Advertisement

The Effect of the Surface Termination on Force Constants and Dispersion Curves

  • Giorgio Benedek
  • Jan Peter Toennies
Chapter
Part of the Springer Series in Surface Sciences book series (SSSUR, volume 63)

Abstract

The Born-von Kármán model introduced in the previous Chapter, is used to illustrate how the invariance properties of the total energy, the equilibrium conditions and the emergence of the surface stress affect the surface force constants in the surface region and the surface phonon dispersion curves. Then the effects of extended surface and interplanar force constant perturbations, which in the case of metals may involve several layers, are discussed.

References

  1. 1.
    A. A. Maradudin, E.W. Montroll, G.H. Weiss, I.P. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation, Solid State Physics, Suppl. 3 (Academic Press, New York, 1971)Google Scholar
  2. 2.
    G. Benedek, Surf. Sci. 61, 603 (1976)ADSCrossRefGoogle Scholar
  3. 3.
    L. Miglio, G. Benedek, in Structure and Dynamics of Surfaces II, ed. by W. Schommers, P. von Blackenhagen. Topics in Current Physics, vol 43 (Springer, Heidelberg, 1987)Google Scholar
  4. 4.
    G. Benedek, Phys. Status Sol. B58, 661 (1973)ADSCrossRefGoogle Scholar
  5. 5.
    G. Benedek, G.P. Brivio, L. Miglio, V.R. Velasco, Phys. Rev. B 26, 497 (1982)ADSCrossRefGoogle Scholar
  6. 6.
    G. Benedek, L. Miglio, in Surface Phonons, eds. by F.W. deWette, W. Kress, Springer Series on Surface Science, vol 27 (Springer, Berlin, Heidelberg, 1991), p. 37Google Scholar
  7. 7.
    B. Lengeler, W. Ludwig, Solid State Comm. 2, 83 (1964)ADSCrossRefGoogle Scholar
  8. 8.
    W. Kress, F.W. de Wette, A.D. Kulkarni, U. Schröder, Phys. Rev. B 35, 5783 (1987)ADSCrossRefGoogle Scholar
  9. 9.
    F.W. de Wette, A.D. Kulkarni, U. Schröder, W. Kress, Phys. Rev. B 35, 2467 (1987)ADSCrossRefGoogle Scholar
  10. 10.
    R. Shuttelworth, Proc. Phys. Soc. London A63, 444 (1950)ADSCrossRefGoogle Scholar
  11. 11.
    C. Herring, in The Physics of Powder Metallurgy, ed. by W.E. Kingston (McGraw-Hill, New York, 1951), p. 162Google Scholar
  12. 12.
    For an introductory treatment of surface stress see A. Zangwill, Physics at Surfaces (Cambridge Univ. Press, Cambridge, 1989), p. 10Google Scholar
  13. 13.
    S. Lehwald, F. Wolf, H. Ibach, B.M. Hall, D.L. Mills, Surf. Sci. 192, 131 (1987)ADSCrossRefGoogle Scholar
  14. 14.
    W. Menezes, P. Knipp, G. Tisdale, S.J. Siebener, Phys. Rev. B 41, 5648 (1990)ADSCrossRefGoogle Scholar
  15. 15.
    W. Wuttig, R. Franchy, H. Ibach, Z. Physik B 65, 71 (1986)ADSCrossRefGoogle Scholar
  16. 16.
    W. Daum, C. Stuhlmann, H. Ibach, Phys. Rev. Lett. 60, 2741 (1988)ADSCrossRefGoogle Scholar
  17. 17.
    S. Müller, P. Bayer, C. reischl, K. Heinz, B. Feldmann, H. Zillgen, W. Wuttig, Phys. Rev. Lett. 74, 765 (1995)ADSCrossRefGoogle Scholar
  18. 18.
    M. Zharnikov, A. Dittschar, W. Kuch, C.M. Schneider, J. Kirschner, Phys. Rev. Lett. 76, 4620 (1996)ADSCrossRefGoogle Scholar
  19. 19.
    G. Benedek, E. Hulpke, W. Steinhögl, Phys. Rev. Lett. 87, 027201 (2001)ADSCrossRefGoogle Scholar
  20. 20.
    G.G. Rusina, S.D. Borisova, S.V. Eremeev, IYu. Sklyadneva, E.V. Chulkov, G. Benedek, J.P. Toennies, J. Phys. Chem. C 120, 22304 (2016)CrossRefGoogle Scholar
  21. 21.
    B.J. Hinch, C. Koziol, J.P. Toennies, G. Zhang, Europhys. Lett. 10, 341 (1989)ADSCrossRefGoogle Scholar
  22. 22.
    J. Braun, J.P. Toennies, Surf. Sci. 384, L858 (1997)Google Scholar
  23. 23.
    M. Schackert, T. Märkl, J. Jandke, M. Hölzer, S. Ostanin, E.K.U. Gross, A. Ernst, W. Wulfhekel, Phys. Rev. Lett. 114, 047002 (2015)Google Scholar
  24. 24.
    J. Braun, J.P. Toennies, Surf. Sci. 368, 226 (1996)Google Scholar
  25. 25.
    G. Benedek, Physicalia Mag. 18, 205 (1996)Google Scholar
  26. 26.
    F.W. de Wette, in Dynamical Properties of Solids, ed. by G.K. Horton, A.A. Maradudin (Elsevier, Amsterdam, 1990) Chap. 5Google Scholar
  27. 27.
    W. Kress, F.W. de Wette, A.D. Kulkarni, U. Schröder, Phys. Rev. B 35, 5783 (1987)ADSCrossRefGoogle Scholar
  28. 28.
    F.W. de Wette, in Surface Phonons, ed. by F.W. de Wette, W. Kress, Springer Series on Surface Science, vol 27 (Springer, Berlin, Heidelberg, 1991), p. 67Google Scholar
  29. 29.
    G. Benedek, M. Miura, W. Kress, H. Bilz, Phys. Rev. Lett. 52, 1907 (1984)ADSCrossRefGoogle Scholar
  30. 30.
    R. Rieger, J. Prade, U. Schröder, F.E. de Wette, A.D. Kulkarni, W. Kress, Phys. Rev. B 39, 7938 (1989)ADSCrossRefGoogle Scholar
  31. 31.
    J. Prade, U. Schröder, W. Kress, A.D. Kulkarni, F.W. de Wette, in Phonons 1989, ed. by S. Hunklinger, W. Ludwig, G. Weiss (World Sci., Singapore, 1990), p. 841Google Scholar
  32. 32.
    R.J. Needs, M.J. Godfrey, M. Mansfield, Surf. Sci. 242, 215 (1991)ADSCrossRefGoogle Scholar
  33. 33.
    R.J. Needs, Phys. Rev. Lett. 58, 53 (1987)ADSCrossRefGoogle Scholar
  34. 34.
    M.C. Payne, N. Roberts, R.J. Needs, N. Needels, J.D. Joannopoulos, Surf. Sci. 211(212), 1 (1989)ADSCrossRefGoogle Scholar
  35. 35.
    R.J. Needs, M.J. Godfrey, Phys. Rev. B 42, 10933 (1990)ADSCrossRefGoogle Scholar
  36. 36.
    M. Mansfield, R.J. Needs, Phys. Rev. B 43, 8829 (1991)ADSCrossRefGoogle Scholar
  37. 37.
    P.J. Feibelman, Phys. Rev. B 56, 2175 (1997)ADSCrossRefGoogle Scholar
  38. 38.
    P.M. Marcus, X. Qian, W. Hübner, Phys. Rev. B 60, 16088 (1999)ADSCrossRefGoogle Scholar
  39. 39.
    P.M. Marcus, X. Qian, W. Hübner, J. Phys.: Condens. Matter 12, 5541 (2000)ADSGoogle Scholar
  40. 40.
    O.L. Alerhand, D. Vanderbilt, R.D. Meade, J.D. Joannopoulos, Phys. Rev. Lett. 61, 1973 (1988)ADSCrossRefGoogle Scholar
  41. 41.
    M. Mansfield, R.J. Needs, J. Phys.: Condens. Matter 2, 2361 (1990)ADSGoogle Scholar
  42. 42.
    D. Wolf, Phys. Rev. Lett. 70 (1993) 627, with comments by R.J. Needs, 71 (1993) 460 and by M.Y. Chou, S. Wei and D. Vanderbilt, 71 (1993) 461 and D. Wolf’s reply 71 (1993) 462Google Scholar
  43. 43.
    J.E. Müller, M. Wuttig, H. Ibach, Phys. Rev. Lett. 56, 1583 (1986)ADSCrossRefGoogle Scholar
  44. 44.
    H. Ibach, Surf. Sci. Rep. 29, 196 (1997)ADSCrossRefGoogle Scholar
  45. 45.
    M. Born, K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, London, New York, 1954)Google Scholar
  46. 46.
    G.W. Farnell, in Physical Acoustics, vol. 6, ed. by W.P. Mason, R.N. Thurston (Academic Press, New York, 1970), p. 109Google Scholar
  47. 47.
    J.E. Black, in Dynamical Properties of Solids, vol. 4, ed. by G.K. Horton, A.A. Maradudin (Elsevier, Amsterdam, 1990), p. 179Google Scholar
  48. 48.
    S.W. Musser, K.H. Rieder, Phys. Rev. B 2, 3034 (1970)ADSCrossRefGoogle Scholar
  49. 49.
    J. DeLaunay, Solid State Phys. 2, 219 (1956)Google Scholar
  50. 50.
    J.M. Morabito, Jr. R.F. Steiger, G.A. Somorjai, Phys. Rev. 179, 638 (1969)ADSCrossRefGoogle Scholar
  51. 51.
    M.H. Mohamed, L.L. Kesmodel, B.M. Hall, D.L. Mills, Phys. Rev. B 37, 2763 (1988)ADSCrossRefGoogle Scholar
  52. 52.
    B.M. Hall, D.L. Mills, M.H. Mohamed, L.L. Kesmodel, Phys. Rev. 38, 5856 (1988)Google Scholar
  53. 53.
    V. Bortolani, A. Franchini, F. Nizzoli, G. Santoro, Phys. Rev. Lett. 52, 429 (1984)ADSCrossRefGoogle Scholar
  54. 54.
    V. Bortolani, A. Franchini, G. Santoro, Surf. Sci. 189(190), 675 (1987)Google Scholar
  55. 55.
    V. Bortolani, G. Santoro, U. Harten, J.P. Toennies, Surf. Sci. 148, 82 (1984)Google Scholar
  56. 56.
    G. Santoro, A. Franchini, V. Bortolani, U. Harten, J.P. Toennies, Ch. Wöll, Surf. Sci. 183, 180 (1987)Google Scholar
  57. 57.
    V. Bortolani, A. Franchini, F. Nizzoli, G. Santoro, in Dynamics of Gas Surface Interactions, ed. by G. Benedek, U. Valbusa (Springer, Heidelberg, 1982), p. 196Google Scholar
  58. 58.
    G. Brusdeylins, R.B. Doak, J.P. Toennies, Phys. Rev. B 27, 3662 (1983)ADSCrossRefGoogle Scholar
  59. 59.
    R.B. Doak, U. Harten, J.P. Toennies, Phys. Rev. Lett. 51, 578 (1983)ADSCrossRefGoogle Scholar
  60. 60.
    C.S. Jayanthi, H. Bilz, W. Kress, G. Benedek, Phys. Rev. Lett. 59, 795 (1987)ADSCrossRefGoogle Scholar
  61. 61.
    C. Kaden, P. Ruggerone, J.P. Toennies, G. Zhang, G. Benedek, Phys. Rev. B 46, 13509 (1992)ADSCrossRefGoogle Scholar
  62. 62.
    S.Y. Tong, Y. Chen, K.P. Bohnen, T. Rodach, K.M. Ho, Surf. Rev. Lett. 1, 97 (1994)ADSCrossRefGoogle Scholar
  63. 63.
    G. Santoro, A. Franchini, V. Bortolani, Phys. Rev. Lett. 80, 2378 (1998)ADSCrossRefGoogle Scholar
  64. 64.
    R. Heid, K.-P. Bohnen, Phys. Rep. 387, 151–213 (2003)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Università di Milano-BicoccaMilanItaly
  2. 2.Max Planck Institute for Dynamics and Self-OrganizationGöttingenGermany
  3. 3.Donostia International Physics CenterDonostia/San SebastianSpain

Personalised recommendations