Advertisement

The European Physical Journal B

, Volume 59, Issue 4, pp 519–525 | Cite as

Vanishing integral relations and expectation values for Bloch functions in finite domains

  • C. PacherEmail author
  • M. Peev
Mathematical Structures in Statistical and Condensed Matter Physics

Abstract.

Integral identities for particular Bloch functions in finite periodic systems are derived. All following statements are proven for a finite domain consisting of an integer number of unit cells. It is shown that matrix elements of particular Bloch functions with respect to periodic differential operators vanish identically. The real valuedness, the time-independence and a summation property of the expectation values of periodic differential operators applied to superpositions of specific Bloch functions are derived.

PACS.

3.65.Fd Algebraic methods 03.65.Nk Scattering theory 3.65.Ge Solutions of wave equations: bound states 73.21.Cd Superlattices 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. F. Bloch, ZS. f. Phys. 52, 555 (1929) CrossRefADSGoogle Scholar
  2. C. Kittel, Quantum Theory of Solids (John Wiley & Sons, Inc., New York, 1987) Google Scholar
  3. Strictly speaking, at the band edge energies (i.e. when q∈0,π/d) the general solution [18] is either ΨB n,(q)(x)=exp (iqx)[α1 u(1) n,q(x)+α2 (xu(1) n,q(x)+u(2) n,q(x))] or ΨB n,(q)(x)=exp (iqx)[α1 u(1) n,q(x)+α2 u(2) n,q(x)], depending on the symmetry of the unit cell. Here u(1) n,q(x) and u(2) n,q(x) are two different periodic functions belonging either both to q=0 or both to q=π/d Google Scholar
  4. D.W.L. Sprung, H. Wu, J. Martorell, Am. J. Phys. 61, 1118 (1993) CrossRefADSGoogle Scholar
  5. D.W.L. Sprung, J.D. Sigetich, H. Wu, J. Martorell, Am. J. Phys. 68, 715 (2000) CrossRefADSGoogle Scholar
  6. S.Y. Ren, Phys. Rev. B 64, 035322 (2001) CrossRefADSGoogle Scholar
  7. P. Pereyra, E. Castillo, Phys. Rev. B 65, 205120 (2002) CrossRefADSGoogle Scholar
  8. S.Y. Ren, Ann. Physics 301, 22 (2002) zbMATHCrossRefADSMathSciNetGoogle Scholar
  9. C. Pacher, E. Gornik, Phys. Rev. B 68, 155319 (2003) CrossRefADSGoogle Scholar
  10. C. Pacher, W. Boxleitner, E. Gornik, Phys. Rev. B 71, 125317 (2005) CrossRefADSGoogle Scholar
  11. P. Pereyra, Ann. Physics 320, 1 (2005) zbMATHCrossRefADSMathSciNetGoogle Scholar
  12. The so-called surface states that arise in systems with CQC lie in general energetically outside the band and are characterized by an imaginary Bloch wave number. The history of the study of surface states together with new insights have been reported recently in [11] Google Scholar
  13. C. Pacher, C. Rauch, G. Strasser, E. Gornik, F. Elsholz, A. Wacker, G. Kießlich, E. Schöll, Appl. Phys. Lett. 79, 1486 (2001) CrossRefADSGoogle Scholar
  14. M. Reed, B. Simon, Analysis of Operators (Methods of Modern Mathematical Physics), Vol. IV (Academic Press, San Diego, 1978) Google Scholar
  15. L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics: Quantum Mechanics (Non-Relativistic Theory), Vol. III 3rd edn. (Pergamon Press, Oxford, 1977) Google Scholar
  16. G.L. Bir, G.E. Pikus, Symmetry and strain-induced effects in semiconductors (John Wiley & Sons, Inc., New York, 1974) Google Scholar
  17. L. Jansen, M. Boon, Theory of Finite Groups (North Holland, Amsterdam, 1967) Google Scholar
  18. M.S.P. Eastham, The Spectral Theory of Periodic Differential Equations (Scottish Academic Press Ltd., Edinburgh, 1973) Google Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Authors and Affiliations

  1. 1.Austrian Research Centers GmbH - ARC, Smart Systems DivisionWienAustria

Personalised recommendations