The Propagation Matrix Method for the Band Problem with a Plane Boundary

  • D. W. Jepsen
  • P. M. Marcus
Part of the The IBM Research Symposia Series book series (IRSS)


Solution of electronic problems involving plane surfaces on crystals requires solution of the band problem for real energy E in complex k space, and superposition of the generalized Bloch functions at the surface. A compact and general formulation of the problem of finding these Bloch functions and matching them across a plane makes use of a numerical matrix, the propagation matrix P, obtained from the Schrodinger equation. The eigenvectors of P are just the desired Bloch functions, and the eigenvalues give all k values at given E, k// (component parallel to the surface). Thus once P is found, the band problem is reduced to an ordinary eigenvalue problem; the bands can be followed along any line in k space parallel to k by varying k//; the potential may be complex (to describe inelastic scattering). A procedure for generating P by integration of a matrix equation has the advantage that a general anistropic potential can be used, but the disadvantage of a Fourier expansion parallel to the surface plane which does not hold well near the nucleus; hence it applies best for potentials that are weak or have a small number of Fourier coefficients. By generating P for a single layer by a two-dimensional version of KKR, this limitation is avoided for muffin-tin potentials.


Wave Function Scattered Wave Evanescent Wave Schrodinger Equation Outgoing Wave 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Beeby, J. L., 1968, “The Diffraction of Low Energy Electrons by Crystals,” J. Phys. C., Proc. Phys. Soc. 1, 82–87.Google Scholar
  2. Ham, F. S., and Segall, B., 1961, “Energy Bands in Periodic Lattices — Green’s Function Method,” Phys. Rev. 124, 1786–96.CrossRefGoogle Scholar
  3. Heine, V., 1963, “On the General Theory of Surface States and Scattering of Electrons in Solids,” Proc. Phys. Soc. 81, 300–10.CrossRefGoogle Scholar
  4. Heine, V., 1964, “Some Theory About Surface States,” Surf. Sci. 29 1–7.CrossRefGoogle Scholar
  5. James, R. W., 1950, The Optical Principles of the Diffraction of X-Rays (G. Bell and Sons, Ltd., London ), Chap. II.Google Scholar
  6. Kambe, K., 1967a, “Theory of Low-Energy Electron Diffraction I. Application of the Cellular Method to Monatomic Layers,” Z. Naturforsch. 22a, 322–30.Google Scholar
  7. Kambe, K., 1967b, “Theory of Electron Diffraction by Crystals I. Green’s Function and Integral Equation,” ibid., 22a 422–31.Google Scholar
  8. Kambe, K., 1968, “Theory of Low-Energy Electron Diffraction II. Cellular Method for Complex Monolayers and Multilayers,” ibid., 23a, 1280–94.Google Scholar
  9. Kambe, K., This Proceedings, previous paper.Google Scholar
  10. Kestner, N. R., Jortner, J., Cohen, M. H., and Rice, S. A., 1965, “Low-Energy Elastic Scattering of Electrons and Positrons from Helium Atoms,” Phys. Rev. 140, A56–66.CrossRefGoogle Scholar
  11. Korringa, J., 1947, “On the Calculation of the Energy of a Bloch Wave in a Metal,” Physica 134, 392–400.CrossRefGoogle Scholar
  12. McRae, E. G., 1968, “Electron Diffraction at Crystal Surfaces I. Generalization of Darwin’s Dynamical Theory,” Surf. Sci. 11, 479–91.CrossRefGoogle Scholar
  13. McRae, E. G., and Jennings, P. J., 1969, “Surface-State Resonances in Low-Energy Diffraction,” ibid., 345–48.Google Scholar
  14. McRae, E. G., and Jennings, P. J., 1970, “Electron Diffraction at Crystal Surfaces IV. Computation of LEED Intensities for Muffin-Tin Models with Application to Tungsten (001),” to be published in Surf. Sci.Google Scholar
  15. Morse, P. M., and Feshbach, H., 1953, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York).Google Scholar
  16. Pendry, J. B., and Forstmann, F., 1970, “Complex Band Structure in the Presence of Bound States and Resonances,” J. Phys., Proc. Phys. Soc. 34, 59–69.Google Scholar
  17. Slater, J. C., 1937, “Damped Electron Waves in Crystals,” Phys. Rev. 51, 840–846.CrossRefGoogle Scholar
  18. Williams, A. R., 1970, “Non-Muffin-Tin Energy Bands for Silicon by the Korringa-Kohn-Rostoker Method,” ibid, B1, 3417–26.Google Scholar
  19. Ziman, J. M., 1965, “The T Matrix, The K Matrix, d Bands, and ℓ-Dependent Pseudo-Potentials in the Theory of Metals,” Proc. Phys. Soc. 86, 337–53.CrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1971

Authors and Affiliations

  • D. W. Jepsen
    • 1
  • P. M. Marcus
    • 1
  1. 1.IBM Thomas J. Watson Research CenterYorktown HeightsUSA

Personalised recommendations