Nonlinear evolution of the morphological instability of a strained epitaxial film

Transcription

1 Nonlinear evolution of the morphological instability of a strained epitaxial film Thomas Frisch, Jean-No Noël Aqua, Alberto Verga, IM2NP,Marseille 1. Self-organization and crystal growth (step dynamics) 2. Asaro-Tiller-Grinfeld (ATG) instability & the wetting-atg 3. Non-linear and non-local evolution 4. Conclusion and Perspective Vendredi 14 mars, 2008

2 1. Self-organization in Crystal growth Rev Modern Physics, Stangl et al, Self-organization via growth instabilities ( step dynamic) Increasing density of electronic components e.g. Nano-Crystal memory Homoepitaxy : 2 basic instabilities on a vicinal surface Step-bunching : Erhlich and Schwoebel 1969, Stoyanov 1988 Mandering : Bales and Zangwill, 1990 KESE effect, Pierre-Louis 1998 Fujita, et al., Phys Rev B, 1999 Neel, Ersnt Maroutian, PRL 2003

3 Step bunching and meandering on Si (001) during MBE growth Lagally,1990 Si(001) désorientée vers [110], STM 10 nm Frisch et Verga, Phys. Rev. L 94, 2005 Frisch et Verga, Phys Rev. L 96, 2006 Berbezier et Ronda, L2MP, AFM (Atomic force microscopy). Si (001) désorienté vers [110] 6 microns

4 Electromigration on Si (111) Effect of an electrical field perpendicular to a vicinal surface Dufay, Frisch, Debierre, Phys. Rev B temps

5 2.Heteroepitaxy : paradigm Si/Ge systems Stransky-Krastanov mode During annealing, islands separated by a wetting layer Dominated by surface diffusion Floro et al., Phys. Rev. Lett. 2002

6 Effets des contraintes élastiques: Croissance Si/Ge Quelle est l évolution 3D du film lors d un recuit en température? E f, ν f germanium E s, ν s silicium contrainte créée par la différence de maille atomique Croissance de Stranski et Krastanov Désacord de maille δa/a de l ordre de 4.2%

7 Asaro-Tiller-Grinfel d Instability Route towards crack or fracture formation in He 4 films : Bodenson, 1986, Balibar, 1991 Finite time singularities for (Chiu, Gao, Spencer, Davis, Voorhees, Meiron, Kassner, Misbah, Xiang, E) Elastic stress concentration What is different in heteroepitaxial systems? Floro

8 Morphological Instabilites (Asaro-Tiller-Grinfeld Instability) Ge Si Les atomes de Germanium sont solubles dans le silicium mais ils sont 4.2% plus gros Formation des ilots Création de dislocation

9 Effect of the elastic constraint on Si/Ge coherent film What is the 3D evolution of the film during thermal annealing? E f, ν f germanium Elastic stress created by the misfit E s, ν s silicium Misfit is δa/a ~ 4.2% Two new features : 1. Elastic response of the substrate 2. Wetting interactions between the film and the substrate Find a model to give qualitative and quantitative description of the film/substrate system

10 2. Asaro-Tiller-Grinfeld (ATG) instability During annealing, strain relaxation via surface diffusion with λ Surface energy Elastic energy h(x,y,t) E F, ν F Typical length : with the flat film energy density Instability : whatevolution? Asaro, Tiller, 1972, Grinfeld, 1986, Srolovitz, 1989

11 λ Surface diffusion during annealing h(x,y,t) E f, ν f E s, ν s Surface energy Elastic energy γ(h) extrapolates between the bulk and the thin film surface energy, e.g. Wetting potential Chiu, Gao, 1993 Spencer, Tekalign, Golovin, Davis, Voorhess, 2003 where is a few layer diameters γ(h) h Ab-initio calculations Lu, Liu, Phys. Rev. Lett., 2005

12 Film and substrate elasticity Linear elasticity Lamé equations α = f (film) or s (substrate) with different film and substrate properties Boundary conditions : 1. coherent film/substrate interface : 2. free upper film surface : 3. infinitely thick substrate :

13 Linear analysis : σ h 0 > h c Rem : k h 0 h 0 < h c h 0 > h c annealing Flat film stable for h 0 < h c Wetting-ATG instability for h 0 > h c What evolution? Finite-time Singularity?

14 3. Non-Linear analysis, in the thin-film approximation In 2D, non-local, non-linear film evolution with the Hilbert transform vs (conserved) Kuramoto-Shivashinsky

15 In 3D : with : Compact form of the elastic free energy Analytical study and large-scale simulations (spectral)

16 NO singularity when BOTH wetting (small, not hardcore) & non-linear non-local effects are considered 0.8 h max 0.6 h(x,t) t x 40 60

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18 Instability t=1 t=4 Ripening Large islands grow at the expense of the small ones wetting layer t=60 t=100 Non-interrupted coarsening towards an equilibrium state made of a single island

19 Equilibrium maximum height : h max Instability threshold : h 0 > h c h c 2D 2D h c 3D First-order like Equilibrium chemical potential 2D 3D Non-interrupted coarsening : larger islands relax strain more efficently

20 Aqua, Frisch, Verga, Phys Rev B, 76, Dynamics of the coarsening 2D 3D 1. roughness 2D 3D 2. Number of islands

21 4. Conclusion and perspectives Basic frame describing the evolution via surface diffusion of a thin strained film during annealing, considering wetting and film/substrate elasticity Compact form of the surface dynamical equation Long-range and non-linear elastic effects coupled to small (and not hard-core like) wetting interactions 1. regularize the ATG singularity 2. lead to a non-interrupted coarsening Perspectives : with unexpected scaling Understand the scaling behavior Facetting interrupted coarsening? Anisotropic elasticty (C11, C12, C22) A. Rastelli et al, PRL. 87, 2002

22 Collaborateurs dans l équipe Théorie-Modélisation et Simulation (TMS), IM2NP, M. Ranganathan, J. N Aqua, A. Verga, M. Robini, M. Dufay et J.-M. Debierre Isabelle berbezier et Antoine Ronda, IM2NP, Si-Ge AFM image, 6 micron scale