Skip to main content
SpringerLink
Log in

The three-dimensional equilibrium crystal shape of Pb: Recent results of theory and experiment

  • Published:
Applied Physics A Aims and scope Submit manuscript

Abstract

The three-dimensional equilibrium crystal shape (ECS) is constructed from a set of 35 orientation-dependent surface energies of fcc Pb which are calculated by density functional theory in the local-density approximation and distributed over the [110] and [001] zones of the stereographic triangle. Surface relaxation has a pronounced influence on the equilibrium shape. The (111), (100), (110), (211), (221), (411), (665), (15,1,1), (410) and (320) facets are present after relaxation of all considered surfaces, while only the low-index facets (111), (100) and (110) exist for the unrelaxed ECS. The result for the relaxed Pb crystal state is in support of the experimental ECS of Pb at 320–350 K. On the other hand, approximating the surface energies of vicinal surfaces by assuming a linear relationship between the Pb(111) first-principles surface energy and the number of broken bonds of surface atoms leads to a trivial ECS that shows only (111) and (100) facets, with a sixfold symmetric (111) facet instead of the correct threefold symmetry. It is concluded that the broken bond rule in this simple linear form is not a suitable approximation for obtaining the proper three-dimensional ECS and correct step formation energies.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E.D. Hondros, Proc. R. Soc. London A 286, 479 (1965)

    Article  ADS  Google Scholar 

  2. E.D. Hondros, M. McLean, “Structure et Propriétés des Surfaces des Solides”, presented at Colloques Intern. du CNRS, Paris (1969)

  3. E.D. Hondros, Surface Energy Measurements, in Physicochemical Measurements in Metals Research, Techniques of Metals Research, ed. by R.A. Rapp (Interscience, New York, 1970), vol. IV, part 2, pp. 293–348

  4. D. Sander, H. Ibach, Surface Free Energy and Surface Stress, in Physics of Covered Solid Surfaces,Vol. III-42-A2, (Landolt-Börnstein), ed. by H.P. Bonzel, New Series N Edn. (Springer, Berlin, 2002), vol. III-42-A2, pp. 4.4-1–49

  5. A.R. Miedema, Z. Metallkd. 69, 287 (1978)

    Google Scholar 

  6. A.R. Miedema, R. Boom, Z. Metallkd. 69, 183 (1978)

    Google Scholar 

  7. I. Galanakis, N. Papanikolaou, P.H. Dederichs, Surf. Sci. 511, 1 (2002)

    Article  ADS  Google Scholar 

  8. M. Mansfield, R.J. Needs, Phys. Rev. B 43, 8829 (1991)

    Article  ADS  Google Scholar 

  9. M. Methfessel, D. Henning, M. Scheffler, Appl. Phys. A 55, 442 (1992)

    Article  ADS  Google Scholar 

  10. M. Methfessel, D. Henning, M. Scheffler, Phys. Rev. B 46, 4816 (1992)

    Article  ADS  Google Scholar 

  11. H.L. Skriver, N.M. Rosengaard, Phys. Rev. B 46, 7157 (1992)

    Article  ADS  Google Scholar 

  12. P. Feibelman, Phys. Rev. B 52, 16845 (1995)

    Article  ADS  Google Scholar 

  13. L. Vitos, A.V. Ruban, H.L. Skriver, J. Kollár, Surf. Sci. 411, 186 (1998)

    Article  ADS  Google Scholar 

  14. P. Feibelman, Phys. Rev. B 62, 17020 (2000)

    Article  ADS  Google Scholar 

  15. I. Galanakis, G. Bihlmayer, V. Bellini, N. Papanikolaou, R. Zeller, S. Blügel, P.H. Dederichs, Eur. Phys. Lett. 58, 751 (2002)

    Article  ADS  Google Scholar 

  16. D. Yu, M. Scheffler, Phys. Rev. B 70, 155417 (2004)

    Article  ADS  Google Scholar 

  17. J.L.F. Da Silva, C. Barreteau, K. Schroeder, S. Blügel, Phys. Rev. B 73, 125402 (2006)

    Article  ADS  Google Scholar 

  18. D. Yu, H.P. Bonzel, M. Scheffler, Phys. Rev. B 74, 115408 (2006)

    Article  ADS  Google Scholar 

  19. C. Bombis, A. Emundts, M. Nowicki, H.P. Bonzel, Surf. Sci. 511, 83 (2002)

    Article  ADS  Google Scholar 

  20. J.L.F. Da Silva, C. Stampfl, M. Scheffler, Surf. Sci. 600, 703 (2006)

    Article  ADS  Google Scholar 

  21. I.N. Stranski, Z. Phys. Chem. B 11, 342 (1931)

    Google Scholar 

  22. I.N. Stranski, R. Kaischew, Z. Kristallogr. 78, 373 (1931)

    Google Scholar 

  23. I.N. Stranski, R. Kaischew, Ann. Phys. 23, 330 (1935)

    Article  Google Scholar 

  24. I.N. Stranski, R. Suhrmann, Z. Kristallogr. 105, 481 (1944)

    Google Scholar 

  25. I.N. Stranski, R. Suhrmann, Ann. Phys. 1, 153 (1947)

    Article  Google Scholar 

  26. I.N. Stranski, Discuss. Faraday Soc. 5, 13 (1949)

    Article  Google Scholar 

  27. C. Herring, Phys. Rev. 82, 87 (1951)

    Article  MATH  ADS  Google Scholar 

  28. C. Herring, The Use of Classical Macroscopic Concepts in Surface Energy Problems, in Structure and Properties of Solid Surfaces, ed. by R. Gomer, C.S. Smith (University of Chicago, Chicago, 1952), p. 5

  29. J. Friedel, B.D. Cullity, C. Crussard, Acta Metall. 1, 79 (1953)

    Article  Google Scholar 

  30. J.K. MacKenzie, A.J.W. Moore, J.F. Nicholas, J. Phys. Chem. Solids 23, 185 (1962)

    Article  ADS  Google Scholar 

  31. I.N. Stranski, Z. Phys. Chem. 136, 259 (1928)

    Google Scholar 

  32. W.K. Burton, N. Cabrera, Discuss. Faraday Soc. 5, 33 (1949)

    Article  Google Scholar 

  33. W.K. Burton, N. Cabrera, F.C. Frank, Philos. Trans. R. Soc. London 243A, 299 (1951)

    Google Scholar 

  34. E.F. Gruber, W.W. Mullins, J. Phys. Chem. Solids 28, 875 (1967)

    Article  ADS  Google Scholar 

  35. W.W. Mullins, Solid Surface Morphologies Governed by Capillarity, in Metal Surfaces: Structure, Energetics and Kinetics, ed. by W.D. Robertson, N.A. Gjostein (American Soc. for Metals, Metals Park, Ohio, USA, 1963), pp. 17–66

  36. N. Cabrera, Surf. Sci. 2, 320 (1964)

    Article  ADS  Google Scholar 

  37. C. Jayaprakash, C. Rottman, W.F. Saam, Phys. Rev. B 30, 6549 (1984)

    Article  ADS  Google Scholar 

  38. C. Jayaprakash, W.F. Saam, Phys. Rev. B 30, 3916 (1984)

    Article  ADS  Google Scholar 

  39. N. Garcia, P.A. Serena, Surf. Sci. 330, L665 (1995)

    Article  ADS  Google Scholar 

  40. J.J. Sáenz, N. García, Surf. Sci. 155, 24 (1985)

    Article  Google Scholar 

  41. C. Rottman, M. Wortis, J.C. Heyraud, J.J. Métois, Phys. Rev. Lett. 52, 1009 (1984)

    Article  ADS  Google Scholar 

  42. J.J. Métois, J.C. Heyraud, Surf. Sci. 180, 647 (1987)

    Article  Google Scholar 

  43. M. Nowicki, C. Bombis, A. Emundts, H.P. Bonzel, P. Wynblatt, New J. Phys. 4, 60 (2002)

    Article  ADS  Google Scholar 

  44. H.P. Bonzel, Phys. Rep. 385, 1 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  45. G. Wulff, Z. Kristallogr. 34, 449 (1901)

    Google Scholar 

  46. A. Dinghas, Z. Kristallogr. 105, 304 (1944)

    MathSciNet  MATH  Google Scholar 

  47. L.D. Landau, E.M. Lifshitz, Statistical Physics (Addison-Wesley, Reading, 1958), vol. V, p. 460

  48. A.R. Roosen, R. McCormack, W.C. Carter, Wulff Construction Software: http://www.ctcms.nist.gov/wulffman/, National Institute of Standards and Technology (1999)

  49. J.C. Heyraud, J.J. Métois, J. Cryst. Growth 50, 571 (1980)

    Article  ADS  Google Scholar 

  50. J.C. Heyraud, J.J. Métois, Acta Metall. 28, 1789 (1980)

    Article  Google Scholar 

  51. J.C. Heyraud, J.J. Métois, Surf. Sci. 128, 334 (1983)

    Article  ADS  Google Scholar 

  52. J.J. Métois, G.D.T. Spiller, J.A. Venables, Philos. Mag. A 46, 1015 (1982)

    ADS  Google Scholar 

  53. H.P. Bonzel, A. Emundts, Phys. Rev. Lett. 84, 5804 (2000)

    Article  ADS  Google Scholar 

  54. M. Wortis, in Chemistry and Physics of Solid Surfaces, ed. by R. Vanselow, R. Howe (Springer, New York, 1988), vol. 7, p. 367–405

  55. D. Yu, H.P. Bonzel, M. Scheffler, New J. Phys. 8, 65 (2006)

    Article  ADS  Google Scholar 

  56. P. Feibelman, Phys. Rev. B 65, 129902 (2002)

    Article  ADS  Google Scholar 

  57. C.M. Wei, M.Y. Chou, Phys. Rev. B 66, 233408 (2002)

    Article  ADS  Google Scholar 

  58. M. Fuchs, M. Scheffler, Comput. Phys. Commun. 119, 67 (1999)

    Article  MATH  ADS  Google Scholar 

  59. S. Wei, M.Y. Chou, Phys. Rev. B 50, 4859 (1994)

    Article  ADS  Google Scholar 

  60. C. Rottman, M. Wortis, Phys. Rev. B 29, 328 (1984)

    Article  ADS  Google Scholar 

  61. R. Smoluchowski, Phys. Rev. 60, 661 (1941)

    Article  MATH  ADS  Google Scholar 

  62. M.W. Finnis, V. Heine, J. Phys. F Met. Phys. 4, L37 (1974)

    Article  ADS  Google Scholar 

  63. M.C. Desjonquères, D. Spanjaard, C. Barreteau, F. Raouafi, Phys. Rev. Lett. 88, 056104 (2002)

    Article  ADS  Google Scholar 

  64. M. Nowicki, A. Emundts, H.P. Bonzel, Prog. Surf. Sci. 74, 123 (2003)

    Article  ADS  Google Scholar 

  65. J.J. Métois, J.C. Heyraud, J. Cryst. Growth 57, 487 (1982)

    Article  Google Scholar 

  66. G. Rao, D.B. Zhang, P. Wynblatt, Acta Metall. Mater. 41, 3331 (1993)

    Article  Google Scholar 

  67. A. Emundts, P. Coenen, G. Pirug, B. Voigtländer, H.P. Bonzel, P. Wynblatt, Rev. Sci. Instrum. 72, 3546 (2001)

    Article  ADS  Google Scholar 

  68. S. Surnev, P. Coenen, B. Voigtländer, H.P. Bonzel, P. Wynblatt, Phys. Rev. B 56, 12131 (1997)

    Article  ADS  Google Scholar 

  69. S. Surnev, K. Arenhold, P. Coenen, B. Voigtländer, H.P. Bonzel, P. Wynblatt, J. Vac. Sci. Technol. A 16, 1059 (1998)

    Article  ADS  Google Scholar 

  70. K. Arenhold, S. Surnev, H.P. Bonzel, P. Wynblatt, Surf. Sci. 424, 271 (1999)

    Article  ADS  Google Scholar 

  71. H.P. Bonzel, Prog. Surf. Sci. 67, 45 (2001)

    Article  ADS  Google Scholar 

  72. H.P. Bonzel, Interface Sci. 9, 21 (2001)

    Article  Google Scholar 

  73. K. Thürmer, J.E. Reutt-Robey, E.D. Williams, M. Uwaha, A. Emundts, H.P. Bonzel, Phys. Rev. Lett. 87, 186102 (2001)

    Article  ADS  Google Scholar 

  74. A. Emundts, M. Nowicki, H.P. Bonzel, Surf. Sci. 496, L35L42 (2002)

  75. M. Nowicki, C. Bombis, A. Emundts, H.P. Bonzel, P. Wynblatt, Eur. Phys. Lett. 59, 239 (2002)

    Article  ADS  Google Scholar 

  76. M. Nowicki, C. Bombis, A. Emundts, H.P. Bonzel, Phys. Rev. B 67, 075405 (2003)

    Article  ADS  Google Scholar 

  77. H.P. Bonzel, M. Nowicki, Phys. Rev. B 70, 245430 (2004)

    Article  ADS  Google Scholar 

  78. W.W. Mullins, G.S. Rohrer, J. Am. Ceram. Soc. 83, 214 (2000)

    Article  Google Scholar 

  79. G.S. Rohrer, C.L. Rohrer, W.W. Mullins, J. Am. Ceram. Soc. 84, 2099 (2001)

    Article  Google Scholar 

  80. J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Flolhais, Phys. Rev. B 46, 6671 (1992)

    Article  ADS  Google Scholar 

  81. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996)

    Article  ADS  Google Scholar 

  82. V.N. Staroverov, G.E. Scuseria, J. Tao, J.P. Perdew, Phys. Rev. B 69, 075102 (2004)

    Article  ADS  Google Scholar 

  83. V.K. Kumikov, K.B. Khokonov, J. Appl. Phys. 54, 1346 (1983)

    Article  ADS  Google Scholar 

  84. D. Chatain, J.J. Métois, Surf. Sci. 291, 1 (1993)

    Article  ADS  Google Scholar 

  85. D. Alfe, M.J. Gillan, J. Phys. C Condens. Matter 18, L435 (2006)

    Article  ADS  Google Scholar 

  86. N. Akutsu, Y. Akutsu, J. Phys. C Condens. Matter 11, 6635 (1999)

    Article  ADS  Google Scholar 

  87. C. Steimer, M. Giesen, L. Verheij, H. Ibach, Phys. Rev. B 64, 085416 (2001)

    Article  ADS  Google Scholar 

  88. D.C. Schlösser, L.K. Verheij, G. Rosenfeld, G. Comsa, Phys. Rev. Lett. 82, 3843 (1999)

    Article  ADS  Google Scholar 

  89. G. Schulze Icking-Konert, M. Giesen, H. Ibach, Phys. Rev. Lett. 83, 3880 (1999)

    Article  ADS  Google Scholar 

  90. P. Feibelman, Phys. Rev. B 60, 11118 (1999)

    Article  ADS  Google Scholar 

  91. A. Pavlovska, K. Faulian, E. Bauer, Surf. Sci. 221, 233 (1989)

    Article  ADS  Google Scholar 

  92. A. Pavlovska, D. Dobrev, E. Bauer, Surf. Sci. 326, 101 (1995)

    Article  ADS  Google Scholar 

  93. U. Gangopadhyay, P. Wynblatt, Met. Mater. Trans. A 25, 607 (1994)

    Google Scholar 

  94. J.C. Heyraud, J.J. Métois, J. Cryst. Growth 82, 269 (1987)

    Article  ADS  Google Scholar 

  95. L. Kuipers, J.W.M. Frenken, Phys. Rev. Lett. 70, 3907 (1993)

    Article  ADS  Google Scholar 

  96. L. Kuipers, M.S. Hoogeman, J.W.M. Frenken, Surf. Sci. 340, 231 (1995)

    Article  ADS  Google Scholar 

  97. K. Thürmer, J.E. Reutt-Robey, E.D. Williams, Surf. Sci. 537, 123 (2003)

    Article  ADS  Google Scholar 

  98. M. Ranganathan, D.B. Dougherty, W.G. Cullen, T. Zhao, J.D. Weeks, E.D. Williams, Phys. Rev. Lett. 95, 225505 (2005)

    Article  ADS  Google Scholar 

  99. M. Degawa, W. Cullen, A. Pimpinelli, T.L. Einstein, E.D. Williams, Phys. Rev. Lett. 97, 080601 (2006)

    Article  ADS  Google Scholar 

  100. N. Moll, A. Kley, E. Pehlke, M. Scheffler, Phys. Rev. B 54, 8844 (1996)

    Article  ADS  Google Scholar 

  101. Q.K.K. Liu, N. Moll, M. Scheffler, E. Pehlke, Phys. Rev. B 60, 17008 (1999)

    Article  ADS  Google Scholar 

  102. E. Pehlke, N. Moll, A. Kley, M. Scheffler, Appl. Phys. A 65, 525 (1997)

    Article  ADS  Google Scholar 

  103. N. Moll, M. Scheffler, E. Pehlke, Phys. Rev. B 58, 4566 (1998)

    Article  ADS  Google Scholar 

  104. M. Degawa, F. Szalma, E.D. Williams, Surf. Sci. 583, 126 (2005)

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Bio- und Nanosysteme, IBN-3, Forschungszentrum Jülich, 52425, Jülich, Germany

    H.P. Bonzel

  2. Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4–6, 14195, Berlin, Germany

    D.K. Yu & M. Scheffler

Authors
  1. H.P. Bonzel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D.K. Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Scheffler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H.P. Bonzel.

Additional information

PACS

05.70.Np; 61.50.Jr; 68.35.Md; 71.15.Mb

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bonzel, H., Yu, D. & Scheffler, M. The three-dimensional equilibrium crystal shape of Pb: Recent results of theory and experiment. Appl. Phys. A 87, 391–397 (2007). https://doi.org/10.1007/s00339-007-3951-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00339-007-3951-7

Keywords

Search

Navigation