Advertisement

Interatomic Forces and Bond Energies in the Tight Binding Approximation

  • A. T. Paxton

Abstract

One might well take the view, that an interatomic potential for atomistic modeling of materials will only be useful if it can correctly reproduce the phase stability of the particular material it is supposed to describe. Yet a glance at the literature will show that prescriptions used for cohesive energies and interatomic forces are rarely tested to destruction in respect of crystal structure prediction. This is particularly regretable, if the potential in question is applied to a study of crystal defects, because of the close connection between competing crystal structures and the energies of defects. A good example of this is the relationship that exists in fcc materials between the intrinsic stacking fault energy, and the difference in energy between hcp and fcc phases. In favorable cases, these are both measurable quantities, and may therefore provide excellent guidance in the choice and testing of interatomic potentials. The present article concentrates on the semi-empirical tight-binding method, which may be thought of as a half-way stage between classical models such as pair potentials or Keating models, and first-principles total energy methods. In the next section, the development of the theory is described, emphasizing its inception as a way of understanding the crystal structures adopted by transition metals. The following two sections focus on the tight-binding theory of Si, describing recent applications of the method to the calculation of bond energies and interatomic forces.

Keywords

Cohesive Energy Pair Potential Maximum Entropy Method Canonical Model Recursion Method 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H.J. Jones, Proc. Phys. Soc. London 49:250 (1937).CrossRefGoogle Scholar
  2. 2.
    S.L. Altmann, C.A. Coulson and W. Hume-Rothery, Proc. Roy. Soc. London, A240:145 (1957).Google Scholar
  3. 3.
    J. Friedel, Trans. AIME, 230:616 (1964).Google Scholar
  4. 4.
    J.C. Slater and G.F. Koster, Phys. Rev. 94:1498 (1954).CrossRefGoogle Scholar
  5. 5.
    F. Cyrot-Lackman, J. Phys. Chem. Solids, 29:1235 (1968).CrossRefGoogle Scholar
  6. 6.
    F. Ducastelle and F. Cyrot-Lackmann, J. Phys. Chem. Solids, 32:285 (1971).CrossRefGoogle Scholar
  7. 7.
    V. Heine, Solid State Physics, 35:1 (1980).CrossRefGoogle Scholar
  8. 8.
    J.P. Gaspard and F. Cyrot-Lackmann, J. Phys. C, 6:3077 (1973).CrossRefGoogle Scholar
  9. 9.
    R. Haydock, V. Heine and M.J. Kelly, J. Phys. C, 5:2845 (1972).CrossRefGoogle Scholar
  10. 10.
    C.C. Paige, J. Inst. Maths Applics 10:373 (1972).CrossRefGoogle Scholar
  11. 11.
    C.M.M. Nex, J. Phys. A, 11:653 (1978).CrossRefGoogle Scholar
  12. 12.
    L.R. Mead and N. Papanicolaou, J. Math. Phys. 25:2404 (1984).CrossRefGoogle Scholar
  13. 13.
    S. Glanville, A.T. Paxton and M.W. Finnis, J. Phys. F 18:693 (1988).CrossRefGoogle Scholar
  14. 14.
    R. Haydock, Solid State Phys 35:215 (1980).CrossRefGoogle Scholar
  15. 15.
    D.G. Pettifor, J. Phys. C 2:1051 (1969).CrossRefGoogle Scholar
  16. 16.
    O.K. Andersen, Solid State Commun 13:133 (1973).CrossRefGoogle Scholar
  17. 17.
    J. Callaway, “Energy Band Theory”, Academic Press, New York (1964).Google Scholar
  18. 18.
    O.K. Andersen, O. Jepsen and D. Glötzel, in: “Highlights of Condensed Matter Theory”, F. Bassani et al. eds., North Holland, Amsterdam (1985).Google Scholar
  19. 19.
    D.G. Pettifor, J. Phys. F 7:613 (1977).CrossRefGoogle Scholar
  20. 20.
    H.L. Skriver, Phys. Rev B 31:1909 (1985).CrossRefGoogle Scholar
  21. 21.
    D.G. Pettifor,, J. Phys. C 3:367 (1970).CrossRefGoogle Scholar
  22. 22.
    B. Legrand, G. Treglia, M.C. Desjonquères and D. Spanjaard, J. Phys. C 19:4463 (1986).CrossRefGoogle Scholar
  23. 23.
    D.G. Pettifor, Commun. Phys. 1:141 (1976); J. Chem. Phys. 69:2930 (1978).Google Scholar
  24. 24.
    O.K. Andersen, H.L. Skriver, H. Nohl and B. Johansson, Pure Appl. Chem. 52:93 (1979).CrossRefGoogle Scholar
  25. 25.
    L. Kaufman, in “Phase Stability in Metals and Alloys”, P.S. Rudman et al. eds. McGraw-Hill, New York (1967); A.R. Miedema and A.K. Niessen, CALPHAD 7:27 (1983).Google Scholar
  26. 26.
    A. Pasturel, D. Nguyen Manh and D. Mayou, J. Phys. Chem. Solids 47:325 (1986).CrossRefGoogle Scholar
  27. 27.
    D. Nguyen Manh, D. Mayou, F. Cyrot-Lackmann and A. Pasturel, J. Phys. F 17:1309 (1987).CrossRefGoogle Scholar
  28. 28.
    B. Legrand, Phil. Mag. A, 52:83 (1985).CrossRefGoogle Scholar
  29. 29.
    M.W. Finnis, J. Phys. F 4:1645 (1974).CrossRefGoogle Scholar
  30. 30.
    D.G. Pettifor and M.A. Ward, Solid State Commun. 49:291 (1984).CrossRefGoogle Scholar
  31. 31.
    D.J. Chadi and M.L. Cohen, Phys. Stat. Sol (b) 68:405 (1975).CrossRefGoogle Scholar
  32. 32.
    Q. Gou-Xin and D.J. Chadi, Phys. Rev. B 35:1288 (1987).CrossRefGoogle Scholar
  33. 33.
    W.A. Harrison, “Electronic Structure”, W.H. Freeman, San Francisco (1980).Google Scholar
  34. 34.
    G. Allan and M. Lannoo, J. Physique 44:1355 (1983).CrossRefGoogle Scholar
  35. 35.
    A.T. Paxton, DPhil Thesis, University of Oxford (1987).Google Scholar
  36. 36.
    A.T. Paxton, A.P. Sutton and C.M.M. Nex, J. Phys. C 20:L263 (1987).CrossRefGoogle Scholar
  37. 37.
    O.K. Andersen and O. Jepsen, Phys. Rev. Lett. 53:2571 (1984).CrossRefGoogle Scholar
  38. 38.
    D.J. Chadi, Phys. Rev. Lett 41:1062 (1978).CrossRefGoogle Scholar
  39. 39.
    D.J. Chadi, Phys. Rev. B 29:785 (1984).CrossRefGoogle Scholar
  40. 40.
    W.A. Harrison, Phys. Rev. B 27:3592 (1983).CrossRefGoogle Scholar
  41. 41.
    M. van Schilfgaarde and A. Sher, Phys. Rev. B 36:4375 (1987).CrossRefGoogle Scholar
  42. 42.
    J.A. Majewski and P. Vogl, Phys. Rev. Lett. 57:1366 (1986).CrossRefGoogle Scholar
  43. 43.
    A.P. Sutton, M.W. Finnis, D.G. Pettifor and Y. Ohta, J. Phys. C 21:35 (1988).CrossRefGoogle Scholar
  44. 44.
    M.T. Yin and M.L. Cohen, Phys. Rev. B 26:5668 (1982).CrossRefGoogle Scholar
  45. 45.
    A.T. Paxton and A.P. Sutton, J. Phys. C 21:L481 (1988).CrossRefGoogle Scholar
  46. 46.
    M. Kohyama, R. Yamamoto, Y. Ebata and M. Kinoshita, J. Phys. C 21:3205 (1988).CrossRefGoogle Scholar
  47. 47.
    M. Kohyama, R. Yamamoto, Y. Watanabe, Y. Ebata and M. Kinoshita, J. Phys. C 21:L695 (1988).CrossRefGoogle Scholar
  48. 48.
    A.P. Sutton, J. Physique Coll. 46:C4–347 (1985).Google Scholar
  49. 49.
    C.A. Coulson, Proc. Roy. Soc. London A169:413 (1939).Google Scholar
  50. 50.
    R. Jones and M.W. Lewis, Phil. Mag. B 49:95 (1984).Google Scholar
  51. 51.
    J. Inoue and Y. Ohta, J. Phys. C 20:1947 (1987).CrossRefGoogle Scholar
  52. 52.
    F.V. Atkinson, “Discrete and Continuous Boundary Problems”, Academic Press, New York (1966).Google Scholar
  53. 53.
    M.G. Krein,, Dokl. Akad. Nauk SSSR 69:125 (1949).Google Scholar
  54. 54.
    C.M.M. Nex, paper presented at MSI workshop, “Practical Iterative Methods for Large-Scale Computations” (1988).Google Scholar
  55. 55.
    I. Turek, J. Phys. C 21:3251 (1988).CrossRefGoogle Scholar
  56. 56.
    A.T. Paxton, Phil. Mag. B 58:603 (1988).Google Scholar
  57. 57.
    P. Pirouz, R. Chaim and J. Samuels, Proc. 5th Int. Symp. “Structure and Properties of Dislocations in Semiconductors”, Moscow (1986). To appear in Izv. Akad. Nauk Fiz. Ser.Google Scholar
  58. 58.
    J.L. Demenet, J. Rabier and H. Garem, “Inst. Phys. Conf. Ser. No. 87”, Adam Hilger, Bristol (1987).Google Scholar
  59. 59.
    A. Bourret, “Inst. Phys. Conf. Ser. No. 87”, Adam Hilger, Bristol (1987).Google Scholar
  60. 60.
    J.P. Hirth and H. Lothe, “Theory of Dislocations”, McGraw-Hill, New York (1982).Google Scholar
  61. 61.
    T.Y. Tan, H. Föll and S.M. Hu, Phil. Mag. A 44:127 (1981).CrossRefGoogle Scholar
  62. 62.
    S.G. Louie, J. Physique Coll 46:C4–335 (1985).Google Scholar
  63. 63.
    V.G. Eremenko and V.I. Nikitenko, Phys. Stat. Sol (a) 14:317 (1972).CrossRefGoogle Scholar
  64. 64.
    A. Bourret, L. Billard and M. Petit “Inst. Phys. Conf. Ser. No. 76”, Adam Hilger, Bristol (1985).Google Scholar
  65. 65.
    F. Komminou, Th. Karakostas, G.L. Bleris and N.A. Economou, J. Physique Coll. 43:C1–C9 (1982).Google Scholar
  66. 66.
    J.A. Kohn, Amer. Mineral. 43:263 (1958).Google Scholar
  67. 67.
    J. Hornstra, Physica 25:409 (1959).CrossRefGoogle Scholar
  68. 68.
    H.J. Möller, Phil. Mag. A 43:1045 (1981).CrossRefGoogle Scholar
  69. 69.
    C. Fontaine and D.A. Smith, Appl. Phys. Lett 40:153 (1982).CrossRefGoogle Scholar
  70. 70.
    D.S. Vlachavas and R.C. Pond, “Inst. Phys. Conf. Ser. No. 60”, Adam Hilger Bristol (1981).Google Scholar
  71. 71.
    R.C. Pond, J. Physique Coll. 43:C1–51 (1982).Google Scholar
  72. 72.
    R.C. Pond, D.J. Bacon and A.M. Bastaweesy, “Inst. Phys. Conf. Ser. No. 67”, Adam Hilger, Bristol (1983).Google Scholar
  73. 73.
    A. Bourret and J.J. Bacmann, “Grain Boundary Structure and Related Phenomena”, Proc. JIMIS-4 (1986).Google Scholar
  74. 74.
    A.M. Papon and M. Petit, Scripta Metall. 19:391 (1985).CrossRefGoogle Scholar
  75. 75.
    A. Bourret and J.J. Bacmann, Surf. Sci. 162:495 (1985).CrossRefGoogle Scholar
  76. 76.
    R.C. Pond and D.S. Vlachavas, Proc. Roy. Soc. London A386:95 (1983).Google Scholar
  77. 77.
    A. Rocher and M. Labidi, Revue Phys. Appl. 21:201 (1986).CrossRefGoogle Scholar
  78. 78.
    P. Ruterana, A. Bary and G. Nouet, J. Physique Coll. 43:C1–27 (1982).Google Scholar
  79. 79.
    R. Sharko, A. Gervais and C. Texier-Hervo, J. Physique Coll. 43:C1–129 (1982).Google Scholar
  80. 80.
    A. Mauger, J.C. Bourgouin, G. Allan, M. Lannoo, A. Bourret and L. Billard, Phys. Rev. B 35:1267 (1986).CrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1989

Authors and Affiliations

  • A. T. Paxton
    • 1
  1. 1.Max-Planck-Institut für FestkörperforschungStuttgart-80West Germany

Personalised recommendations