Il Nuovo Cimento D

, Volume 15, Issue 2–3, pp 375–392 | Cite as

Renormalization and envelope function formalism for incommensurate systems

  • R. Farchioni
  • G. Grosso
  • G. Pastori Parravicini


We study the localization-delocalization transition in one-dimensional incommensurate crystals both numerically and analytically. From the numerical point of view we provide an implementation of the renormalization method, which allows to process with high accuracy millions of sites (whenever necessary). From the analytic point of view we extend the envelope function concepts to incommensurate potentials, smoothly varying on lattice constant scale. The control of the transition is made by numerical calculation of the Lyapunov exponent: it presents surprising aspects of universality and simplicity, with plateaux, linear regions and, at times, much more complicated behaviours. The envelope function method, together with semi-analytic considerations, allows to understand, in a number of situations, the presence of mobility edges, pseudo-mobility edges, and gaps in the energy spectrum.

PACS 71.50

Localized single-particle electronic states (excluding impurities) 

PACS 71.30

Metal-insulator transitions 

PACS 73.20.Dx

Electron states in low-dimensional structures (including quantum wells, superlattices, layer structures, and intercalation compounds) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    F. Bassani, F. Fumi andM. P. Tosi (Editors):Highlights in Condensed Matter Theory, Proceedings of the International School of Physics «Enrico Fermi», Course LXXXIX (North-Holland, Amsterdam, 1983).Google Scholar
  2. [2]
    L. Esaki:IEEE J. Quantum Electron.,QE22, 1611 (1986).CrossRefADSGoogle Scholar
  3. [3]
    J. B. Sokoloff:Phys. Rep.,126, 189 (1985).CrossRefADSGoogle Scholar
  4. [4]
    See for instance,H. Z. Cummins:Phys. Rep.,185, 211 (1990);R. Currat andT. Janssen:Solid State Phys.,41, 201 (1988); edited byH. Ehrenreich andD. Turnbull (Wiley, New York, N.Y., 1988);T. Jannsen andA. Janner:Adv. Phys.,36, 519 (1987);P. Bak:Rep. Prog. Phys.,45, 587 (1982).CrossRefADSGoogle Scholar
  5. [5]
    R. Farchioni, G. Grosso andG. Pastori Parravicini:Phys. Rev. B,45, 6383 (1992).CrossRefADSGoogle Scholar
  6. [6]
    G. Grosso andG. Pastori Parravicini:Adv. Chem. Phys.,62, 81 (1986);62, 131 (1986);P. Giannozzi, G. Grosso, S. Moroni andG. Pastori Parravicini:Appl. Num. Math.,4, 273 (1988);R. D. Graft, G. Grosso, D. J. Lohrmann, L. Martinelli, S. Moroni, G. Pastori Parravicini andL. Resca: inProgress in Electronic Properties of Solids, edited byE. Doni, R. Girlanda, G. Pastori Parravicini andA. Quattropani (Kluwer, Dordrecht, 1989).MathSciNetGoogle Scholar
  7. [7]
    See, for instance, the review article:P. Giannozzi, G. Grosso andG. Pastori Parravicini:Riv. Nuovo Cimento,13, No. 3 (1990) and references quoted therein.Google Scholar
  8. [8]
    For the envelope functions in superlattices see, for instance,G. Bastard, J. A. Brum andR. Ferreira:Solid State Phys.,44, 229 (1991), edited byH. Ehrenreich andD. Turnbull (Academic, Boston, Mass.).Google Scholar
  9. [9]
    T. J. Godin andR. Haydock:Europhys. Lett.,14, 137 (1991).ADSGoogle Scholar
  10. [10]
    M. Griniasty andS. Fishman:Phys. Rev. Lett.,60, 1334 (1988).CrossRefADSGoogle Scholar
  11. [11]
    S. Das Sarma, Song He andX. C. Xie:Phys. Rev. Lett.,61, 2144 (1988);Phys. Rev. B,41, 5544 (1990).CrossRefADSGoogle Scholar
  12. [12]
    S. Aubry andG. André:Ann. Israel. Phys. Soc.,3, 133 (1979).Google Scholar
  13. [13]
    R. Farchioni, G. Grosso andG. Pastori Parravicini:Phys. Rev. B,47, 2394 (1993).CrossRefADSGoogle Scholar
  14. [14]
    D. J. Thouless:Phys. Rev. Lett.,61, 2141 (1988).CrossRefADSGoogle Scholar
  15. [15]
    D. R. Hofstadter:Phys. Rev. B,14, 2239 (1976).CrossRefADSGoogle Scholar
  16. [16]
    J. Bellisard, D. Lima andD. Testard: inMathematical Problems in Theoretical Physics, Lecture Notes in Physics, Vol.153, edited byR. Schrader, R. Seiler andD. A. Uhlenbrok (Springer-Verlag, Berlin, 1982), pp. 356–363.Google Scholar
  17. [17]
    B. Simon:Adv. Appl. Math.,3, 463 (1982) and references therein;J. Avron andB. Simon:Duke Math. J.,50, 369 (1983).MATHCrossRefGoogle Scholar
  18. [18]
    C. M. Soukoulis andE. N. Economou:Phys. Rev. Lett.,48, 1043 (1982).CrossRefADSGoogle Scholar
  19. [19]
    C. Wiecko andE. Roman:Phys. Rev. B,30, 1603 (1984).CrossRefADSGoogle Scholar
  20. [20]
    K. A. Chao, R. Riklund andY. Y. Liu:Phys. Rev. B,32, 5979 (1985).CrossRefADSGoogle Scholar
  21. [21]
    Y. Y. Liu, R. Riklund andK. A. Chao:Phys. Rev. B,32, 8387 (1985);Y. Y. Liu andK. A. Chao:Phys. Rev. B,34, 1603 (1986).CrossRefADSGoogle Scholar
  22. [22]
    J. Sun:Phys. Rev. B,40, 8270 (1989);J. Sun andC. Wang;Phys Rev. B,43, 8587 (1991).CrossRefADSGoogle Scholar
  23. [23]
    M. Johansson andR. Riklund:Phys. Rev. B,43, 13468 (1991).CrossRefADSGoogle Scholar

Copyright information

© Società Italiana di Fisica 1993

Authors and Affiliations

  • R. Farchioni
    • 1
    • 2
  • G. Grosso
    • 1
  • G. Pastori Parravicini
    • 2
  1. 1.Dipartimento di Fisica dell'Università Consorzio Interuniversitario Nazionale Fisica della MateriaGruppo Nazionale Struttura della MateriaPisaItalia
  2. 2.Dipartimento di Fisica «A. Volta» dell'Università Consorzio Interuniversitario Nazionale Fisica della MateriaGruppo Nazionale Struttura della MateriaPaviaItalia

Personalised recommendations