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

Journal of Engineering Mathematics - Tập 104 - Trang 77-92 - 2016
Mikhail Khenner1
1Department of Mathematics and Applied Physics Institute, Western Kentucky University, Bowling Green, USA

Tóm tắt

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.

Tài liệu tham khảo

Zhang Z, Niu Q, Shih CK (1998) Electronic growth of metallic overlayers on semiconductor substrates. Phys Rev Lett 80:5381 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 Hirayama H (2009) Growth of atomically flat ultra-thin Ag films on Si surfaces. Surf Sci 603:1492–1497 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 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 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 Chiu CH (2004) Stable and uniform arrays of self-assembled nanocrystalline islands. Phys Rev B 69:165413 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 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 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 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 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 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 Srolovitz DJ, Safran SA (1986) Capillary instabilities in thin films. II. Kinetics. J Appl Phys 60:255 Khenner M (2008) Dewetting of an ultrathin solid film on a lattice matched or amorphous substrate. Phys Rev B 77:165414 Khenner M (2008) Morphologies and kinetics of a dewetting ultrathin solid film. Phys Rev B 77:245445 Thompson CV (2012) Solid-state dewetting of thin films. Annu Rev Mater Res 42:399 Wong H, Voorhees PW, Miksis MJ, Davis SH (2000) Periodic mass shedding of a retracting solid film step. Acta Mater 48:1719–1728 Kan W, Wong H (2005) Fingering instability of a retracting solid film edge. J Appl Phys 97:043515 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 Hamilton JC, Wolfer WG (2009) Theories of surface elasticity for nanoscale objects. Surf Sci 603:1284–1291 Gurtin ME, Murdoch AI (1978) Surface stress in solids. Int J Solids Struct 14:431–440 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 Tyson WR, Miller W (1977) Surface free energies of solid metals: estimation from liquid surface tension measurements. Surf Sci 62:267–276 McFadden GB, Coriell SR, Sekerka RF (1988) Effect of surface tension anisotropy on cellular morphologies. J Cryst Growth 91:180–198 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 Herring C (1951) Some theorems on the free energies of crystal surfaces. Phys Rev 82:87 Di Carlo A, Gurtin ME, Podio-Guidugli P (1992) A regularized equation for anisotropic motion-by-curvature. SIAM J Appl Math 52:1111–1119 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 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 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 Liu F, Metiu H (1993) Dynamics of phase separation of crystal surfaces. Phys Rev B 48:5808 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 Khenner M (2007) Tailoring of crystal surface morphology by induced spatio-temporal oscillations of temperature. Phys Rev E 75:021605 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 Wolfram Research Inc (2014) Mathematica, version 10.0. Wolfram Research Inc, Champaign, IL Pierre-Louis O, private communication