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Interplay of quantum size effect, anisotropy and surface stress shapes the instability of thin metal films

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Abstract

Morphological instability of a planar surface ([111], [011], or [001]) of an ultra-thin metal film is studied in a parameter space formed by three major effects (the quantum size effect, the surface energy anisotropy and the surface stress) that influence a film dewetting. The analysis is based on the extended Mullins equation, where the effects are cast as functions of the film thickness. The formulation of the quantum size effect (Zhang et al., Phys Rev Lett 80:5381, 1998) includes the oscillation of the surface energy with thickness caused by electron confinement. By systematically comparing the effects, their contributions into the overall stability (or instability) is highlighted.

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Notes

  1. Herring–Mullins theory has been used to study the instabilities and morphological evolution of a comparatively thick films (>100 nm). However, the theory does not have a limitation on a thickness per se. The base assumption in the theory is that there is a mobile adsorbate on the crystal surface, i.e., the top layer of the crystal is “fluid”, and the geometrical quantities such as the curvature can be defined for the surface. The top layer is usually considered to be one to three atoms deep. Thus, the theory is applicable to thinner films. It must be noted here that the dewetting of the ultra-thin metallic films is apparently driven by surface diffusion, as the films develop faceted pinholes that extend from the surface down to the substrate, and the pinholes are surrounded at their surface perimeter by the characteristic high rims formed out of the film material that diffused from the pinhole bottom [5].

  2. It was argued recently [20] that in addition to this conventional surface stress of a macroscopic thin film, the ultra-thin metal films also are influenced by the quantum oscillations in the surface stress that are phase shifted with respect to the oscillations of the surface energy (the latter oscillations are termed the QSE oscillations in this paper). The authors of Ref. [20] propose that the coupling of the two oscillations may be responsible for the extension of the surface energy oscillations to thick films (>30 ML).

  3. In Section 5 of his paper, Hirayama [3] cites the experiments where the shift of the critical thickness is arguably achieved through enhancement of charge spillage at the interface after the layer of Al is inserted between Ag film and Si substrate.

References

  1. Zhang Z, Niu Q, Shih CK (1998) Electronic growth of metallic overlayers on semiconductor substrates. Phys Rev Lett 80:5381

    Article  ADS  Google Scholar 

  2. Smith AR, Chao KJ, Qiu N, Shih CK (1996) Formation of atomically flat silver films on GaAs with a silver mean quasi periodicity. Science 273:226

    Article  ADS  Google Scholar 

  3. Hirayama H (2009) Growth of atomically flat ultra-thin Ag films on Si surfaces. Surf Sci 603:1492–1497

    Article  ADS  Google Scholar 

  4. Ozer MM, Wang CZ, Zhang Z, Weitering HH (2009) Quantum size effects in the growth, coarsening, and properties of ultra-thin metal films and related nanostructures. J Low Temp Phys 157:221–251

    Article  ADS  Google Scholar 

  5. Sanders CE, Zhang C, Kellogg GL, Shih CK (2014) Role of thermal processes in dewetting of epitaxial Ag(111) film on Si(111). Surf Sci 630:168–173

    Article  ADS  Google Scholar 

  6. Han Y, Unal B, Jing D, Thiel PA, Evans JW, Liu DJ (2010) Nanoscale quantum islands on metal substrates: microscopy studies and electronic structure analyzes. Materials 3:3965

    Article  ADS  Google Scholar 

  7. Chiu CH (2004) Stable and uniform arrays of self-assembled nanocrystalline islands. Phys Rev B 69:165413

    Article  ADS  Google Scholar 

  8. Golovin AA, Levine MS, Savina TV, Davis SH (2004) Faceting instability in the presence of wetting interactions: a mechanism for the formation of quantum dots. Phys Rev B 70:235342

    Article  ADS  Google Scholar 

  9. Levine MS, Golovin AA, Davis SH, Voorhees PW (2007) Self-assembly of quantum dots in a thin epitaxial film wetting an elastic substrate. Phys Rev B 75:205312

    Article  ADS  Google Scholar 

  10. Korzec MD, Münch A, Wagner B (2012) Anisotropic surface energy formulations and their effect on stability of a growing thin film. Interfaces Free Bound 14:545–567

    Article  MathSciNet  MATH  Google Scholar 

  11. Korzec MD, Evans PL (2010) From bell shapes to pyramids: a reduced continuum model for self-assembled quantum dot growth. Physica D 239:465–474

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Khenner M, Tekalign WT, Levine M (2011) Stability of a strongly anisotropic thin epitaxial film in a wetting interaction with elastic substrate. Eur Phys Lett 93:26001

    Article  ADS  Google Scholar 

  13. Han Y, Liu DJ (2009) Quantum size effects in metal nanofilms: comparison of an electron-gas model and density functional theory calculations. Phys Rev B 80:155404

    Article  ADS  Google Scholar 

  14. Srolovitz DJ, Safran SA (1986) Capillary instabilities in thin films. II. Kinetics. J Appl Phys 60:255

    Article  ADS  Google Scholar 

  15. Khenner M (2008) Dewetting of an ultrathin solid film on a lattice matched or amorphous substrate. Phys Rev B 77:165414

    Article  ADS  Google Scholar 

  16. Khenner M (2008) Morphologies and kinetics of a dewetting ultrathin solid film. Phys Rev B 77:245445

    Article  ADS  Google Scholar 

  17. Thompson CV (2012) Solid-state dewetting of thin films. Annu Rev Mater Res 42:399

    Article  ADS  Google Scholar 

  18. Wong H, Voorhees PW, Miksis MJ, Davis SH (2000) Periodic mass shedding of a retracting solid film step. Acta Mater 48:1719–1728

    Article  Google Scholar 

  19. Kan W, Wong H (2005) Fingering instability of a retracting solid film edge. J Appl Phys 97:043515

    Article  ADS  Google Scholar 

  20. Liu M, Han Y, Tang L, Jia J-F, Xue Q-K, Liu F (2012) Interplay between quantum size effect and strain effect on growth of nanoscale metal thin films. Phys Rev B 86:125427

    Article  ADS  Google Scholar 

  21. Hamilton JC, Wolfer WG (2009) Theories of surface elasticity for nanoscale objects. Surf Sci 603:1284–1291

    Article  ADS  Google Scholar 

  22. Gurtin ME, Murdoch AI (1978) Surface stress in solids. Int J Solids Struct 14:431–440

    Article  MATH  Google Scholar 

  23. McCarty KF, Hamilton JC, Sato Y, Saa A, Stumpf R, Figuera J, Thurmer K, Jones F, Schmid AK, Talin AA, Bartelt NC (2009) How metal films de-wet substrates—identifying the kinetic pathways and energetic driving forces. New J Phys 11:043001

    Article  Google Scholar 

  24. Tyson WR, Miller W (1977) Surface free energies of solid metals: estimation from liquid surface tension measurements. Surf Sci 62:267–276

    Article  ADS  Google Scholar 

  25. McFadden GB, Coriell SR, Sekerka RF (1988) Effect of surface tension anisotropy on cellular morphologies. J Cryst Growth 91:180–198

    Article  ADS  Google Scholar 

  26. Golovin AA, Davis SH, Nepomnyashchy AA (1998) A convective Cahn–Hilliard model for the formation of facets and corners in crystal growth. Physica D 122:202

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Herring C (1951) Some theorems on the free energies of crystal surfaces. Phys Rev 82:87

    Article  ADS  MATH  Google Scholar 

  28. Di Carlo A, Gurtin ME, Podio-Guidugli P (1992) A regularized equation for anisotropic motion-by-curvature. SIAM J Appl Math 52:1111–1119

    Article  MathSciNet  MATH  Google Scholar 

  29. Rabkin E, Amram D, Alster E (2014) Solid state dewetting and stress relaxation in a thin single crystalline Ni film on sapphire. Acta Mater 74:30–38

    Article  Google Scholar 

  30. Wall D, Tikhonov S, Sindermann S, Spoddig D, Hassel C, Horn-von Hoegen M, Meyer zu Heringdorf F-J (2011) Shape, orientation, and crystalline composition of silver islands on Si(111). IBM J Res Dev 55(4):2158761

    Article  Google Scholar 

  31. Bennett RA, Mulley JS, Etman HA, Sparkes A, Eralp T, Held G, Cavill SA, Dhesi SS (2012) Chromium nanostructures formed by dewetting of heteroepitaxial films on W(100). Phys Rev B 86:045454

    Article  ADS  Google Scholar 

  32. Liu F, Metiu H (1993) Dynamics of phase separation of crystal surfaces. Phys Rev B 48:5808

    Article  ADS  Google Scholar 

  33. Savina TV, Golovin AA, Davis SH, Nepomnyashchy AA, Voorhees PW (2003) Faceting of a growing crystal surface by surface diffusion. Phys Rev E 67:021606

    Article  ADS  Google Scholar 

  34. Khenner M (2007) Tailoring of crystal surface morphology by induced spatio-temporal oscillations of temperature. Phys Rev E 75:021605

    Article  ADS  MathSciNet  Google Scholar 

  35. Li B, Lowengrub J, Ratz A, Voigt A (2009) Geometric evolution laws for thin crystalline films: modeling and numerics. Commun Comput Phys 6:433–482

    MathSciNet  Google Scholar 

  36. Wolfram Research Inc (2014) Mathematica, version 10.0. Wolfram Research Inc, Champaign, IL

  37. Pierre-Louis O, private communication

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Acknowledgments

The author thanks Olivier Pierre-Louis (University of Lyon) for making available his unpublished recent derivation of the electronic contribution to the surface energy. The anonymous Referee is acknowledged for the expert and very thorough multi-stage review of the paper’s manuscript.

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  1. Department of Mathematics and Applied Physics Institute, Western Kentucky University, Bowling Green, KY, 42101, USA

    Mikhail Khenner

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Khenner, M. Interplay of quantum size effect, anisotropy and surface stress shapes the instability of thin metal films. J Eng Math 104, 77–92 (2017). https://doi.org/10.1007/s10665-016-9874-6

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