, 1169 (1987).ADSCrossRefMathSciNetGoogle Scholar
  50. 48.
    Thomas C. Halsey, Morgens H. Jensen, Leo P. Kadanoff, Itamar Procaccia and Boris J. Shraiman., Phys. Rev. A 33 1141 (1986).ADSCrossRefMATHMathSciNetGoogle Scholar
  51. 49.
    For further refinements and generalizations see L.P. Kadanoff, “Fractal singularities on a measure and how to measure singularities on a fractal.” U. of Chicago preprint, and M. Kohnioto, “Entropy Function for Multifractals.”, U. of Utah preprint.Google Scholar
  52. 50.
    The fractal dimension is defined, roughly, by, M ∝ W-F where M is the number of subbands and W is the width of each subband, For an elementary introduction to the fractional dimension concept see B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983).Google Scholar
  53. 51.
    dT*/dq = α > 0 so T* increases monotonically with q.Google Scholar
  54. 52.
    D. Grecu, Phys. Rev. B 8, 1958 (1973).ADSCrossRefGoogle Scholar
  55. 53.
    A.L. Fetter, Ann. Phys. (NY) 88, 1 (1974).ADSCrossRefGoogle Scholar
  56. 54.
    S. Das Sarma and J.J. Quinn, Phys. Rev. B 25, 7603 (1982).ADSCrossRefGoogle Scholar
  57. 55.
    S. Das Sarma, A. Kobayashi and R.E. Prange, Phys. Rev. Lett. 56 1280 (1986).ADSCrossRefGoogle Scholar
  58. 56.
    S. Das Sarma, A. Kobayashi and R.E. Prange, (preprint).Google Scholar
  59. 57.
    P. Hawvrylak and J.J. Quinn, Phys. Rev. Lett. 57, 380 (1986).ADSCrossRefGoogle Scholar
  60. 58.
    P. Hawvrylak, Gunnar Eliasson and J.J. Quinn, Phys. Rev. B 36, 6501 (1987).ADSCrossRefGoogle Scholar
  61. 59.
    We will assume that the relevant elastic constants may be taken to have the same value in each layer of the Fibonacci semiconductor superlattice so that the change in the local sound velocity is due entirely to the change in mass density.Google Scholar
  62. 60.
    In our previous discussion of plasmons, we took the dielectric constant to the same in each layer of the Fibonacci superlattice. Both effects could be included together.Google Scholar
  63. 61a.
    M. Kohmoto, Phys. Rev. B 34, 5043 (1986)ADSCrossRefGoogle Scholar
  64. 61a.
    M. Kohmoto, Phys. Rev. Lett., 58, 2436 (1987).ADSCrossRefGoogle Scholar
  65. 62.
    B.Y. Tong, preprint (1987).Google Scholar
  66. 63.
    G. Bastard and J.A. Brum, IEEE J. Quan. Elec. QE-22, 1625 (1986).ADSCrossRefGoogle Scholar
  67. 64.
    A.H. MacDonald and G.C. Aers, Phys. Rev. B 36, 9142 (1987).ADSCrossRefGoogle Scholar
  68. 65.
    M.Holzer, preprint (1987).Google Scholar
  69. 66.
    G.C. Aers, unpublished.Google Scholar

Copyright information

© Plenum Press, New York 1988

Authors and Affiliations

  • A. H. MacDonald
    • 1
  1. 1.National Research Council of CanadaOttawaCanada

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