Quantum Dots and Dislocations: Dynamics of Materials Defects

Transcription

1 Quantum Dots and Dislocations: Dynamics of Materials Defects with N. Fusco, G. Leoni, M. Morini, A. Pratelli, B. Zwicknagl Irene Fonseca Department of Mathematical Sciences Center for Nonlinear Analysis Carnegie Mellon University Supported by the National Science Foundation (NSF) Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 1 / 51(NS

2 Outline Quantum Dots: Wetting and zero contact angle. Shapes of islands Surface Diffusion in epitaxially strained solids: Existence and regularity Nucleation of Dislocations: Release of energy... and film becomes flat! Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 2 / 51(NS

3 Outline Quantum Dots: Wetting and zero contact angle. Shapes of islands Surface Diffusion in epitaxially strained solids: Existence and regularity Nucleation of Dislocations: Release of energy... and film becomes flat! Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 2 / 51(NS

4 Outline Quantum Dots: Wetting and zero contact angle. Shapes of islands Surface Diffusion in epitaxially strained solids: Existence and regularity Nucleation of Dislocations: Release of energy... and film becomes flat! Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 2 / 51(NS

5 Quantum Dots. The Context Strained epitaxial films on a relatively thick substrate; the thin film wets the substrate Islands develop without forming dislocations Stranski-Krastanow growth plane linear elasticity (In-GaAs/GaAs or SiGe/Si) free surface of film is flat until reaching a critical thikness lattice misfits between substrate and film induce strains in the film Complete relaxation to bulk equilibrium crystalline structure would be discontinuous at the interface Strain flat layer of film morphologically unstable or metastable after a critical value of the thickness is reached (competition between surface and bulk energies) Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 3 / 51(NS

6 Quantum Dots. The Context Strained epitaxial films on a relatively thick substrate; the thin film wets the substrate Islands develop without forming dislocations Stranski-Krastanow growth plane linear elasticity (In-GaAs/GaAs or SiGe/Si) free surface of film is flat until reaching a critical thikness lattice misfits between substrate and film induce strains in the film Complete relaxation to bulk equilibrium crystalline structure would be discontinuous at the interface Strain flat layer of film morphologically unstable or metastable after a critical value of the thickness is reached (competition between surface and bulk energies) Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 3 / 51(NS

7 Quantum Dots. The Context Strained epitaxial films on a relatively thick substrate; the thin film wets the substrate Islands develop without forming dislocations Stranski-Krastanow growth plane linear elasticity (In-GaAs/GaAs or SiGe/Si) free surface of film is flat until reaching a critical thikness lattice misfits between substrate and film induce strains in the film Complete relaxation to bulk equilibrium crystalline structure would be discontinuous at the interface Strain flat layer of film morphologically unstable or metastable after a critical value of the thickness is reached (competition between surface and bulk energies) Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 3 / 51(NS

8 Quantum Dots. The Context Strained epitaxial films on a relatively thick substrate; the thin film wets the substrate Islands develop without forming dislocations Stranski-Krastanow growth plane linear elasticity (In-GaAs/GaAs or SiGe/Si) free surface of film is flat until reaching a critical thikness lattice misfits between substrate and film induce strains in the film Complete relaxation to bulk equilibrium crystalline structure would be discontinuous at the interface Strain flat layer of film morphologically unstable or metastable after a critical value of the thickness is reached (competition between surface and bulk energies) Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 3 / 51(NS

9 Islands To release some of the elastic energy due to the strain: atoms on the free surface rearrange and morphologies such as formation of islands (quatum dots) of pyramidal shapes are energetically more economical. Kinetics of Stranski- Krastanow depend on initial thickness of film, competition between strain and surface energies, anisotropy, ETC. 3D photonic crystal template partially filled with GaAs by epitaxy. Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 4 / 51(NS

10 Frank der Merwe growth mode (FM) Volmer-Weber growth mode (VW) Stranski-Krastanov growth mode (SK) Island formation Si 0.8 Ge 0.2 onto Si. Courtesy of Floro et. al. ( 98) Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 5 / 51(NS

11 Why Do We Care? Quantum Dots: "semiconductors whose characteristics are closely related to size and shape of crystals" transistors, solar cells, optical and optoelectric devices (quantum dot laser), medical imaging, information storage, nanotechnology... electronic properties depend on the regularity of the dots, size, spacing, etc. 3D Printing: New additive manufacturing technology the mathematical understanding of the theory of dislocations will be central to address the energy balance between laser beam power (laser beams are used to melt the powder of the material into a specific shape) and the energy required to form a given geometrical shape Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 6 / 51(NS

12 Why Do We Care? Quantum Dots: "semiconductors whose characteristics are closely related to size and shape of crystals" transistors, solar cells, optical and optoelectric devices (quantum dot laser), medical imaging, information storage, nanotechnology... electronic properties depend on the regularity of the dots, size, spacing, etc. 3D Printing: New additive manufacturing technology the mathematical understanding of the theory of dislocations will be central to address the energy balance between laser beam power (laser beams are used to melt the powder of the material into a specific shape) and the energy required to form a given geometrical shape Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 6 / 51(NS

13 Continuum models in epitaxial growth above roughening transition Guyer & Voorhees ( 94) Kukta & Freund ( 97) Spencer & Meiron ( 94) Spencer & Tersoff ( 97) Spencer ( 99) Freund & Suresh ( 03)... Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 7 / 51(NS

14 Epitaxial films: equilibrium configurations Γ h Ω h 0 Substrate b 1( E(u) = u + T u )... strain 2 W (E) = 1 2E CE... energy density C... positive definite fourth-order tensor ψ =... (anisotropic) surface energy density u(x, 0) = e0 (x, 0), u(, t)... Q-periodic rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 8 / 51(NS

15 Epitaxial films: equilibrium configurations Γ h Ω h F (u, h) := W (E(u)) dx + ψ(ν)dh 1 Ω h Γ h 0 Substrate b 1( E(u) = u + T u )... strain 2 W (E) = 1 2E CE... energy density C... positive definite fourth-order tensor ψ =... (anisotropic) surface energy density u(x, 0) = e0 (x, 0), u(, t)... Q-periodic rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 8 / 51(NS

16 Epitaxial films: equilibrium configurations Γ h Ω h F (u, h) := W (E(u)) dx + ψ(ν)dh 1 Ω h Γ h 0 Substrate b 1( E(u) = u + T u )... strain 2 W (E) = 1 2E CE... energy density C... positive definite fourth-order tensor ψ =... (anisotropic) surface energy density u(x, 0) = e0 (x, 0), u(, t)... Q-periodic rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 8 / 51(NS

17 Epitaxial films: equilibrium configurations Γ h Ω h F (u, h) := W (E(u)) dx + ψ(ν)dh 1 Ω h Γ h 0 Substrate b 1( E(u) = u + T u )... strain 2 W (E) = 1 2E CE... energy density C... positive definite fourth-order tensor ψ =... (anisotropic) surface energy density u(x, 0) = e0 (x, 0), u(, t)... Q-periodic inf {F (u, h) : (u, h) admissible, Ω h = d} rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 8 / 51(NS

18 F (u, h) := W (E(u)) dx + ψ(ν)dh 1 Ω h Γ h Brian Spencer, Bonnetier and Chambolle, Chambolle and Larsen; Caflish, W. E, Otto, Voorhees, et. al. epitaxial thin films: Gao and Nix, Spencer and Meiron, Spencer and Tersoff, Chambolle, Braides, Bonnetier, Solci, F., Fusco, Leoni, Morini anisotropic surface energies: Herring, Taylor, Ambrosio, Novaga, and Paolini, Fonseca and Müller, Morgan mismatch strain (at which minimum energy is attained) { e0 i i if y 0, E 0 (y) = 0 if y < 0, e 0 > 0 i the unit vector along the x direction elastic energy per unit area: W (E E 0 (y)) W (E) := 1 2 E C E, E(u) := 1 2 ( u + ( u)t ) C... positive definite fourth-order tensor film and substrate have similar material properties, share the same homogeneous elasticity tensor C rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, by the Pisa, National September Science 2016 Foundation 9 / 51(NS

19 { γfilm if y > 0, ψ (y) := γ sub if y = 0. Total energy of the system: F (u, h) := W (E(u)(x, y) E 0 (y)) dx + Ω h ψ (y) dh 1 (x), Γ h Γ h := Ω h ((0, b) R)... free surface of the film Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 10 / 51(NS

20 Hard to Implement... Sharp interface model is difficult to be implemented numerically Instead: boundary-layer model; discontinuous transition is regularized over a thin transition region of width δ ( smearing parameter ) E δ (y) := 1 2 e 0 ( ( y )) 1 + f i i, δ y R ψ δ (y) := γ sub + (γ film γ sub ) f ( y δ ), y 0 f (0) = 0, lim f (y) = 1, lim y f (y) = 1 y Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 11 / 51(NS

21 smooth transition total energy of the system: F δ (u, h) := W (E(u)(x, y) E δ (y)) dx + Ω h ψ δ (y) dh 1 (x) Γ h or Dirichlet B.C. u(x, 0) = (e 0 x, 0)... mismatch F (u, h) := W (E(u)) dx + Ω h ψ (y) dh 1 (x) Γ h Two regimes: { γfilm γ sub γ film < γ sub Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 12 / 51(NS

22 Wetting, etc. asymptotics as δ 0 + / relaxation γ film < γ sub relaxed surface energy density is no longer discontinuous: it is constantly equal to γ film... WETTING! F rel (u, h) = W (E (u)) dx + γ film length Γ h Ω h more favorable to cover the substrate with an infinitesimal layer of film atoms (and pay surface energy with density γ film ) rather than to leave any part of the substrate exposed (and pay surface energy with density γ sub ) wetting regime: regularity of local minimizers (u, Ω) of the limiting functional F rel under a volume constraint Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 13 / 51(NS

23 Wetting, etc. asymptotics as δ 0 + / relaxation γ film < γ sub relaxed surface energy density is no longer discontinuous: it is constantly equal to γ film... WETTING! F rel (u, h) = W (E (u)) dx + γ film length Γ h Ω h more favorable to cover the substrate with an infinitesimal layer of film atoms (and pay surface energy with density γ film ) rather than to leave any part of the substrate exposed (and pay surface energy with density γ sub ) wetting regime: regularity of local minimizers (u, Ω) of the limiting functional F rel under a volume constraint Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 13 / 51(NS

24 Wetting, etc. asymptotics as δ 0 + / relaxation γ film < γ sub relaxed surface energy density is no longer discontinuous: it is constantly equal to γ film... WETTING! F rel (u, h) = W (E (u)) dx + γ film length Γ h Ω h more favorable to cover the substrate with an infinitesimal layer of film atoms (and pay surface energy with density γ film ) rather than to leave any part of the substrate exposed (and pay surface energy with density γ sub ) wetting regime: regularity of local minimizers (u, Ω) of the limiting functional F rel under a volume constraint Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 13 / 51(NS

25 Relaxation/Existence: Bonnetier & Chambolle ( 03), F., Fusco, Leoni, & Morini ( 07) 2-D, Braides, Chambolle, & Solci ( 07), Chambolle & Solci ( 07) 3-D, etc. Minimality of flat configuration: Fusco & Morini ( 10) 2-D, isotropic, Bonacini ( 13, 15) 2-D and 3-D anisotropic, etc. Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 14 / 51(NS

26 Regularity: Cusps and Vertical Cuts The profile h of the film for a locally minimizing configuration is regular except for at most a finite number of cusps and vertical cuts which correspond to vertical cracks in the film [Spencer and Meiron]: steady state solutions exhibit cusp singularities, time-dependent evolution of small disturbances of the flat interface result in the formation of deep grooved cusps (also [Chiu and Gao]); experimental validation of sharp cusplike features in SI 0.6 Ge 0.4 Regularity: optimal h is analytic except for a finite number of cusps and jump points, F., Fusco, Leoni, & Morini ( 07), F., Fusco, Leoni, & Millot ( 11) anisotropic surface energy γ film length Γ h Γ h ψ(ν) dh 1, ν normal to Γ h, zero contact-angle condition between the wetting layer and islands vertical slope cusp contact angle =zero Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 15 / 51(NS

27 Regularity: Cusps and Vertical Cuts The profile h of the film for a locally minimizing configuration is regular except for at most a finite number of cusps and vertical cuts which correspond to vertical cracks in the film [Spencer and Meiron]: steady state solutions exhibit cusp singularities, time-dependent evolution of small disturbances of the flat interface result in the formation of deep grooved cusps (also [Chiu and Gao]); experimental validation of sharp cusplike features in SI 0.6 Ge 0.4 Regularity: optimal h is analytic except for a finite number of cusps and jump points, F., Fusco, Leoni, & Morini ( 07), F., Fusco, Leoni, & Millot ( 11) anisotropic surface energy γ film length Γ h Γ h ψ(ν) dh 1, ν normal to Γ h, zero contact-angle condition between the wetting layer and islands vertical slope cusp contact angle =zero Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 15 / 51(NS

28 Regularity... conclude that the graph of h is a Lipschitz continuous curve away from a finite number of singular points (cusps, vertical cuts)... and more: Lipschitz continuity of h +blow up argument+classical results on corner domains for solutions of Lamé systems of h decay estimate for the gradient of the displacement u near the boundary C 1,α regularity of h and u; bootstrap... this leads us to linearly isotropic materials Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation / 51(NS

29 Regularity... conclude that the graph of h is a Lipschitz continuous curve away from a finite number of singular points (cusps, vertical cuts)... and more: Lipschitz continuity of h +blow up argument+classical results on corner domains for solutions of Lamé systems of h decay estimate for the gradient of the displacement u near the boundary C 1,α regularity of h and u; bootstrap... this leads us to linearly isotropic materials rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation / 51(NS

30 Regularity... conclude that the graph of h is a Lipschitz continuous curve away from a finite number of singular points (cusps, vertical cuts)... and more: Lipschitz continuity of h +blow up argument+classical results on corner domains for solutions of Lamé systems of h decay estimate for the gradient of the displacement u near the boundary C 1,α regularity of h and u; bootstrap... this leads us to linearly isotropic materials rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation / 51(NS

31 Linearly Isotropic Elastic Materials W (E) = 1 2 λ [tr (E)]2 + µ tr ( E 2) λ and µ are the (constant) Lamé moduli µ > 0, µ + λ > 0. Euler-Lagrange system of equations associated to W µ u + (λ + µ) (div u) = 0 in Ω. Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 17 / 51(NS

32 Regularity of Γ: No Corners Already know that Γ sing is finite Γ sing := Γ cusps {(x, h(x)) : h(x) < h (x)} Theorem (u, Ω) X... local minimizer for the functional F reg. Then Γ \ Γ sing is of class C 1,α for all 0 < α < 1 2. If z 0 = (x 0, 0) Γ \ Γ sing then h (x 0 ) = 0. Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 18 / 51(NS

33 Regularity of Γ: No Corners Already know that Γ sing is finite Γ sing := Γ cusps {(x, h(x)) : h(x) < h (x)} Theorem (u, Ω) X... local minimizer for the functional F reg. Then Γ \ Γ sing is of class C 1,α for all 0 < α < 1 2. If z 0 = (x 0, 0) Γ \ Γ sing then h (x 0 ) = 0. Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 18 / 51(NS

34 Shapes of Islands [With A. Pratelli and B. Zwicknagl] We proved that the shape of the island evolves with the size (and size varies with misfit!... later... ): small islands always have the half-pyramid shape, and as the volume increases the island evolves through a sequence of shapes that include more facets with increasing steepness half pyramid, pyramid, half dome, dome, half barn, barn This validates what was experimentally and numerically obtained in the physics and materials science literature Scaling laws and shapes of islands: Fonseca, Pratelli, & Zwicknagl ( 14), Goldman & Zwicknagl ( 14), Bella, Goldman, & Zwicknagl ( 15), etc. Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 19 / 51(NS

35 Shapes of Islands [With A. Pratelli and B. Zwicknagl] We proved that the shape of the island evolves with the size (and size varies with misfit!... later... ): small islands always have the half-pyramid shape, and as the volume increases the island evolves through a sequence of shapes that include more facets with increasing steepness half pyramid, pyramid, half dome, dome, half barn, barn This validates what was experimentally and numerically obtained in the physics and materials science literature Scaling laws and shapes of islands: Fonseca, Pratelli, & Zwicknagl ( 14), Goldman & Zwicknagl ( 14), Bella, Goldman, & Zwicknagl ( 15), etc. Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 19 / 51(NS

36 Another Incompatibility: Miscut Small slope approximation of a geometrically linear elastic strain energy ([Tersoff & Tromp, 1992; Spencer & Tersoff, 2010]) fully facetted model: E(u) L L 0 height profile u, supp(u) = [0, L] 0 log x y u (x)u (y) dydx + length(graph(u)) L, u A := {tan( θ m + nθ) : n N Z} θ m describes miscut. If θ m 0, wetting not admissible Figure: Sketch of a faceted height profile function u with support [0, L]. The profile is Lipschitz and the derivative lies almost everywhere in a discrete set. The miscut angle is denoted by θm 0, i.e., the preferred orientation of the film is not parallel to the substrate surface. Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation / 51(NS

37 Compactness: Bounds on the Support of u F(d) := inf{e(u) : u = d} Theorem For every d, r > 0 there exists L such that if E(u) F(d) + r, then L L If d 0 and r 0, then L 0 no wetting effect for small volumes; wetting optimal profiles tend to be extremely large and flat when the mass is small. The flat profile is not admissible Theorem Every minimizer satisfies the quantized zero contact angle property: the island meets the substrate at the smallest angle possible There is a volume d > 0 such that the half pyramid is the unique minimizer for every d (0, d) Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 21 / 51(NS

38 barn half barn dome half dome pyramid half pyramid Figure: Shape transitions with increasing volume at miscut angle 3. Numerical simulation. Courtesy of B. Spencer and J. Tersoff, Appl. Phys. Lett. bf 96/7, (2010) rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 22 / 51(NS

39 Surface Diffusion in Epitaxially Strained Solids [With N. Fusco, G. Leoni, M. Morini] Einstein-Nernst Law : surface flux of atoms Γ µ µ= chemical potential V = c Γ(t) µ (volume preserving) }{{} Laplace-Beltrami operator µ= first variation of energy = div Γ Dψ(ν) }{{} +W (E(u)) + λ anisotropic curvature ) V = Γ (div Γ Dψ(ν) + W (E(u)) Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 23 / 51(NS

40 Surface Diffusion in Epitaxially Strained Solids [With N. Fusco, G. Leoni, M. Morini] Einstein-Nernst Law : surface flux of atoms Γ µ µ= chemical potential V = c Γ(t) µ (volume preserving) }{{} Laplace-Beltrami operator µ= first variation of energy = div Γ Dψ(ν) }{{} +W (E(u)) + λ anisotropic curvature ) V = Γ (div Γ Dψ(ν) + W (E(u)) rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 23 / 51(NS

41 Surface Diffusion in Epitaxially Strained Solids [With N. Fusco, G. Leoni, M. Morini] Einstein-Nernst Law : surface flux of atoms Γ µ µ= chemical potential V = c Γ(t) µ (volume preserving) }{{} Laplace-Beltrami operator µ= first variation of energy = div Γ Dψ(ν) }{{} +W (E(u)) + λ anisotropic curvature ) V = Γ (div Γ Dψ(ν) + W (E(u)) rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 23 / 51(NS

42 Surface Diffusion in Epitaxially Strained Solids [With N. Fusco, G. Leoni, M. Morini] Einstein-Nernst Law : surface flux of atoms Γ µ µ= chemical potential V = c Γ(t) µ (volume preserving) }{{} Laplace-Beltrami operator µ= first variation of energy = div Γ Dψ(ν) }{{} +W (E(u)) + λ anisotropic curvature ) V = Γ (div Γ Dψ(ν) + W (E(u)) Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 23 / 51(NS

43 Surface Diffusion in Epitaxially Strained Solids [With N. Fusco, G. Leoni, M. Morini] Einstein-Nernst Law : surface flux of atoms Γ µ µ= chemical potential V = c Γ(t) µ (volume preserving) }{{} Laplace-Beltrami operator µ= first variation of energy = div Γ Dψ(ν) }{{} +W (E(u)) + λ anisotropic curvature ) V = Γ (div Γ Dψ(ν) + W (E(u)) Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 23 / 51(NS

44 Surface Diffusion in Epitaxially Strained Solids [With N. Fusco, G. Leoni, M. Morini] Einstein-Nernst Law : surface flux of atoms Γ µ µ= chemical potential V = c Γ(t) µ (volume preserving) }{{} Laplace-Beltrami operator µ= first variation of energy = div Γ Dψ(ν) }{{} +W (E(u)) + λ anisotropic curvature ) V = Γ (div Γ Dψ(ν) + W (E(u)) Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 23 / 51(NS

45 Surface Diffusion in Epitaxially Strained Solids [With N. Fusco, G. Leoni, M. Morini] Einstein-Nernst Law : surface flux of atoms Γ µ µ= chemical potential V = c Γ(t) µ (volume preserving) }{{} Laplace-Beltrami operator µ= first variation of energy = div Γ Dψ(ν) }{{} +W (E(u)) + λ anisotropic curvature ) V = Γ (div Γ Dψ(ν) + W (E(u)) Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 23 / 51(NS

46 Highly Anisotropic Surface Energies For highly anisotropic ψ it may happen D 2 ψ(ν)[τ, τ] < 0 for some τ ν the evolution becomes backward parabolic Idea: add a curvature regularization Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 24 / 51(NS

47 Highly Anisotropic Surface Energies For highly anisotropic ψ it may happen D 2 ψ(ν)[τ, τ] < 0 for some τ ν the evolution becomes backward parabolic Idea: add a curvature regularization Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 24 / 51(NS

48 Highly Anisotropic Surface Energies For highly anisotropic ψ it may happen D 2 ψ(ν)[τ, τ] < 0 for some τ ν the evolution becomes backward parabolic Idea: add a curvature regularization Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 24 / 51(NS

49 Highly Anisotropic Surface Energies For highly anisotropic ψ it may happen D 2 ψ(ν)[τ, τ] < 0 for some τ ν the evolution becomes backward parabolic Idea: add a curvature regularization ( ε F (u, h) := W (E(u)) dx + ψ(ν) + Ω h Γ h p H p) dh 1, p > 2, ε > 0 Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 24 / 51(NS

50 Highly Anisotropic Surface Energies For highly anisotropic ψ it may happen D 2 ψ(ν)[τ, τ] < 0 for some τ ν the evolution becomes backward parabolic Idea: add a curvature regularization ( ε F (u, h) := W (E(u)) dx + ψ(ν) + Ω h Γ h p H p) dh 1, p > 2, ε > 0 Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 24 / 51(NS

51 Highly Anisotropic Surface Energies For highly anisotropic ψ it may happen D 2 ψ(ν)[τ, τ] < 0 for some τ ν the evolution becomes backward parabolic Idea: add a curvature regularization ( ε F (u, h) := W (E(u)) dx + ψ(ν) + Ω h Γ h p H p) dh 1, p > 2, ε > 0 V = Γ [div Γ (Dψ(ν)) + W (E(u)) ( ε Γ ( H p 2 H) H p 2 H (κ κ p H2))] Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 24 / 51(NS

52 Highly Anisotropic Surface Energies in 2D Regularized energy: ( ε F (u, h) := W (E(u)) dx + ψ(ν) + Ω h Γ h 2 k2) dh 1 V = ( ) div σ Dψ(ν) + W (E(u)) ε(k σσ k3 ) σσ F., Fusco, Leoni, and Morini (ARMA 2012): evolution of films in 2D F., Fusco, Leoni, and Morini (Analysis & PDE 2015): evolution of films in 3D Gradient flow: Cahn & Taylor ( 94), in L 2 : Piovano ( 14) Short time existence and regularity in H 1 : Siegel, Miksis & Voorhees ( 04), F., Fusco, Leoni, & Morini ( 12) Short time existence and regularity Why here p > 2: technical... in two dimensions, the Sobolev space W 2,p embeds into C 1,(p 2)/p if p > 2 Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 25 / 51(NS

53 The Evolution Law Curvature dependent energies Herring (1951) In the context of grain growth, curvature regularization was proposed by Di Carlo, Gurtin, Podio-Guidugli (1992) In the context of epitaxial growth, Gurtin & Jabbour (2002) Given Q, find h: R 2 [0, T 0 ] (0, + ) s.t. [ 1 h J t = Γ div Γ (Dψ(ν)) + W (E(u)) ( ( ε Γ ( H p 2 H) H p 2 H κ κ p H2))], in R 2 (0, T 0 ) div CE(u) = 0 in Ω h CE(u)[ν] = 0 on Γ h, u(x, 0, t) = e 0 (x, 0) h(, t) and Du(, t) are Q-periodic h(, 0) = h 0 Here J := 1 + Dh 2 Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 26 / 51(NS

54 The Evolution Law Curvature dependent energies Herring (1951) In the context of grain growth, curvature regularization was proposed by Di Carlo, Gurtin, Podio-Guidugli (1992) In the context of epitaxial growth, Gurtin & Jabbour (2002) Given Q, find h: R 2 [0, T 0 ] (0, + ) s.t. [ 1 h J t = Γ div Γ (Dψ(ν)) + W (E(u)) ( ( ε Γ ( H p 2 H) H p 2 H κ κ p H2))], in R 2 (0, T 0 ) div CE(u) = 0 in Ω h CE(u)[ν] = 0 on Γ h, u(x, 0, t) = e 0 (x, 0) h(, t) and Du(, t) are Q-periodic h(, 0) = h 0 Here J := 1 + Dh 2 rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 26 / 51(NS

55 The Evolution Law Curvature dependent energies Herring (1951) In the context of grain growth, curvature regularization was proposed by Di Carlo, Gurtin, Podio-Guidugli (1992) In the context of epitaxial growth, Gurtin & Jabbour (2002) Given Q, find h: R 2 [0, T 0 ] (0, + ) s.t. [ 1 h J t = Γ div Γ (Dψ(ν)) + W (E(u)) ( ( ε Γ ( H p 2 H) H p 2 H κ κ p H2))], in R 2 (0, T 0 ) div CE(u) = 0 in Ω h CE(u)[ν] = 0 on Γ h, u(x, 0, t) = e 0 (x, 0) h(, t) and Du(, t) are Q-periodic h(, 0) = h 0 Here J := 1 + Dh 2 rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 26 / 51(NS

56 The Evolution Law Curvature dependent energies Herring (1951) In the context of grain growth, curvature regularization was proposed by Di Carlo, Gurtin, Podio-Guidugli (1992) In the context of epitaxial growth, Gurtin & Jabbour (2002) Given Q, find h: R 2 [0, T 0 ] (0, + ) s.t. [ 1 h J t = Γ div Γ (Dψ(ν)) + W (E(u)) ( ( ε Γ ( H p 2 H) H p 2 H κ κ p H2))], in R 2 (0, T 0 ) div CE(u) = 0 in Ω h CE(u)[ν] = 0 on Γ h, u(x, 0, t) = e 0 (x, 0) h(, t) and Du(, t) are Q-periodic h(, 0) = h 0 Here J := 1 + Dh 2 rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 26 / 51(NS

57 Some Related Results Siegel, Miksis, Voorhees (2004): numerical experiments in the case of evolving curves Rätz, Ribalta, Voigt (2006): numerical results for the diffuse interface version of the evolution Garcke: analytical results concerning some diffuse interface versions of the evolution equation Bellettini, Mantegazza, Novaga (2007): analytical results concerning the L 2 -gradient flow of higher order geometric functionals (without elasticity) Elliott & Garcke (1997), Escher, Mayer & Simonett (1998): existence results for the surface diffusion equation without elasticity and without curvature regularization, via semigroups techniques no analytical results for the sharp interface evolution with elasticity Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 27 / 51(NS

58 The Gradient Flow Structure Consider the reduced functional F (u, h) F (Ω h ) := F (u h, h) where u h is the elastic equilibrium in Ω h The evolution law is formally equivalent to g Ωh(t) (V, Ṽ ) = F (Ω h(t))[ṽ ] for all Ṽ T Ω h(t) M, where F (Ω h(t) )[Ṽ ] = first variation of F at Ω h(t) in the direction Ṽ. First observed by Cahn &Taylor (1994) in the context of surface diffusion Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 28 / 51(NS

59 The Gradient Flow Structure Consider the reduced functional F (u, h) F (Ω h ) := F (u h, h) where u h is the elastic equilibrium in Ω h The evolution law is formally equivalent to g Ωh(t) (V, Ṽ ) = F (Ω h(t))[ṽ ] for all Ṽ T Ω h(t) M, where F (Ω h(t) )[Ṽ ] = first variation of F at Ω h(t) in the direction Ṽ. First observed by Cahn &Taylor (1994) in the context of surface diffusion Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 28 / 51(NS

60 The Gradient Flow Structure Consider the reduced functional F (u, h) F (Ω h ) := F (u h, h) where u h is the elastic equilibrium in Ω h The evolution law is formally equivalent to g Ωh(t) (V, Ṽ ) = F (Ω h(t))[ṽ ] for all Ṽ T Ω h(t) M, where F (Ω h(t) )[Ṽ ] = first variation of F at Ω h(t) in the direction Ṽ. First observed by Cahn &Taylor (1994) in the context of surface diffusion Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 28 / 51(NS

61 Minimizing Movements Approach to Gradient Flows H Hilbert space F : H R, F of class C 1 { u = H F (u) u(0) = u 0 Semi-implicit time-discretization: Set w 0 := u 0 and let w i the solution to { min F (w) + 1 } w H 2τ w w i 1 2 H The discrete evolution converges to the continuous evolution as τ 0 This approach can be generalized to metric spaces De Giorgi s minimizing movements In the context of geometric flows Almgren-Taylor-Wang Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 29 / 51(NS

62 Minimizing Movements Approach to Gradient Flows H Hilbert space F : H R, F of class C 1 { u = H F (u) u(0) = u 0 Semi-implicit time-discretization: Set w 0 := u 0 and let w i the solution to { min F (w) + 1 } w H 2τ w w i 1 2 H The discrete evolution converges to the continuous evolution as τ 0 This approach can be generalized to metric spaces De Giorgi s minimizing movements In the context of geometric flows Almgren-Taylor-Wang Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 29 / 51(NS

63 Minimizing Movements Approach to Gradient Flows H Hilbert space F : H R, F of class C 1 { u = H F (u) u(0) = u 0 Semi-implicit time-discretization: Set w 0 := u 0 and let w i the solution to { min F (w) + 1 } w H 2τ w w i 1 2 H The discrete evolution converges to the continuous evolution as τ 0 This approach can be generalized to metric spaces De Giorgi s minimizing movements In the context of geometric flows Almgren-Taylor-Wang Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 29 / 51(NS

64 Minimizing Movements Approach to Gradient Flows H Hilbert space F : H R, F of class C 1 { u = H F (u) u(0) = u 0 Semi-implicit time-discretization: Set w 0 := u 0 and let w i the solution to { min F (w) + 1 } w H 2τ w w i 1 2 H The discrete evolution converges to the continuous evolution as τ 0 This approach can be generalized to metric spaces De Giorgi s minimizing movements In the context of geometric flows Almgren-Taylor-Wang Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 29 / 51(NS

65 Minimizing Movements Approach to Gradient Flows H Hilbert space F : H R, F of class C 1 { u = H F (u) u(0) = u 0 Semi-implicit time-discretization: Set w 0 := u 0 and let w i the solution to { min F (w) + 1 } w H 2τ w w i 1 2 H The discrete evolution converges to the continuous evolution as τ 0 This approach can be generalized to metric spaces De Giorgi s minimizing movements In the context of geometric flows Almgren-Taylor-Wang Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 29 / 51(NS

66 Minimizing Movements Approach to Gradient Flows H Hilbert space F : H R, F of class C 1 { u = H F (u) u(0) = u 0 Semi-implicit time-discretization: Set w 0 := u 0 and let w i the solution to { min F (w) + 1 } w H 2τ w w i 1 2 H The discrete evolution converges to the continuous evolution as τ 0 This approach can be generalized to metric spaces De Giorgi s minimizing movements In the context of geometric flows Almgren-Taylor-Wang Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 29 / 51(NS

67 The Minimizing Movements Scheme in Our Case Given T > 0, N N, we set τ := T N. For i = 1,..., N we define inductively (h i, u i ) as the solution of the incremental minimum problem min (h, u) admissible Dh C F (h, u) + 1 D Γhi 1 w h 2 dh 2 2τ Γ hi 1 Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 30 / 51(NS

68 The Minimizing Movements Scheme in Our Case Given T > 0, N N, we set τ := T N. For i = 1,..., N we define inductively (h i, u i ) as the solution of the incremental minimum problem min (h, u) admissible Dh C F (h, u) + 1 D Γhi 1 w h 2 dh 2 2τ Γ hi 1 rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 30 / 51(NS

69 The Minimizing Movements Scheme in Our Case Given T > 0, N N, we set τ := T N. For i = 1,..., N we define inductively (h i, u i ) as the solution of the incremental minimum problem min (h, u) admissible Dh C F (h, u) + 1 D Γhi 1 w h 2 dh 2 2τ Γ hi 1 rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 30 / 51(NS

70 The Minimizing Movements Scheme in Our Case Given T > 0, N N, we set τ := T N. For i = 1,..., N we define inductively (h i, u i ) as the solution of the incremental minimum problem min (h, u) admissible Dh C F (h, u) + 1 D Γhi 1 w h 2 dh 2 2τ Γ hi 1 where h h Γhi 1 w h = i Dhi 1 =: V 2 Ω h, w h dh 2 = 0. Γ hi 1 rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 30 / 51(NS

71 The Minimizing Movements Scheme in Our Case Given T > 0, N N, we set τ := T N. For i = 1,..., N we define inductively (h i, u i ) as the solution of the incremental minimum problem min (h, u) admissible Dh C F (h, u) + 1 D Γhi 1 w h 2 dh 2 2τ Γ hi 1 where h h Γhi 1 w h = i Dhi 1 =: V 2 Ω h, w h dh 2 = 0. Γ hi 1 1 D Γhi 1 w h 2 dh 2 h h i 1 2 H 2τ 1 (Γ hi 1 ) Γ hi 1 rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 30 / 51(NS

72 The Minimizing Movements Scheme in Our Case Given T > 0, N N, we set τ := T N. For i = 1,..., N we define inductively (h i, u i ) as the solution of the incremental minimum problem min (h, u) admissible Dh C F (h, u) + 1 D Γhi 1 w h 2 dh 2 2τ Γ hi 1 where h h Γhi 1 w h = i Dhi 1 =: V 2 Ω h, w h dh 2 = 0. Γ hi 1 1 D Γhi 1 w h 2 dh 2 h h i 1 2 H 2τ 1 (Γ hi 1 ) Γ hi 1 rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 30 / 51(NS

73 The Discrete Euler-Lagrange Equation The Euler-Lagrange equation of the incremental problem is 1 τ w h i = div Γhi (Dψ(ν)) + W (E(u i )) ( ε ( Γhi ( H i p 2 H i ) H i p 2 H i (κ i 1) 2 + (κ i 2) 2 1 )) p H2 i By applying Γhi 1 to both sides, we formally get 1 h i h i 1 = Γhi 1 [div Γhi (Dψ(ν)) + W (E(u i )) J i 1 τ ( ε ( Γhi ( H i p 2 H i ) H i p 2 H i (κ i 1) 2 + (κ i 2) 2 1 p H2 i ))] this is a discrete version of the continuous evolution law Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 31 / 51(NS

74 The Discrete Euler-Lagrange Equation The Euler-Lagrange equation of the incremental problem is 1 τ w h i = div Γhi (Dψ(ν)) + W (E(u i )) ( ε ( Γhi ( H i p 2 H i ) H i p 2 H i (κ i 1) 2 + (κ i 2) 2 1 )) p H2 i By applying Γhi 1 to both sides, we formally get 1 h i h i 1 = Γhi 1 [div Γhi (Dψ(ν)) + W (E(u i )) J i 1 τ ( ε ( Γhi ( H i p 2 H i ) H i p 2 H i (κ i 1) 2 + (κ i 2) 2 1 p H2 i ))] this is a discrete version of the continuous evolution law Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 31 / 51(NS

75 The Discrete Euler-Lagrange Equation The Euler-Lagrange equation of the incremental problem is 1 τ w h i = div Γhi (Dψ(ν)) + W (E(u i )) ( ε ( Γhi ( H i p 2 H i ) H i p 2 H i (κ i 1) 2 + (κ i 2) 2 1 )) p H2 i By applying Γhi 1 to both sides, we formally get 1 h i h i 1 = Γhi 1 [div Γhi (Dψ(ν)) + W (E(u i )) J i 1 τ ( ε ( Γhi ( H i p 2 H i ) H i p 2 H i (κ i 1) 2 + (κ i 2) 2 1 p H2 i ))] this is a discrete version of the continuous evolution law Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 31 / 51(NS

76 Local in Time Existence of Weak Solutions Previous estimates+interpolation inequalities+higher regularity+ compactnes argument h N h (up to a subsequence) h is a weak solution in the following sense: Theorem (Local existence) h L (0, T 0 ; W 2,p # (Q)) H1 (0, T 0 ; H 1 # (Q)) is a weak solution in [0, T 0] in the following sense: ) (i) div Γ (Dψ(ν)) + W (E(u)) ε ( Γ ( H p 2 H) 1p H p H + H p 2 H B 2 L 2 (0, T 0 ; H 1 # (Q)), (ii) for a.e. t (0, T 0 ) [ 1 h J t = Γ div Γ (Dψ(ν)) + W (E(u)) ( ))] ε Γ ( H p 2 H) H p 2 H (κ κ 2 2 1p H2 in H 1 # (Q). Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 32 / 51(NS

77 Local in Time Existence of Weak Solutions Previous estimates+interpolation inequalities+higher regularity+ compactnes argument h N h (up to a subsequence) h is a weak solution in the following sense: Theorem (Local existence) h L (0, T 0 ; W 2,p # (Q)) H1 (0, T 0 ; H 1 # (Q)) is a weak solution in [0, T 0] in the following sense: ) (i) div Γ (Dψ(ν)) + W (E(u)) ε ( Γ ( H p 2 H) 1p H p H + H p 2 H B 2 L 2 (0, T 0 ; H 1 # (Q)), (ii) for a.e. t (0, T 0 ) [ 1 h J t = Γ div Γ (Dψ(ν)) + W (E(u)) ( ))] ε Γ ( H p 2 H) H p 2 H (κ κ 2 2 1p H2 in H 1 # (Q). Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 32 / 51(NS

78 Local in Time Existence of Weak Solutions Previous estimates+interpolation inequalities+higher regularity+ compactnes argument h N h (up to a subsequence) h is a weak solution in the following sense: Theorem (Local existence) h L (0, T 0 ; W 2,p # (Q)) H1 (0, T 0 ; H 1 # (Q)) is a weak solution in [0, T 0] in the following sense: ) (i) div Γ (Dψ(ν)) + W (E(u)) ε ( Γ ( H p 2 H) 1p H p H + H p 2 H B 2 L 2 (0, T 0 ; H 1 # (Q)), (ii) for a.e. t (0, T 0 ) [ 1 h J t = Γ div Γ (Dψ(ν)) + W (E(u)) ( ))] ε Γ ( H p 2 H) H p 2 H (κ κ 2 2 1p H2 in H 1 # (Q). Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 32 / 51(NS

79 Uniqueness and regularity in 2D Theorem In two dimensions: (i) The weak solution is unique. (ii) If h 0 H 3, ψ C 4, then the solution is in H 1 (0, T 0 ; L 2 ) L 2 (0, T 0 ; H 6 ). Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 33 / 51(NS

80 Uniqueness and regularity in 2D Theorem In two dimensions: (i) The weak solution is unique. (ii) If h 0 H 3, ψ C 4, then the solution is in H 1 (0, T 0 ; L 2 ) L 2 (0, T 0 ; H 6 ). rene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 33 / 51(NS

81 Global in Time Existence and Asymptotic Stability Consider the regularized surface diffusion equation [ 1 h J t = Γ div Γ (Dψ(ν)) + W (E(u)) ( ε Γ ( H p 2 H) H p 2 H (κ κ p H2))] Detailed analysis of Asaro-Tiller-Grinfeld morphological stability/instability by Bonacini, and F., Fusco, Leoni and Morini: if d is sufficiently small, then the flat configuration (d, u d ) is a volume constrained local minimizer for the functional G(h, u) := W (E(u)) dz + Ω h ψ(ν) dh 2. Γ h d small enough the second variation 2 G(d, u d ) is positive definite local minimality property. Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 34 / 51(NS

82 Global in Time Existence and Asymptotic Stability Main Result Theorem Assume that D 2 ψ(e 3 ) > 0 on (e 3 ) and 2 G(d, u 0 ) > 0. There exists ε > 0 s.t. if h 0 d W 2,p ε and h 0 = d, then: (i) any variational solution h exists for all times; (ii) h(, t) d in W 2,p as t +. Q Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 35 / 51(NS

83 Liapunov Stability in the Highly Non-Convex Case Consider the Wulff shape W ψ := {z R 3 : z ν < ψ(ν) for all ν S 2 } Theorem (F.-Fusco-Leoni-Morini) Assume that W ψ contains a horizontal facet. Then for every d > 0 the flat configuration (d, u d ) is Liapunov stable, that is, for every σ > 0 there exists δ(σ) > 0 s.t. Q h 0 = d, h 0 d W 2,p δ(σ) = h(t) d W 2,p σ for all t > 0. Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 36 / 51(NS

84 Liapunov Stability in the Highly Non-Convex Case Consider the Wulff shape W ψ := {z R 3 : z ν < ψ(ν) for all ν S 2 } Theorem (F.-Fusco-Leoni-Morini) Assume that W ψ contains a horizontal facet. Then for every d > 0 the flat configuration (d, u d ) is Liapunov stable, that is, for every σ > 0 there exists δ(σ) > 0 s.t. Q h 0 = d, h 0 d W 2,p δ(σ) = h(t) d W 2,p σ for all t > 0. Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 36 / 51(NS

85 And Now... Epitaxy and Dislocations lattice-mismatched semiconductors formation of a periodic dislocation network at the substrate/layer interface nucleation of dislocations is a mode of strain relief for sufficiently thick films when a cusp-like morphology is approached as the result of an increasingly greater stress in surface valleys, it is energetically favorable to nucleate a dislocation in the surface valley dislocations migrate to the film/substrate interface and the film surface relaxes towards a planar-like morphology. Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 37 / 51(NS

86 And Now... Epitaxy and Dislocations lattice-mismatched semiconductors formation of a periodic dislocation network at the substrate/layer interface nucleation of dislocations is a mode of strain relief for sufficiently thick films when a cusp-like morphology is approached as the result of an increasingly greater stress in surface valleys, it is energetically favorable to nucleate a dislocation in the surface valley dislocations migrate to the film/substrate interface and the film surface relaxes towards a planar-like morphology. Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 37 / 51(NS

87 Nucleation of dislocations - Line defects in crystalline materials Orowan ( 34), Polanyi ( 34), Taylor ( 34), Frank ( 49) Figure: Courtesy of Elder & al. ( 07) Dislocations in epitaxial growth: Matthews & Blakeslee ( 74) Roux & Van der Merwe ( 79) Dong, Schnitker, Smith & Srolovitz ( 98) Gao & Nix ( 99) Haataja, Muller, Rutenberg & Grant ( 02) Sreekala & Haataja ( 07),... Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 38 / 51(NS

88 Microscopic Level Perfect crystals Figure: Courtesy of James Hedberg Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 39 / 51(NS

89 Microscopic Level Defects in crystalline materials Figure: Courtesy of Helmut Föll Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 40 / 51(NS

90 Microscopic Level Line defects in crystalline materials. Orowon (1934); Polanyi (1934), Taylor (1934). Figure: Courtesy of NTD Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 41 / 51(NS

91 Microscopic Level Edge dislocations, Burgers vector, Burgers (1939) Dislocation line Figure: Courtesy of J. W. Morris, Jr Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 42 / 51(NS

92 Microscopic Level Edge dislocations, Burgers vector, Burgers (1939) Dislocation line Figure: Courtesy of J. W. Morris, Jr Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 42 / 51(NS

93 Microscopic Level Edge dislocations, Burgers vector, Burgers (1939) Dislocation line Figure: Courtesy of J. W. Morris, Jr Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 42 / 51(NS

94 Microscopic Level Screw dislocations Burgers vector Figure: Courtesy of Helmut Föll Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 43 / 51(NS

95 Microscopic Level Screw dislocations Burgers vector Figure: Courtesy of Helmut Föll Irene Fonseca ( Department of Mathematical Sciences Center for Nonlinear EpitaxyAnalysis and Dislocations Carnegie Mellon University Supported De Giorgi, bypisa, the National September Science 2016Foundation 43 / 51(NS