Phase field models for step flow.

The relation between phase field and discontinuous models for crystal steps is analyzed. Different formulations of the kinetic boundary conditions of the discontinuous model are first presented. We show that (i) step transparency, usually interpreted as the possibility for adatoms to jump through steps, may be seen as a modification of the equilibrium concentration engendered by step motion. (ii) The interface definition (i.e., the position of the dividing line) intervenes in the expression of the kinetic coefficients only in the case of fast attachment kinetics. (iii) We also identify the thermodynamically consistent reference state for kinetic boundary conditions. Asymptotic expansions of the phase field models in the limit where the interface width is small, lead to various discontinuous models. (1) A phase field model with one global concentration field and variable mobility is shown to lead to a discontinuous model with fast step kinetics. (2) A phase field model with one concentration field per terrace allows one to recover arbitrary step kinetics (i.e., arbitrarily strong Ehrlich-Schwoebel effect and step transparency). Quantitative agreement is found, in both the linear and nonlinear regimes, between the numerical solution of the phase field models and the analytical solution of the discontinuous model.

[1]  A. Karma,et al.  Quantitative phase-field modeling of dendritic growth in two and three dimensions , 1996 .

[2]  Yukio Saito,et al.  New Nonlinear Evolution Equation for Steps during Molecular Beam Epitaxy on Vicinal Surfaces , 1998 .

[3]  O. Pierre-Louis Step bunching with general step kinetics: stability analysis and macroscopic models , 2003 .

[4]  J. Métois,et al.  Impact of the growth on the stability–instability transition at Si (111) during step bunching induced by electromigration , 1999 .

[5]  A. Karma,et al.  Step motions on high-temperature vicinal surfaces , 1994 .

[6]  A. Karma Phase-field formulation for quantitative modeling of alloy solidification. , 2001, Physical review letters.

[7]  Zangwill,et al.  Morphological equilibration of a facetted crystal. , 1992, Physical review. B, Condensed matter.

[8]  B. Merriman,et al.  Kinetic model for a step edge in epitaxial growth. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  Liu,et al.  Stability and kinetics of step motion on crystal surfaces. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  W. K. Burton,et al.  The growth of crystals and the equilibrium structure of their surfaces , 1951, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[11]  G. McFadden,et al.  Morphological instability in phase-field models of solidification. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[12]  G. Caginalp,et al.  Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations. , 1989, Physical review. A, General physics.

[13]  Misbah,et al.  Nonlinear evolution of a terrace edge during step-flow growth. , 1993, Physical review. B, Condensed matter.

[14]  N. Bartelt,et al.  Step Permeability and the Relaxation of Biperiodic Gratings on Si(001) , 1997 .

[15]  J. Langer,et al.  Theory of departure from local equilibrium at the interface of a two-phase diffusion couple , 1975 .

[16]  Eshel Ben-Jacob,et al.  Dynamics of Interfacial Pattern Formation , 1983 .

[17]  Joachim Krug,et al.  Island nucleation in the presence of step-edge barriers: Theory and applications , 1999, cond-mat/9912410.

[18]  Alain Karma,et al.  Spiral Surface Growth without Desorption , 1998, cond-mat/9809358.

[19]  O Pierre-Louis Continuum model for low temperature relaxation of crystal steps. , 2001, Physical review letters.

[20]  Zangwill,et al.  Morphological instability of a terrace edge during step-flow growth. , 1990, Physical review. B, Condensed matter.

[21]  A. Saúl,et al.  Experimental evidence for an Ehrlich-Schwoebel effect on Si(111) , 2002 .

[22]  C. Trautmann,et al.  Swelling effects in lithium fluoride induced by swift heavy ions , 2000 .

[23]  F. Hudda,et al.  Atomic View of Surface Self‐Diffusion: Tungsten on Tungsten , 1966 .

[24]  Maria R. D'Orsogna,et al.  The Kink Ehrlich-Schwoebel Effect and Resulting Instabilities , 1999 .

[25]  Karma,et al.  Numerical Simulation of Three-Dimensional Dendritic Growth. , 1996, Physical review letters.

[26]  C. Misbah,et al.  Dynamics and fluctuations during MBE on vicinal surfaces. I. Formalism and results of linear theory , 1998 .