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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. [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. [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. [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. [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. [6] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19 (1998), American Mathematical Society. [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. [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. [9] N. Israeli and D. Kandel, Decay of one-dimensional surface modulations, Physical Review B, 62 (2000), 13707. doi: 10.1103/PhysRevB.62.13707. [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. [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. [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. [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. [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. [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. [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. [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. [18] W. W. Mullins, Theory of thermal grooving, Journal of Applied Physics, 28 (1957), 333-339. doi: 10.1063/1.1722742. [19] P. Nozières, On the motion of steps on a vicinal surface, Journal de Physique, 48 (1987), 1605-1608. [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. [21] A. Pimpinelli and J. Villain, Physics of Crystal Growth, vol. 19, Cambridge University Press, 1998. [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. [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. [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. [25] C. Villani, Topics in Optimal Transportation, 58, American Mathematical Soc., 2003. doi: 10.1007/b12016. [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.

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. [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. [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. [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. [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. [6] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19 (1998), American Mathematical Society. [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. [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. [9] N. Israeli and D. Kandel, Decay of one-dimensional surface modulations, Physical Review B, 62 (2000), 13707. doi: 10.1103/PhysRevB.62.13707. [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. [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. [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. [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. [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. [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. [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. [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. [18] W. W. Mullins, Theory of thermal grooving, Journal of Applied Physics, 28 (1957), 333-339. doi: 10.1063/1.1722742. [19] P. Nozières, On the motion of steps on a vicinal surface, Journal de Physique, 48 (1987), 1605-1608. [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. [21] A. Pimpinelli and J. Villain, Physics of Crystal Growth, vol. 19, Cambridge University Press, 1998. [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. [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. [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. [25] C. Villani, Topics in Optimal Transportation, 58, American Mathematical Soc., 2003. doi: 10.1007/b12016. [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.
(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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