March  2009, 11(2): 443-457. doi: 10.3934/dcdsb.2009.11.443

The role of processing speed in determining step patterns during directional epitaxy


Department of Mathematics, University of Tennessee, Knoxville, TN, 37996, United States


Department of Mathematics, University of Tennessee, 121 Ayres Hall, 1403 Circle Drive, Knoxville, TN 37996, United States

Received  January 2008 Revised  August 2008 Published  December 2008

We consider the growth of an epitaxial thin film on a continuously supplied substrate using both the Burton-Cabrara-Frank (BCF) mean-field model and kinetic Monte-Carlo (KMC) simulation. Of particular interest are effects due to the finite size of the deposition zone, which is modeled by imposing an up- and downwind adatom density equal to the adatom density on an infinite terrace in equilibrium with a step. For the BCF model, we find this scenario admits a steady-state pattern with a specific number of steps separated by alternating widths. The specific spacing between the steps depends sensitively on the processing speed and on whether the number of steps is odd or even, with the range of velocities admitting an odd number of steps typically much narrower. These predictions are only partially confirmed by KMC simulations, however, with particularly poor agreement for an odd number of steps. To investigate further, we consider alternative KMC simulations with the interactions between random walkers on the terraces neglected so as to conform more closely with the mean field model. The latter simulations also more readily allow one to disable the step detachment mechanism, in which case they agree well with the predictions of the BCF model.
Citation: Michael A. Saum, Tim Schulze. The role of processing speed in determining step patterns during directional epitaxy. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 443-457. doi: 10.3934/dcdsb.2009.11.443

Zhiyan Ding, Qin Li. Constrained Ensemble Langevin Monte Carlo. Foundations of Data Science, 2022, 4 (1) : 37-70. doi: 10.3934/fods.2021034


Giacomo Dimarco. The moment guided Monte Carlo method for the Boltzmann equation. Kinetic and Related Models, 2013, 6 (2) : 291-315. doi: 10.3934/krm.2013.6.291


Guillaume Bal, Ian Langmore, Youssef Marzouk. Bayesian inverse problems with Monte Carlo forward models. Inverse Problems and Imaging, 2013, 7 (1) : 81-105. doi: 10.3934/ipi.2013.7.81


Ajay Jasra, Kody J. H. Law, Yaxian Xu. Markov chain simulation for multilevel Monte Carlo. Foundations of Data Science, 2021, 3 (1) : 27-47. doi: 10.3934/fods.2021004


Theodore Papamarkou, Alexey Lindo, Eric B. Ford. Geometric adaptive Monte Carlo in random environment. Foundations of Data Science, 2021, 3 (2) : 201-224. doi: 10.3934/fods.2021014


Michael B. Giles, Kristian Debrabant, Andreas Rössler. Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3881-3903. doi: 10.3934/dcdsb.2018335


Jiakou Wang, Margaret J. Slattery, Meghan Henty Hoskins, Shile Liang, Cheng Dong, Qiang Du. Monte carlo simulation of heterotypic cell aggregation in nonlinear shear flow. Mathematical Biosciences & Engineering, 2006, 3 (4) : 683-696. doi: 10.3934/mbe.2006.3.683


Yuan Gao, Hangjie Ji, Jian-Guo Liu, Thomas P. Witelski. A vicinal surface model for epitaxial growth with logarithmic free energy. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4433-4453. doi: 10.3934/dcdsb.2018170


Chjan C. Lim, Joseph Nebus, Syed M. Assad. Monte-Carlo and polyhedron-based simulations I: extremal states of the logarithmic N-body problem on a sphere. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 313-342. doi: 10.3934/dcdsb.2003.3.313


Joseph Nebus. The Dirichlet quotient of point vortex interactions on the surface of the sphere examined by Monte Carlo experiments. Discrete and Continuous Dynamical Systems - B, 2005, 5 (1) : 125-136. doi: 10.3934/dcdsb.2005.5.125


Olli-Pekka Tossavainen, Daniel B. Work. Markov Chain Monte Carlo based inverse modeling of traffic flows using GPS data. Networks and Heterogeneous Media, 2013, 8 (3) : 803-824. doi: 10.3934/nhm.2013.8.803


Mazyar Zahedi-Seresht, Gholam-Reza Jahanshahloo, Josef Jablonsky, Sedighe Asghariniya. A new Monte Carlo based procedure for complete ranking efficient units in DEA models. Numerical Algebra, Control and Optimization, 2017, 7 (4) : 403-416. doi: 10.3934/naco.2017025


Juntao Yang, Viet Ha Hoang. Multilevel Markov Chain Monte Carlo for Bayesian inverse problem for Navier-Stokes equation. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022033


Reiner Henseler, Michael Herrmann, Barbara Niethammer, Juan J. L. Velázquez. A kinetic model for grain growth. Kinetic and Related Models, 2008, 1 (4) : 591-617. doi: 10.3934/krm.2008.1.591


Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252


Yang Liu, Wenke Li. A class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4367-4381. doi: 10.3934/dcdss.2021112


Christopher Oballe, Alan Cherne, Dave Boothe, Scott Kerick, Piotr J. Franaszczuk, Vasileios Maroulas. Bayesian topological signal processing. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 797-817. doi: 10.3934/dcdss.2021084


Xiaoming Zheng, Gou Young Koh, Trachette Jackson. A continuous model of angiogenesis: Initiation, extension, and maturation of new blood vessels modulated by vascular endothelial growth factor, angiopoietins, platelet-derived growth factor-B, and pericytes. Discrete and Continuous Dynamical Systems - B, 2013, 18 (4) : 1109-1154. doi: 10.3934/dcdsb.2013.18.1109


Jianhong (Jackie) Shen, Sung Ha Kang. Quantum TV and applications in image processing. Inverse Problems and Imaging, 2007, 1 (3) : 557-575. doi: 10.3934/ipi.2007.1.557


Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017

2021 Impact Factor: 1.497


  • PDF downloads (47)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]