Skip to main content

Introduction to Step Dynamics and Step Instabilities

  • Conference paper

Part of the book series: ISNM International Series of Numerical Mathematics ((ISNM,volume 149))

Abstract

This paper provides an elementary introduction to the basic concepts used in describing epitaxial crystal growth in terms of the thermodynamics and kinetics of atomic steps. Selected applications to morphological instabilities of stepped surfaces are reviewed, and some open problems are outlined.

This work was supported in part by DFG within SFB 237 Unordnung und grosse Fluktuationen and SFB 616 Energiedissipation an Oberflächen.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. W.K. Burton, N. Cabrera, F.C. Frank, The Growth of Crystals and the Equilibrium Structure of their Surfaces. Phil. Trans. Roy. Soc. A 243 (1951), 299–358.

    Google Scholar 

  2. A. Pimpinelli, J. Villain, Physics of Crystal Growth. Cambridge University Press, 1998

    Google Scholar 

  3. P. Politi, G. Grenet, A. Marty, A. Ponchet, J. Villain, Instabilities in Crystal Growth by Atomic or Molecular Beams. Phys. Rep. 324 (2000), 271–404.

    Article  Google Scholar 

  4. T. Michely, J. Krug, Islands, Mounds and Atoms. Patterns and Processes in Crystal Growth Far from Equilibrium. Springer, Heidelberg 2004.

    Google Scholar 

  5. H. van Beijeren, I. Nolden, The Roughening Transition. In W. Schommers, P. von Blanckenhagen (Eds.), Structure and Dynamics of Surfaces II. (Springer, Heidelberg 1987), pp. 259–300.

    Google Scholar 

  6. P. Nozières, Shape and Growth of Crystals. In C. Godrèche (Ed.), Solids Far from Equilibrium (Cambridge University Press, 1991), pp. 1–154.

    Google Scholar 

  7. H.-C. Jeong, E.D. Williams, Steps on Surfaces: Experiment and Theory. Surf. Sci. Rep. 34 (1999), 171–294.

    Article  Google Scholar 

  8. M. Giesen, Step and island dynamics at solid/vacuum and solid/liquid interfaces. Prog. Surf. Sci. 68 (2001), 1–153.

    Article  Google Scholar 

  9. H.P. Bonzel, 3D Equilibrium Crystal Shapes in the New Light of STM and AFM. Phys. Rep. 385 (2003), 1–67.

    MathSciNet  Google Scholar 

  10. E.E. Gruber, W.W. Mullins, On the Theory of Anisotropy of Crystalline Surface Tension. J. Phys. Chem. Solids 28 (1967), 875–887.

    Article  Google Scholar 

  11. B. Joós, T.L. Einstein, N.C. Bartelt, Distribution of terrace widths on a vicinal surface within the one-dimensional free-fermion model. Phys. Rev. B 43 (1991), 8153–8162.

    Article  Google Scholar 

  12. S. Stoyanov, Heating Current Induced Conversion between 2×1 and 1×2 Domains at Vicinal (001) Si Surfaces — Can it be Explained by Electromigration of Si Adatoms? Jap. J. Appl. Phys. 29 (1990), L659–L662.

    Article  Google Scholar 

  13. K. Yagi, H. Minoda, M. Degawa, Step bunching, step wandering and faceting: self-organization at Si surfaces. Surf. Sci. Repts. 43 (2001) 45–126.

    Article  Google Scholar 

  14. H. Minoda, Direct current heating effects on Si(111) vicinal surfaces. J. Phys.-Cond. Matter 15 (2003), S3255–S3280.

    Article  Google Scholar 

  15. R. Ghez, S.S. Iyer, The Kinetics of Fast Steps on Crystal Surfaces and its Application to the Molecular Beam Epitaxy of Silicon. IBM J. Res. Dev. 32, 804 (1988)

    Google Scholar 

  16. O. Pierre-Louis, Step bunching with general kinetics: stability analysis and macroscopic models. Surf. Sci. 529 (2003), 114–134.

    Article  Google Scholar 

  17. C. Roland, G.H. Gilmer, Epitaxy on surfaces vicinal to Si(001). II. Growth properties of Si(001) steps. Phys. Rev. B 46 (1992) 13437–13451.

    Article  Google Scholar 

  18. R.E. Caflisch, W. E, M.F. Gyure, B. Merriman, C. Ratsch, Kinetic model for a step edge in epitaxial growth. Phys. Rev. B 59 (1999), 6879–6887.

    Google Scholar 

  19. S.N. Filimonov, Yu.Yu. Hervieu, Terrace-edge-kink model of atomic processes at the permeable steps. Surf. Sci. 553 (2004), 133–144.

    Article  Google Scholar 

  20. V.V. Voronkov, The movement of an elementary step by means of the formation of one-dimensional nuclei. Sov. Phys. Crystallogr. 15 (1970), 8–13.

    Google Scholar 

  21. M.C. Bartelt, J.W. Evans, Scaling analysis of diffusion-mediated island growth in surface adsorption processes. Phys. Rev. B 46 (1992), 12675–12687.

    Article  Google Scholar 

  22. J. Villain, A. Pimpinelli, L. Tang, D. Wolf, Terrace sizes in molecular beam epitaxy. J. Phys. I France 2 (1992), 2107–2121.

    Article  Google Scholar 

  23. J. Kallunki, J. Krug, Competing mechanisms for step meandering in unstable growth. Phys. Rev. B 65 (2002), 205411.

    Article  Google Scholar 

  24. O. Pierre-Louis, M.R. D’Orsogna, T.L. Einstein, Edge Diffusion during Growth: The Kink Ehrlich-Schwoebel Effect and Resulting Instabilities. Phys. Rev. Lett. 82 (1999), 3661–3664.

    Article  Google Scholar 

  25. M.V. Ramana Murty, B.H. Cooper, Instability in Molecular Beam Epitaxy due to Fast Edge Diffusion and Corner Diffusion Barriers. Phys. Rev. Lett 83 (1999), 352–355.

    Article  Google Scholar 

  26. J. Kallunki, J. Krug, Effect of kink-rounding barriers on step edge fluctuations. Surf. Sci. 523 (2003) L53–L58.

    Article  Google Scholar 

  27. J. Krug, H.T. Dobbs, S. Majaniemi, Adatom mobility for the solid-on-solid model. Z. Phys. B 97 (1995), 281–291.

    Article  Google Scholar 

  28. P. Politi, J. Krug, Crystal symmetry, step-edge diffusion, and unstable growth. Surf. Sci. 446 (2000), 89–97.

    Article  Google Scholar 

  29. M. Rusanen, I.T. Koponen, T. Ala-Nissila, C. Ghosh, T.S. Rahman, Morphology of ledge patterns during step flow growth of metal surfaces vicinal to fcc(001). Phys. Rev. B 65 (2002), 041404.

    Article  Google Scholar 

  30. T. Zhao, J.D. Weeks, D. Kandel, A unified treatment of current-induced instabilities on Si surfaces. (preprint, cond-mat/0403488).

    Google Scholar 

  31. R.L. Schwoebel, Step Motion on Crystal Surfaces. II. J. Appl. Phys. 40 (1969), 614–618.

    Article  Google Scholar 

  32. R.L. Schwoebel, E.J. Shipsey, Step Motion on Crystal Surfaces. J. Appl. Phys. 37 (1966), 3682–3686.

    Article  Google Scholar 

  33. G. Ehrlich, F.G. Hudda, Atomic View of Surface Self-Diffusion: Tungsten on Tungsten. J. Chem. Phys. 44 (1966), 1039–1055.

    Article  Google Scholar 

  34. W.F. Chung, M.S. Altman, Kinetic length, step permeability, and kinetic coefficient asymmetry on the Si(111) (7 × 7) surface. Phys. Rev. B 66 (2002), 075338.

    Article  Google Scholar 

  35. A. Saúl, J.-J. Métois, A. Ranguis, Experimental evidence for an Ehrlich-Schwoebel effect on Si(111). Phys. Rev. B 65 (2002) 075409.

    Article  Google Scholar 

  36. M. Ozdemir, A. Zangwill, Morphological equilibration of a facetted crystal. Phys. Rev. B 45 (1992), 3718–3729.

    Article  Google Scholar 

  37. S. Tanaka, N.C. Bartelt, C.C. Umbach, R.M. Tromp, J.M. Blakely, Step Permeability and the Relaxation of Biperiodic Gratings on Si(001). Phys. Rev. Lett. 78 (1997) 3342–3345.

    Article  Google Scholar 

  38. F. Buatier de Mongeot, W. Zhu, A. Molle, R. Buzio, C. Boragno, U. Valbusa, E. G. Wang, Z. Zhang, Nanocrystal Formation and Faceting Instability in Al(110) Homoepitaxy: True Upward Adatom Diffusion at Step Edges and Island Corners. Phys. Rev. Lett. 91 (2003), 016102.

    PubMed  Google Scholar 

  39. O. Pierre-Louis, T.L. Einstein, Electromigration of single-layer clusters. Phys. Rev. B 62 (2000), 13697–13706.

    Article  Google Scholar 

  40. N. Néel, T. Maroutian, L. Douillard, H.-J. Ernst, From Meandering to Faceting, Is Step Flow Growth Ever Stable? Phys. Rev. Lett. 91 (2003) 226103.

    PubMed  Google Scholar 

  41. N. Néel, T. Maroutian, L. Douillard, H.-J. Ernst, Spontaneous structural pattern formation at the nanometer scale in kinetically restricted homoepitaxy on vicinal surfaces. J. Phys.: Condens. Matter 15 (2003), S3227–S3240.

    Article  Google Scholar 

  42. J. Krug, M. Schimschak, Metastability of Step Flow Growth in 1 + 1 Dimensions. J. Phys. I France 5 (1995), 1065–1086.

    Article  Google Scholar 

  43. O. Pierre-Louis, C. Misbah, Dynamics and fluctuations during MBE on vicinal surfaces. I. Formalism and results of linear theory. Phys. Rev. B 58 (1998), 2259–2275.

    Article  Google Scholar 

  44. N. Cabrera, D.A. Vermilyea, The Growth of Crystals from Solution. In: Growth and Perfection of Crystals, ed. by R. Doremus, B. Roberts, D. Turnbull (Wiley, New York 1958) pp. 393–408.

    Google Scholar 

  45. D. Kandel, J. Weeks: Theory of impurity-induced step bunching. Phys. Rev. B 49 (1994), 5554–5564.

    Article  Google Scholar 

  46. J. Krug, New mechanism for impurity-induced step bunching. Europhys. Lett. 60 (2002), 788–794.

    Article  Google Scholar 

  47. A. Pimpinelli, A. Videcoq, Novel mechanism for the onset of morphological instabilities during chemical vapour epitaxial growth. Surf. Sci. 445 (2000), L21–L28.

    Article  Google Scholar 

  48. M. Vladimirova, A. De Vita, A. Pimpinelli, Dimer diffusion as a driving mechanism of the step bunching instability during homoepitaxial growth. Phys. Rev. B 64 (2001), 245420.

    Article  Google Scholar 

  49. J. Mysliveček, C. Schelling, F. Schäffler, G. Springholz, P. Šmilauer, J. Krug, B. Voigtländer, On the microscopic origin of the kinetic step bunching instability on vicinal Si(001). Surf. Sci. 520 (2002), 193–206.

    Article  Google Scholar 

  50. G.S. Bales, A. Zangwill, Morphological instability of a terrace edge during step-flow growth. Phys. Rev. B 41 (1990), 5500–5508.

    Article  Google Scholar 

  51. A. Pimpinelli, I. Elkinani, A. Karma, C. Misbah, J. Villain, Step motions on high-temperature vicinal surfaces. J. Phys.: Condens. Matter 6 (1994), 2661–2680.

    Article  Google Scholar 

  52. B. Caroli, C. Caroli, B. Roulet, Instabilities of planar solidification fronts. In C. Godrèche (Ed.), Solids Far from Equilibrium (Cambridge University Press, 1991), pp.155–296.

    Google Scholar 

  53. F. Gillet, O. Pierre-Louis, C. Misbah, Non-linear evolution of the step meander during growth of a vicinal surface with no desorption. Eur. Phys. J. B 18 (2000), 519–534.

    Article  Google Scholar 

  54. J. Kallunki, Growth instabilities of vicinal crystal surfaces during Molecular Beam Epitaxy. (PhD dissertation, University of Duisburg-Essen, 2003).

    Google Scholar 

  55. J. Krug, Four lectures on the physics of crystal growth. Physica A 313 (2002), 47–82.

    Google Scholar 

  56. Y. Homma, P. Finnie, M. Uwaha, Morphological instability of atomic steps observed on Si(111) surfaces. Surf. Sci. 492 (2001), 125–136.

    Article  Google Scholar 

  57. R. Kato, M. Uwaha, Y. Saito, Step wandering due to the structural difference of the upper and lower terraces. Surf. Sci. 550 (2004), 149–165.

    Article  Google Scholar 

  58. T. Maroutian, L. Douillard, H.-J. Ernst, Morphological instability of Cu vicinal surfaces during step-flow growth. Phys. Rev. B 64 (2001), 165401.

    Article  Google Scholar 

  59. T. Maroutian, Étude expérimentale d’instabilités de croissance des faces vicinales (PhD dissertation, Université Paris 7, 2001).

    Google Scholar 

  60. M. Rusanen, I. T. Koponen, J. Heinonen, T. Ala-Nissila, Instability and wavelength selection during step flow growth of metal surfaces vicinal to fcc(001). Phys. Rev. Lett. 86 (2001), 5317–5320.

    PubMed  Google Scholar 

  61. J. Kallunki (unpublished).

    Google Scholar 

  62. J. Krug, Kinetic Pattern Formation at Solid Surfaces. In G. Radons, P. Häussler, W. Just (Eds.), Collective Dynamics of Nonlinear and Disordered Systems (Springer, Berlin 2004).

    Google Scholar 

  63. M. Rost, (this volume)

    Google Scholar 

  64. M. Rost, P. Šmilauer, J. Krug, Unstable epitaxy on vicinal surfaces. Surf. Sci. 369 (1996), 393–402.

    Article  Google Scholar 

  65. A. Videcoq, Auto-organisation de surfaces cristallines pendant la croissance épitaxiale: une étude théorique (PhD dissertation, Université Blaise Pascal, Clermont-Ferrand 2002).

    Google Scholar 

  66. M. Avignon, B.K. Chakraverty, Morphological stability of a two-dimensional nucleus. Proc. Roy. Soc. A 310 (1969), 277–296.

    Google Scholar 

  67. G.S. Bales, D.C. Chrzan, Transition from Compact to Fractal Islands during Submonolayer Epitaxial Growth. Phys. Rev. Lett. 74 (1995), 4879–4882.

    PubMed  Google Scholar 

  68. A.V. Latyshev, A.L. Aseev, A.B. Krasilnikov, S.I. Stenin, Transformations on clean Si(111) stepped surface during sublimation. Surf. Sci. 213 (1989) 157–169.

    Article  Google Scholar 

  69. P. Kuhn, J. Krug, (this volume)

    Google Scholar 

  70. S. Stoyanov, Electromigration Induced Step Bunching on Si Surfaces — How Does It Depend on the Temperature and Heating Current Direction? Jap. J. Appl. Phys. 30 (1991), 1–6.

    Article  Google Scholar 

  71. D. Kandel, E. Kaxiras, Microscopic Theory of Electromigration on Semiconductor Surfaces. Phys. Rev. Lett. 76 (1996), 1114–1117.

    PubMed  Google Scholar 

  72. M. Degawa, H. Minoda, Y. Tanishiro, K. Yagi, Direct-current-induced drift direction of silicon adatoms on Si(111)-(1 × 1) surfaces. Surf. Sci. 461 (2000), L528–L536.

    Article  Google Scholar 

  73. C. Misbah, O. Pierre-Louis, A. Pimpinelli, Advacancy-induced step bunching on vicinal surfaces. Phys. Rev. B 51 (1995), 17283–17286.

    Google Scholar 

  74. S. Stoyanov, Current-induced step bunching at vicinal surfaces during crystal sublimation. Surf. Sci. 370 (1997), 345–354.

    Article  Google Scholar 

  75. S. Stoyanov, New type of step bunching instability at vicinal surfaces in crystal evaporation affected by electromigration. Surf. Sci. 416 (1998), 200–213.

    Article  Google Scholar 

  76. J.J. Métois, S. Stoyanov, Impact of growth on the stability-instability transition at Si(111) during step bunching induced by electromigration. Surf. Sci. 440 (1999), 407–419.

    Article  Google Scholar 

  77. H. Minoda, I. Morishima, M. Degawa, Y. Tanishiro, K. Yagi, Time evolution of DC heating-induced in-phase step wandering on Si(111) vicinal surfaces. Surf. Sci. 493 (2001), 487–493.

    Article  Google Scholar 

  78. M. Sato, M. Uwaha, Y. Saito, Instabilities of steps induced by the drift of adatoms and effect of the step permeability. Phys. Rev. B 62 (2000), 8452–8472.

    Article  Google Scholar 

  79. N. Suga, J. Kimpara, N.-J. Wu, H. Yasunaga, A. Natori: Novel Transition Mechanism of Surface Electromigration Induced Step Structure on Vicinal Si(111) Surfaces. Jpn. J. Appl. Phys. 39 (2000), 4412–4416.

    Article  Google Scholar 

  80. H. Dobbs, J. Krug, Current Induced Faceting in Theory and Simulation. J. Phys. I France 6 (1996), 413–430.

    Article  Google Scholar 

  81. M. Degawa, H. Minoda, Y. Tanishiro, K. Yagi, In-phase step wandering on Si(111) vicinal surfaces: Effect of direct current heating tilted from the step-down direction. Phys. Rev. B 63 (2001), 045309.

    Article  Google Scholar 

  82. S. Stoyanov, V. Tonchev, Properties and dynamic interaction of step density waves at a crystal surface during electromigration affected sublimation. Phys. Rev. B 58 (1998), 1590–1600.

    Article  Google Scholar 

  83. D.-J. Liu, J.D. Weeks, Quantitative theory of current-induced step bunching on Si(111). Phys. Rev. B 57 (1998), 14891–14900.

    Article  Google Scholar 

  84. M. Sato, M. Uwaha, Growth of step bunches formed by the drift of adatoms. Surf. Sci. 442 (1999), 318–328.

    Article  Google Scholar 

  85. M. Sato, M. Uwaha, Growth law of step bunches induced by the Ehrlich-Schwoebel effect in growth. Surf. Sci. 493 (2001), 494–498.

    Article  Google Scholar 

  86. Y.-N. Yang, E.S. Fu, E.D. Williams, An STM study of current-induced step bunching on Si(111). Surf. Sci. 356 (1996), 101–111.

    Article  Google Scholar 

  87. K. Fujita, M. Ichikawa, S.S. Stoyanov, Size-scaling exponents of current-induced step bunching on silicon surfaces. Phys. Rev. B 60 (1999), 16006–16012.

    Article  Google Scholar 

  88. Y. Homma, N. Aizawa, Electric-current-induced step bunching on Si(111). Phys. Rev. B 62 (2000), 8323–8329.

    Article  Google Scholar 

  89. J. Krug, Continuum Equations for Step Flow Growth. In D. Kim, H. Park, B. Kahng (Eds.), Dynamics of Fluctuating Interfaces and Related Phenomena (World Scientific, Singapore 1997), pp. 95–113.

    Google Scholar 

  90. J. Krug, V. Tonchev, S. Stoyanov, A. Pimpinelli, Scaling properties of step bunches induced by Ehrlich-Schwoebel barriers during sublimation. (in preparation)

    Google Scholar 

  91. P. Nozières, On the motion of steps on a vicinal surface. J. Physique 48 (1987), 1605–1608.

    Google Scholar 

  92. A. Pimpinelli, V. Tonchev, A. Videcoq, M. Vladimirova, Scaling and Universality of Self-Organized Patterns on Unstable Vicinal Surfaces. Phys. Rev. Lett. 88 (2002), 206103.

    PubMed  Google Scholar 

  93. F. Gillet, Z. Csahok, C. Misbah, Continuum nonlinear surface evolution equation for conserved step-bunching dynamics. Phys. Rev. B 63 (2001), 241401.

    Article  Google Scholar 

  94. P. Šmilauer, M. Rost, J. Krug, Fast coarsening in unstable expitaxy with desorption. Phys. Rev. E 59 (1999), R6263–R6266.

    Article  Google Scholar 

  95. M. Sato, M. Uwaha, Step Bunching as Formation of Soliton-like Pulses in Benney Equation. Europhys. Lett. 32 (1995), 639–644.

    Google Scholar 

  96. I. Bena, C. Misbah, A. Valance, Nonlinear evolution of a terrace edge during step-flow growth. Phys. Rev. B 47 (1993), 7408–7419.

    Article  Google Scholar 

  97. O. Pierre-Louis, C. Misbah, Dynamics and fluctuations during MBE on vicinal surfaces. II. Nonlinear analysis. Phys. Rev. B 58 (1998), 2276–2288.

    Article  Google Scholar 

  98. O. Pierre-Louis, C. Misbah, Y. Saito, J. Krug, P. Politi, New Nonlinear Evolution Equation for Steps during Molecular Beam Epitaxy on Vicinal Surfaces. Phys. Rev. Lett. 80 (1998), 4221–4224.

    Article  Google Scholar 

  99. J. Kallunki, J. Krug, Asymptotic step profiles from a nonlinear growth equation for vicinal surfaces. Phys. Rev. E 62 (2000), 6229–6232.

    Article  Google Scholar 

  100. P. Politi, C. Misbah, When Does Coarsening Occur in the Dynamics of One-Dimensional Fronts? Phys. Rev. Lett. 92 (2004), 090601.

    PubMed  Google Scholar 

  101. G. Danker, O. Pierre-Louis, K. Kassner, C. Misbah, Interrupted coarsening of anisotropic step meander. Phys. Rev. E 68 (2003), 020601(R).

    Article  Google Scholar 

  102. J. Kallunki, J. Krug, Breakdown of step-flow growth in unstable homoepitaxy. Europhys. Lett. 66 (2004), 749–755.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Krug, J. (2005). Introduction to Step Dynamics and Step Instabilities. In: Voigt, A. (eds) Multiscale Modeling in Epitaxial Growth. ISNM International Series of Numerical Mathematics, vol 149. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7343-1_6

Download citation

Publish with us

Policies and ethics