Crystal surface morphological evolution below the roughening temperature:
From microscopic models to macroscopic limits (and back)

 Small structures and devices, with sizes ranging from the order of nanometer (1 nm = 1/100,000 of human hair thickness) up to a few microns, have properties that may revolutionize modern communications and information technology. The design and fabrication of such devices requires a precise control over their morphological evolution, especially their decay. When the dominant mechanism of mass transport is surface diffusion, as it happens in many situations of experimental interest, the lifetime of small surface features scales as a large power (usually fourth) of their size and typically varies as an Arrhenius function of temperature. (Hence, the lifetime increases with decreasing temperature). It follows that fabricating smaller and smaller devices would imply using lower and lower temperatures.

Above the roughening transition temperature, crystal surfaces can be treated as smooth, and their morphological evolution is well understood via the pioneering works of Mullins, Herring and others in the 1950s and 1960s. By contrast, below the roughening transition temperature, crystal surfaces develop distinct atomic steps separated by nanoscale terraces, may have facets (macroscopic planar regions), and thus cannot be treated always as smooth. The study of crystal surface morphological evolution below the roughening transition is an area of active research, posing numerous analytical challenges.

My research focuses on the quantitative description of crystal morphological evolution below roughening. A theme that I have been exploring is the use and limitations of macroscopic theories and equations, such as PDEs, based on the fundamentals of step motion, and even on models at smaller scales when possible. I am especially interested in microscale effects that leave their signatures in the macroscopic limit.

For example, a basic problem is how to solve the requisite 4th-order, nonlinear parabolic PDEs on sloping surfaces by finding and applying suitable boundary conditions at edges of facets, which are "free boundaries" for such PDEs. These boundary conditions in principle involve microscale features of the motion of individual steps on top of facets yet do affect surface profiles macroscopically. Solutions to the related PDEs develop singularities at these moving boundaries. Addressing these issues requires study of coupled differential equations for the step positions, first for simple morphologies of steps that are everywhere parallel, and then for more general cases in two independent space dimensions. The central issue is how to link solutions to the coupled ODE system with the PDE solution near the facet edge.
Most recently, I have been interested in quantum effects, especially electronic surface state effects, on the macroscopic free energy of crystal surfaces.

Topics and papers:

  • I. Numerical simulation by the finite-element method of PDE for crystal surface morphological relaxation in 2+1 dimensions with periodic boundary conditions. We numerically observe sharp topographic transitions from 2D to almost 1D profiles.
    Collaborators: A. Bonito, R. H. Nochetto. Grad. student: J. Quah.

    Numerical experiments: Surface height profile: Initial aspect ratio 3:2; zero line tension (no facets).
    Anisotropic PDE (Margetis-Kohn theory; see publication II.1 below):
    photo photo photo










    (Time flows from left to right)
    To see the corresponding movie, click here

    Simulations from previous isotropic theory (e.g., Shenoy et al., Phys. Rev. Lett. Vol. 92, art. 256101 (2004), and also publication V.2 below) for comparison purposes:
    photo photo photo










    To see the corresponding movie, click here

    Anisotropic PDE theory with ``small'' line tension (facets form):
    The FEM results here do NOT account for microscale effects on top of facets.
    (For the radial case, see publication V.1 below.)
    photo photo photo










  • II. Derivation of macroscopic PDE for crystal surface morphological relaxation from step motion (BCF-type models) in 2+1 dimensions:
    Anisotropy of macroscopic surface flux;
    plausible unified treatment of observed, different decay laws for height profile in periodic settings of Si(001) and Ag(110).

    1. D. Margetis and R. V. Kohn, Continuum relaxation of interacting steps on crystal surfaces in (2+1) dimensions (PDF), (SIAM Journal on) Multiscale Modeling & Simulation, Vol. 5, pp. 729-758 (2006).

    2. D. Margetis, Unified continuum approach to crystal surface morphological relaxation (PDF), Physical Review B, Vol. 76, art. 193403, pp. 1-4 (2007).

    3. J. Quah and D. Margetis, Anisotropic diffusion in continuum relaxation of stepped crystal surfaces (PDF), Journal of Physics A: Mathematical & Theoretical, Vol. 41, art. 235004, pp. 1-18 (2008).

  • III. Derivation of macroscopic PDE for material deposition on stepped crystal surfaces in 2+1 dimensions.

    Related paper:

    1. D. Margetis and R. V. Kohn, Continuum theory of material deposition on stepped surfaces in 2+1 dimensions, in preparation.

  • IV. Kinetic theory approach to studying relation of nonlinear evolution PDEs to ``particle'' schemes for steps (BCF-type models); use of kinetic (BBGKY) hierarchies for step correlations.

    Related paper:

    1. D. Margetis and A. E. Tzavaras, Kinetic hierarchies and macroscopic limits for crystalline steps in 1+1 dimensions, 26 pages, submitted to Multiscale Modeling & Simulation, 2008.

  • V. Radial setting: Step motion laws versus continuum theory: Theoretical predictions are derived using a PDE and comparisons are made with discrete simulations for axisymmetric profiles. In particular, the issue of boundary conditions at facet edges is addressed. (Note: In V.2 below, a phenomenological PDE is written more generally in 2+1 dimensions but applied only to radial setting.)

    Related papers:

    1. D. Margetis, P.-W. Fok, M. J. Aziz, and H. A. Stone, Continuum theory of nanostructure decay via a microscale condition (PDF), Physical Review Letters, Vol. 97, art. 096102, pp. 1-4 (2006).
    Also selected to appear in the Virtual Journal of Nanoscale Science & Technology, September 11, 2006 ( http://www.vjnano.org/nano/ ).

    2. D. Margetis, M. J. Aziz, and H. A. Stone, Continuum approach to self-similarity and scaling in morphological relaxation of a crystal with a facet (PDF), Physical Review B, Vol. 71, art. 165432, pp. 1-22 (2005).

    3. D. Margetis, M. J. Aziz, and H. A. Stone, Continuum description of profile scaling in nanostructure decay (PDF), Physical Review B (Rapid Communications), Vol. 69, art. 041404(R), pp. 1-4 (2004).

  • VI. Derivation of mesoscopic and macroscopic material parameters, such as step stiffness and permeability, from a kinetic model of epitaxial growth; out-of-equilibrium kinetics:

    1. D. Margetis and R. E. Caflisch, Anisotropic step stiffness from a kinetic model of epitaxial growth (PDF); Multiscale Modeling & Simulation, Vol. 7, pp. 242-273 (2008).

  • VII. Continuum description of step motion via continuum Lagrangian coordinates and shocks; study of formation of step bunches and of boundary conditions at facet edges:

    1. P.-W. Fok, R. R. Rosales, and D. Margetis, Unification of step bunching phenomena on vicinal surfaces (PDF), Physical Review B, Vol. 76, art. 033408, pp. 1-4 (2007).
    Also selected to appear in the Virtual Journal of Nanoscale Science & Technology, July 23, 2007 ( http://www.vjnano.org/nano/ ).

    2. P.-W. Fok, R. R. Rosales, and D. Margetis, Facet evolution on supported nanostructures: the effect of finite height, 15 pages, submitted to Physical Review B, 2008.

  • VIII. Analytical study of grooving of grain boundaries by evaporation-condensation or surface-diffusion processes via self-similar shapes below the roughening transition temperature.

    1. H. A. Stone, M. J. Aziz, and D. Margetis, Grooving of a grain boundary by evaporation-condensation below the roughening transition (PDF); Journal of Applied Physics, Vol. 97, art. 113535, pp. 1-6 (2005).


  • IX. Short expository review article on continuum descriptions of crystal surface morphology evolution.

    H. A. Stone and D. Margetis, Continuum descriptions of crystal surface evolution (PDF); review article in the book series Handbook of Materials Modeling, Vol. I: Methods and Models, pp. 1389-1401, S. Yip (Editor), Springer Verlag, Netherlands (2005).