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December  2018, 23(10): 4433-4453. doi: 10.3934/dcdsb.2018170

## A vicinal surface model for epitaxial growth with logarithmic free energy

 Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong Department of Mathematics‡, University of California, Los Angeles, Los Angeles, CA 90095, USA Department of Mathematics#, Duke University, Durham, NC 27708, USA Department of Physics†, Duke University, Durham, NC 27708, USA

* Corresponding author: Hangjie Ji (hangjie@math.ucla.edu)

Received  April 2017 Revised  December 2017 Published  June 2018

We study a continuum model for solid films that arises from the modeling of one-dimensional step flows on a vicinal surface in the attachment-detachment-limited regime. The resulting nonlinear partial differential equation, $u_t = -u^2(u^3+α u)_{hhhh}$, gives the evolution for the surface slope $u$ as a function of the local height $h$ in a monotone step train. Subject to periodic boundary conditions and positive initial conditions, we prove the existence, uniqueness and positivity of global strong solutions to this PDE using two Lyapunov energy functions. The long time behavior of $u$ converging to a constant that only depends on the initial data is also investigated both analytically and numerically.

Citation: Yuan Gao, Hangjie Ji, Jian-Guo Liu, Thomas P. Witelski. A vicinal surface model for epitaxial growth with logarithmic free energy. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4433-4453. doi: 10.3934/dcdsb.2018170
##### References:
 [1] H. Al Hajj Shehadeh, R. V. Kohn and J. Weare, The evolution of a crystal surface: Analysis of a one-dimensional step train connecting two facets in the ADL regime, Physica D, 240 (2011), 1771-1784.  doi: 10.1016/j.physd.2011.07.016.  Google Scholar [2] F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, Journal of Differential Equations, 83 (1990), 179-206.  doi: 10.1016/0022-0396(90)90074-Y.  Google Scholar [3] W. K. Burton, N. Cabrera and F. C. Frank, The growth of crystals and the equilibrium structure of their surfaces, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 243 (1951), 299-358.  doi: 10.1098/rsta.1951.0006.  Google Scholar [4] W. L. Chan, A. Ramasubramaniam, V. B. Shenoy and E. Chason, Relaxation kinetics of nano-ripples on Cu (001) surface, Physical Review B, 70 (2004), 245403. doi: 10.1103/PhysRevB.70.245403.  Google Scholar [5] W. E and N. K. Yip, Continuum theory of epitaxial crystal growth. Ⅰ, Journal of Statistical Physics, 104 (2001), 221-253.  doi: 10.1023/A:1010361711825.  Google Scholar [6] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19 (1998), American Mathematical Society.  Google Scholar [7] Y. Gao, J.-G. Liu and J. Lu, Continuum limit of a mesoscopic model with elasticity of step motion on vicinal surfaces, Journal of Nonlinear Science, 27 (2017), 873-926.  doi: 10.1007/s00332-016-9354-1.  Google Scholar [8] Y. Gao, J.-G. Liu and J. Lu, Weak solution of a continuum model for vicinal surface in the attachment-detachment-limited regime, SIAM Journal on Mathematical Analysis, 49 (2017), 1705-1731.  doi: 10.1137/16M1094543.  Google Scholar [9] N. Israeli and D. Kandel, Decay of one-dimensional surface modulations, Physical Review B, 62 (2000), 13707. doi: 10.1103/PhysRevB.62.13707.  Google Scholar [10] H.-C. Jeong and E. D. Williams, Steps on surfaces: Experiment and theory, Surface Science Reports, 34 (1999), 171-294.  doi: 10.1016/S0167-5729(98)00010-7.  Google Scholar [11] R. V. Kohn, Surface relaxation below the roughening temperature: Some recent progress and open questions, Nonlinear Partial Differential Equations, Springer, 7 (2012), 207–221. doi: 10.1007/978-3-642-25361-4_11.  Google Scholar [12] R. V. Kohn, T. S. Lo and N. K. Yip, Continuum limit of a step flow model of epitaxial growth, MRS Proceedings, vol. 701, Cambridge Univ Press, 2001, 1–7. Google Scholar [13] R. V. Kohn and H. M. Versieux, Numerical analysis of a steepest-descent PDE model for surface relaxation below the roughening temperature, SIAM Journal on Numerical Analysis, 48 (2010), 1781-1800.  doi: 10.1137/090750378.  Google Scholar [14] R. Kohn and Y. Giga, Scale-invariant extinction time estimates for some singular diffusion equations, Discrete and Continuous Dynamical Systems, 30 (2011), 509-535.  doi: 10.3934/dcds.2011.30.509.  Google Scholar [15] B. Li and J.-G. Liu, Thin film epitaxy with or without slope selection, European Journal of Applied Mathematics, 14 (2003), 713-743.  doi: 10.1017/S095679250300528X.  Google Scholar [16] B. Li and J.-G. Liu, Epitaxial growth without slope selection: Energetics, coarsening, and dynamic scaling, Journal of Nonlinear Science, 14 (2004), 429-451.  doi: 10.1007/s00332-004-0634-9.  Google Scholar [17] D. Margetis and R. V. Kohn, Continuum relaxation of interacting steps on crystal surfaces in $2+1$ dimensions, Multiscale Modeling & Simulation, 5 (2006), 729-758.  doi: 10.1137/06065297X.  Google Scholar [18] W. W. Mullins, Theory of thermal grooving, Journal of Applied Physics, 28 (1957), 333-339.  doi: 10.1063/1.1722742.  Google Scholar [19] P. Nozières, On the motion of steps on a vicinal surface, Journal de Physique, 48 (1987), 1605-1608.   Google Scholar [20] M. Ozdemir and A. Zangwill, Morphological equilibration of a corrugated crystalline surface, Physical Review B 42 (1990), 5013. doi: 10.1103/PhysRevB.42.5013.  Google Scholar [21] A. Pimpinelli and J. Villain, Physics of Crystal Growth, vol. 19, Cambridge University Press, 1998. Google Scholar [22] A. A. Rettori and J. Villain, Flattening of grooves on a crystal surface: A method of investigation of surface roughness, Journal de Physique, 49 (1988), 257-267.  doi: 10.1051/jphys:01988004902025700.  Google Scholar [23] V. B. Shenoy, A. Ramasubramaniam and L. B. Freund, A variational approach to nonlinear dynamics of nanoscale surface modulations, Surface Science, 529 (2003), 365-383.  doi: 10.1016/S0039-6028(03)00276-0.  Google Scholar [24] V. B. Shenoy, A. Ramasubramaniam, H. Ramanarayan, D. T. Tambe, W. L. Chan and E. Chason, Influence of step-edge barriers on the morphological relaxation of nanoscale ripples on crystal surfaces, Physical Review Letters, 92 (2004), 256101. doi: 10.1103/PhysRevLett.92.256101.  Google Scholar [25] C. Villani, Topics in Optimal Transportation, 58, American Mathematical Soc., 2003. doi: 10.1007/b12016.  Google Scholar [26] Y. Xiang, Derivation of a continuum model for epitaxial growth with elasticity on vicinal surface, SIAM Journal on Applied Mathematics, 63 (2002), 241-258.  doi: 10.1137/S003613990139828X.  Google Scholar

show all references

##### References:
 [1] H. Al Hajj Shehadeh, R. V. Kohn and J. Weare, The evolution of a crystal surface: Analysis of a one-dimensional step train connecting two facets in the ADL regime, Physica D, 240 (2011), 1771-1784.  doi: 10.1016/j.physd.2011.07.016.  Google Scholar [2] F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, Journal of Differential Equations, 83 (1990), 179-206.  doi: 10.1016/0022-0396(90)90074-Y.  Google Scholar [3] W. K. Burton, N. Cabrera and F. C. Frank, The growth of crystals and the equilibrium structure of their surfaces, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 243 (1951), 299-358.  doi: 10.1098/rsta.1951.0006.  Google Scholar [4] W. L. Chan, A. Ramasubramaniam, V. B. Shenoy and E. Chason, Relaxation kinetics of nano-ripples on Cu (001) surface, Physical Review B, 70 (2004), 245403. doi: 10.1103/PhysRevB.70.245403.  Google Scholar [5] W. E and N. K. Yip, Continuum theory of epitaxial crystal growth. Ⅰ, Journal of Statistical Physics, 104 (2001), 221-253.  doi: 10.1023/A:1010361711825.  Google Scholar [6] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19 (1998), American Mathematical Society.  Google Scholar [7] Y. Gao, J.-G. Liu and J. Lu, Continuum limit of a mesoscopic model with elasticity of step motion on vicinal surfaces, Journal of Nonlinear Science, 27 (2017), 873-926.  doi: 10.1007/s00332-016-9354-1.  Google Scholar [8] Y. Gao, J.-G. Liu and J. Lu, Weak solution of a continuum model for vicinal surface in the attachment-detachment-limited regime, SIAM Journal on Mathematical Analysis, 49 (2017), 1705-1731.  doi: 10.1137/16M1094543.  Google Scholar [9] N. Israeli and D. Kandel, Decay of one-dimensional surface modulations, Physical Review B, 62 (2000), 13707. doi: 10.1103/PhysRevB.62.13707.  Google Scholar [10] H.-C. Jeong and E. D. Williams, Steps on surfaces: Experiment and theory, Surface Science Reports, 34 (1999), 171-294.  doi: 10.1016/S0167-5729(98)00010-7.  Google Scholar [11] R. V. Kohn, Surface relaxation below the roughening temperature: Some recent progress and open questions, Nonlinear Partial Differential Equations, Springer, 7 (2012), 207–221. doi: 10.1007/978-3-642-25361-4_11.  Google Scholar [12] R. V. Kohn, T. S. Lo and N. K. Yip, Continuum limit of a step flow model of epitaxial growth, MRS Proceedings, vol. 701, Cambridge Univ Press, 2001, 1–7. Google Scholar [13] R. V. Kohn and H. M. Versieux, Numerical analysis of a steepest-descent PDE model for surface relaxation below the roughening temperature, SIAM Journal on Numerical Analysis, 48 (2010), 1781-1800.  doi: 10.1137/090750378.  Google Scholar [14] R. Kohn and Y. Giga, Scale-invariant extinction time estimates for some singular diffusion equations, Discrete and Continuous Dynamical Systems, 30 (2011), 509-535.  doi: 10.3934/dcds.2011.30.509.  Google Scholar [15] B. Li and J.-G. Liu, Thin film epitaxy with or without slope selection, European Journal of Applied Mathematics, 14 (2003), 713-743.  doi: 10.1017/S095679250300528X.  Google Scholar [16] B. Li and J.-G. Liu, Epitaxial growth without slope selection: Energetics, coarsening, and dynamic scaling, Journal of Nonlinear Science, 14 (2004), 429-451.  doi: 10.1007/s00332-004-0634-9.  Google Scholar [17] D. Margetis and R. V. Kohn, Continuum relaxation of interacting steps on crystal surfaces in $2+1$ dimensions, Multiscale Modeling & Simulation, 5 (2006), 729-758.  doi: 10.1137/06065297X.  Google Scholar [18] W. W. Mullins, Theory of thermal grooving, Journal of Applied Physics, 28 (1957), 333-339.  doi: 10.1063/1.1722742.  Google Scholar [19] P. Nozières, On the motion of steps on a vicinal surface, Journal de Physique, 48 (1987), 1605-1608.   Google Scholar [20] M. Ozdemir and A. Zangwill, Morphological equilibration of a corrugated crystalline surface, Physical Review B 42 (1990), 5013. doi: 10.1103/PhysRevB.42.5013.  Google Scholar [21] A. Pimpinelli and J. Villain, Physics of Crystal Growth, vol. 19, Cambridge University Press, 1998. Google Scholar [22] A. A. Rettori and J. Villain, Flattening of grooves on a crystal surface: A method of investigation of surface roughness, Journal de Physique, 49 (1988), 257-267.  doi: 10.1051/jphys:01988004902025700.  Google Scholar [23] V. B. Shenoy, A. Ramasubramaniam and L. B. Freund, A variational approach to nonlinear dynamics of nanoscale surface modulations, Surface Science, 529 (2003), 365-383.  doi: 10.1016/S0039-6028(03)00276-0.  Google Scholar [24] V. B. Shenoy, A. Ramasubramaniam, H. Ramanarayan, D. T. Tambe, W. L. Chan and E. Chason, Influence of step-edge barriers on the morphological relaxation of nanoscale ripples on crystal surfaces, Physical Review Letters, 92 (2004), 256101. doi: 10.1103/PhysRevLett.92.256101.  Google Scholar [25] C. Villani, Topics in Optimal Transportation, 58, American Mathematical Soc., 2003. doi: 10.1007/b12016.  Google Scholar [26] Y. Xiang, Derivation of a continuum model for epitaxial growth with elasticity on vicinal surface, SIAM Journal on Applied Mathematics, 63 (2002), 241-258.  doi: 10.1137/S003613990139828X.  Google Scholar
(Top) A typical PDE simulation for (1.10) with $\alpha = 1$ on $0\le x\le 1$ and (bottom) the corresponding plot of $u(t,h)$ with boundary conditions (1.15) and $H = 2$, clearly showing the convergence of $h$ to a straight line, with the slope $u$ approaching to a spatially-uniform profile $u = 2$.
A numerical simulation of PDE (1.6) plotted in semi-log coordinates starting from the initial condition (5.1) (plotted with the dashed line): (top) early stage near-rupture is approached as the global minimum decreases from $0.07$ to $0.007$ for $0<t< 0.0032$; (bottom) later stage behavior for $t>0.0032$ as the solution approaches a constant $u^{\star} = 0.27$.
(Top) A numerical simulation of PDE (1.13) starting from identical initial conditions used in Fig. 2 showing convergence to a spatially-uniform solution $u = u^{\star}$ as $t \to \infty$. (Bottom) A plot showing that $u_{m}(t) = \min_h u(t,h)$ is bounded below by $\mathcal{J}(E(t))$ given by (3.32) which is in line with the conclusion of Theorem 1, and that the asymptotic lower bound $\mathcal{J}(E(t)) \to {1}/{(2L)}$ for $t \to \infty$ as in (3.8).
Plots of corresponding energy $E$ in (2.5) and (5.7) for PDE simulations in Fig. 2 and Fig. 3. The energy $E(t)$ decays exponentially to zero following (5.6) with $k = 2\pi$.
Evolution of the surface height $h(t,x)$ and slope $h_x(t,x)$ following equation (6.2) with $\alpha = 0$ starting from initial condition $h_0(x) = \sin(2\pi x)$ on $0\le x\le 1$, showing convergence to spatially-uniform solution $h \equiv 0$ as $t \to \infty$.
Evolution of the surface height $h(t,x)$ and slope $h_x(t,x)$ for equation (6.2) with $\alpha = 1$ starting from identical initial data used in Fig. 5, showing convergence to a piece-wise constant profile in $h$ and jump in $h_x$.
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