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Equilibrium shapes of strained islands with non-zero contact angles. (English) Zbl 1120.74619
Summary: The equilibrium morphology of a strained island on an elastic substrate is determined. The island is assumed to partially wet the substrate (Volmer-Weber growth) and thus makes a non-zero contact angle with the surface. Both isotropic and anisotropic misfit strain are allowed. Two- and three-dimensional equilibrium island shapes are determined by using expressions for the elastic strain energy in the small-slope approximation. In this limit, the problem can be reduced to a singular integral-differential equation for the island thickness. We find that when there is a non-zero contact angle, all island shapes, for a given ratio of the elastic stress to surface energy, attain a form that is independent of the specific contact angle under an appropriate scaling. We show that for islands with non-zero contact angles, as the island volume increases, the shape approaches the geometry of a completely wetting island. But when the volume decreases, these islands approach a point while islands with a zero contact angle, approach a finite length line segment of zero volume. Multiple-hump equilibrium shapes are found. Single-humped islands are shown to have a lower chemical potential than multiple-humped islands, implying that they are the most stable. This conclusion is shown to be consistent with a stability analysis of the two-dimensional case. The effects of a tetragonal misfit strain on the three-dimensional island shape is investigated.

MSC:
74K35 Thin films
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[1] Chen, Y.; Ohlberg, D.A.; Medeiros-Ribeiro, G.; Chang, Y.A.; Williams, R.S., Self-assembled growth of epitaxial erbium disilicide nanowires on silicon (001), Appl. phys. lett., 76, 4004, (2000)
[2] Eaglesham, D.J.; Cerullo, M., Dislocation-free stranski – krastanow growth of ge on si(100), Phys. rev. lett., 64, 16, 1943-1946, (1990)
[3] Gao, H., Stress concentration at slightly undulating surfaces, J. mech. phys. solids, 39, 4, 443-458, (1991) · Zbl 0734.73007
[4] Grandos, D.; Garcia, J.M., In(ga)As self-assembled quantum ring formation by molecular beam epitaxy, J. appl. phys. lett., 82, 15, 2401-2403, (2003)
[5] Guyer, J.E., Voorhees, P.W., 1996. The stability of lattice mismatched thin films. Material Research Society Symposium Proceedings, vol. 399, Material Research Society, pp. 351-357.
[6] Kukta, R.V.; Frend, L.B., Minimum energy configuration of epitaxial material clusters on a lattice-mismatched substrate, J. mech. phys. solids, 45, 11/12, 1835-1860, (1997)
[7] Liu, B.Z.; Nogami, J., Growth of parallel rare-Earth silicide nanowire arrays on vicinal si(001), Nanotechnology, 14, 873-877, (2003)
[8] Mura, T., Micromechanics of defects in solids, (1987), Kluwer Academic Publishers Dordrecht
[9] Nogami, J.; Liu, B.Z.; Katkov, M.V.; Ohbuchi, C.; Birge, N.O, Self-assembled rare-Earth silicide nanowires on silicon (001), Phys. rev. B, 63, 233305, (2001)
[10] Ross, F.M.; Tersoff, J.; Tromp, R.M., Coarsening of self-assembled ge quantum dots on si(001), Phys. rev. lett., 80, 5, 984-986, (1998)
[11] Rudin, C.D.; Spencer, B.J., Equilibrium island ridge arrays in strained solid films, J. appl. phys., 86, 10, 5530-5536, (1999)
[12] Shanahan, L.L.; Spencer, B.J., A codimension-two free boundary problem for the equilibrium shapes of a small three-dimensional island in an epitaxially strained solid film, Interfaces free boundaries, 4, 1-25, (2002) · Zbl 0998.35060
[13] Shchukin, V.A.; Bimberg, D., Spontaneous ordering of nanostructures on crystal surfaces, Rev. mod. phys., 71, 4, 1125-1170, (1999)
[14] Spencer, B.J.; Tersoff, J., Equilibrium shapes and properties of epitaxially strained islands, Phys. rev. lett., 79, 24, 4858-4861, (1997)
[15] Spencer, B.J.; Voorhees, P.W.; Davis, S.H., Morphological instability in epitaxially strained dislocation-free solid films: linear stability theory, J. appl. phys., 73, 10, 4955-4970, (1993)
[16] Tsatsul’nikov, A.F.; Kovsh, A.R.; Zhukov, A.E.; Shernyakov, Y.M.; Musikhin, Y.G.; Ustinov, V.M.; Bert, N.A.; Kop’ev, P.S.; Alferov, Z.I.; Mintairov, A.M.; Merz, J.L.; Ledentsov, N.N.; Bimberg, D., Volmer – weber and stranski – krastanov inas-algaas quantum dots emitting at \(1.4 \operatorname{\mu} \operatorname{m}\), J. appl. phys., 88, 11, 6272-6275, (2000)
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