A Step-Flow Model for the Heteroepitaxial Growth of Strained, Substitutional, Binary Alloy Films with Phase Segregation: I. Theory

We develop a step‐flow model for the heteroepitaxy of a generic, strained, substitutional, binary alloy. The underlying theory is based on the fundamental principles of modern continuum thermodynamics. In order to resolve the inherent disparity in the spatial scales—continuous in the lateral directions vs. atomistically discrete along the epitaxial axis—we represent the film as a layered structure, with the layer height equal to the lattice parameter along the growth direction, thus extending the classical BCF framework [W. K. Burton, N. Cabrera, and F. C. Frank, Philos. Trans. Roy. Soc. London Ser. A, 243 (1951), pp. 299–358] to growth situations in which the bulk behavior impacts the surface evolution. Our discrete‐continuum model takes the form of a free‐boundary problem for the evolution of monoatomic steps on a vicinal surface, in which interfacial effects on the terraces and along the step edges couple to their bulk counterparts (i.e., within both film and, indirectly, substrate). In particular, the...

[1]  Morton E. Gurtin,et al.  The role of the configurational force balance in the nonequilibrium epitaxy of films , 2003 .

[2]  W. Lu,et al.  Self-organized nanostructures in multi-phase epilayers , 2004 .

[3]  J. Cahn,et al.  An invariant formulation of multicomponent diffusion in crystals , 1983 .

[4]  Morton E. Gurtin,et al.  The nature of configurational forces , 1995 .

[5]  M. Gurtin.  Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance , 1996 .

[6]  W. Tiller,et al.  Interface morphology development during stress corrosion cracking: Part I. Via surface diffusion , 1972 .

[7]  Process of irreversible nucleation in multilayer growth. I. Failure of the mean-field approach. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  James K. Knowles,et al.  On the driving traction acting on a surface of strain discontinuity in a continuum , 1990 .

[9]  A multispecies step-flow model of growth of compound thin films by MOCVD , 1999 .

[10]  Ross,et al.  Instability-driven SiGe island growth , 2000, Physical review letters.

[11]  J. W. Cahn,et al.  The Interactions of Composition and Stress in Crystalline Solids , 1999 .

[12]  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.

[13]  Morton E. Gurtin,et al.  Multiphase thermomechanics with interfacial structure , 1990 .

[14]  Ellen D. Williams,et al.  Steps on surfaces: experiment and theory , 1999 .

[15]  Andrew N. Norris,et al.  The energy of a growing elastic surface , 1998 .

[16]  J. Villain,et al.  Physics of crystal growth , 1998 .

[17]  Introduction to Step Dynamics and Step Instabilities , 2004, cond-mat/0405066.

[18]  L. Freund,et al.  A continuum description of the energetics and evolution of stepped surfaces in strained nanostructures , 2002, cond-mat/0203514.

[19]  W. Heidug,et al.  Thermodynamics of coherent phase transformations in nonhydrostatically stressed solids , 1985 .

[20]  Conyers Herring,et al.  Surface Tension as a Motivation for Sintering , 1999 .

[21]  L. B. Freund.  A surface chemical potential for elastic solids , 1998 .

[22]  Axel Voigt,et al.  A 2 + 1-Dimensional Terrace-Step-Kink Model for Epitaxial Growth Far from Equilibrium , 2006, Multiscale Model. Simul..

[23]  Zhigang Suo,et al.  Dynamics of nanoscale pattern formation of an epitaxial monolayer , 2001 .

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

[25]  Guyer,et al.  Morphological Stability of Alloy Thin Films. , 1996, Physical review letters.

[26]  R. Caflisch,et al.  Theory of strain relaxation in heteroepitaxial systems , 2003 .

[27]  J. D. Eshelby.  Energy Relations and the Energy-Momentum Tensor in Continuum Mechanics , 1999 .

[28]  Interfacial mixing in heteroepitaxial growth. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  Olivier Pierre-Louis,et al.  Dynamics of crystal steps , 2005 .

[30]  A. Voigt,et al.  Kinetic model for step flow growth of [100] steps. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  L. Truskinovskii,et al.  Dynamics of non-equilibrium phase boundaries in a heat conducting non-linearly elastic medium☆ , 1987 .

[32]  Yukio Saito,et al.  Statistical physics of crystal growth , 1996 .

[33]  Richard L. Schwoebel,et al.  Step Motion on Crystal Surfaces. II , 1966 .

[34]  Yuhai Tu,et al.  Origin of apparent critical thickness for island formation in heteroepitaxy. , 2004, Physical review letters.

[35]  L. Freund,et al.  Minimum energy configuration of epitaxial material clusters on a lattice-mismatched substrate , 1997 .

[36]  F. Léonard,et al.  Molecular beam epitaxy in the presence of phase separation , 1997 .

[37]  M. Jabbour.  Epitaxy of Binary Compounds and Alloys , 2005 .

[38]  N. Bartelt,et al.  Identifying the forces responsible for self-organization of nanostructures at crystal surfaces , 1999, Nature.

[39]  M. Gurtin.  Thermomechanics of Evolving Phase Boundaries in the Plane , 1993 .

[40]  M. Biehl,et al.  Interplay of strain relaxation and chemically induced diffusion barriers: Nanostructure formation in 2D alloys , 2004, cond-mat/0411271.

[41]  Spencer,et al.  Enhanced instability of strained alloy films due to compositional stresses , 2000, Physical review letters.

[42]  Process of irreversible nucleation in multilayer growth. II. Exact results in one and two dimensions. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  Davis,et al.  Morphological instability in epitaxially strained dislocation-free solid films. , 1991, Physical review letters.

[44]  F. Léonard,et al.  Elastic effects and phase segregation during the growth of thin alloy layers by molecular-beam epitaxy , 1997 .

[45]  M. Copel,et al.  Critical role of surface steps in the alloying of Ge on Si(001). , 2004, Physical review letters.

[46]  L. Freund,et al.  Thin Film Materials: Stress, Defect Formation and Surface Evolution , 2004 .

[47]  F. Léonard,et al.  Alloy decomposition and surface instabilities in thin films , 1998 .

[48]  J. D. Eshelby.  The elastic energy-momentum tensor , 1975 .

[49]  J. D. Eshelby.  The Continuum Theory of Lattice Defects , 1956 .

[50]  Sutter,et al.  Nucleationless three-dimensional island formation in low-misfit heteroepitaxy , 2000, Physical review letters.

[51]  Chien H. Wu.  The chemical potential for stress-driven surface diffusion , 1996 .

[52]  J. Tsao,et al.  Materials Fundamentals of Molecular Beam Epitaxy , 1992 .

[53]  A Continuum Theory of Multispecies Thin Solid Film Growth by Chemical Vapor Deposition , 2003 .

[54]  Possible mechanism for the onset of step-bunching instabilities during the epitaxy of single-species crystalline films , 2007 .

[55]  D. Chambliss,et al.  SELF-ASSEMBLED LATERAL MULTILAYERS FROM THIN FILM ALLOYS OF IMMISCIBLE METALS , 1998 .

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

[57]  David J. Srolovitz,et al.  ON THE STABILITY OF SURFACES OF STRESSED SOLIDS , 1989 .

[58]  J. P. Eerden.  CHAPTER 6 – Crystal Growth Mechanisms , 1993 .

[59]  L. Freund.  The mechanics of electronic materials , 2000 .

[60]  N. Bartelt,et al.  Role of stress in thin film alloy thermodynamics: competition between alloying and dislocation formation. , 2001, Physical review letters.

[61]  J. D. Eshelby,et al.  The force on an elastic singularity , 1951, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[62]  A. Pimpinelli,et al.  Kinetic surface patterning in two-particle models of epitaxial growth , 2001 .

[63]  Morton E. Gurtin,et al.  A Unified Treatment of Evolving Interfaces Accounting for Small Deformations and Atomic Transport with Emphasis on Grain-Boundaries and Epitaxy , 2004 .

[64]  Z. Suo,et al.  Symmetry breaking in self-assembled monolayers on solid surfaces. II. Anisotropic substrate elasticity , 2002 .

[65]  A. Pimpinelli,et al.  Two-particle surface diffusion-reaction models of vapour-phase epitaxial growth on vicinal surfaces , 2003 .

[66]  Multispecies epitaxial growth on vicinal surfaces with chemical reactions and diffusion , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[67]  M. Peach,et al.  THE FORCES EXERTED ON DISLOCATIONS AND THE STRESS FIELDS PRODUCED BY THEM , 1950 .