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A scaling law for a model of epitaxially strained elastic films with dislocations

Lukas Abel, Janusz Ginster, Barbara Zwicknagl

Abstract

A static variational model for shape formation in heteroepitaxial crystal growth is considered. The energy functional takes into account surface energy, elastic misfit-energy and nucleation energy of dislocations. A scaling law for the infimal energy is proven. The results quantify the expectation that in certain parameter regimes, island formation or topological defects are favorable. This generalizes results in the purely elastic setting from [Goldman and Zwicknagl (2014)]. To handle dislocations in the lower bound, a new variant of a ball-construction combined with thorough local estimates is presented.

A scaling law for a model of epitaxially strained elastic films with dislocations

Abstract

A static variational model for shape formation in heteroepitaxial crystal growth is considered. The energy functional takes into account surface energy, elastic misfit-energy and nucleation energy of dislocations. A scaling law for the infimal energy is proven. The results quantify the expectation that in certain parameter regimes, island formation or topological defects are favorable. This generalizes results in the purely elastic setting from [Goldman and Zwicknagl (2014)]. To handle dislocations in the lower bound, a new variant of a ball-construction combined with thorough local estimates is presented.
Paper Structure (9 sections, 10 theorems, 112 equations, 7 figures)

This paper contains 9 sections, 10 theorems, 112 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. The model
  4. Presence of Dislocations in Flat Films
  5. Heuristics of the proof
  6. Upper Bound
  7. Lower Bound
  8. Preliminaries
  9. Proof of the lower bound in Theorem \ref{['thm: main_intro']}

Key Result

Theorem 1.1

There is a constant $c_s>0$ with the following property: For all $\gamma, {e_0},b,{d}>0$ and $r_0\in (0,1]$ with $b/{e_0} \geq 64^4 r_0$ there holds where $s(\gamma, e_0,b,d,r_0) = \gamma(1 + {d}) + \min\left\{\gamma^{2/3}{e_0}^{2/3} {d}^{2/3}, \left[\gamma {e_0}b{d}\left(1+\log\left(\frac{b}{{e_0}r_0}\right)\right)\right]^{1/2}\right\}$.

Figures (7)

  • Figure 1: sketch of local length scales $\ell_i$ with (gray-shaded) local volumes $d_i$
  • Figure 2: profile $h$ with indicated dislocations for the upper bound of the scaling law (not to scale)
  • Figure 3: detail of the construction of the displacement $u_1$, the greyscale represents $u_1(x,y)$, white represents $u_1(x,y)=0$, black represents $u_1(x,y)=b$.
  • Figure 4: possible configuration for Lemma \ref{['isoperineq2']}
  • Figure 5: sketch of ball construction for five balls with equal starting radii
  • ...and 2 more figures

Theorems & Definitions (22)

  • Theorem 1.1: Scaling Law
  • proof
  • Proposition 2.1
  • Remark 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 12 more