Dynamics of Surface Pattern Evolution in Thin Films

1. Dynamics of Surface Pattern Evolution in Thin Films Rui Huang Center for Mechanics of Solids, Structures and Materials Department of Aerospace Engineering and Engineering Mechanics The University of Texas at Austin

2. Self-Assembled Surface Patterns Granadosand Garcia, 2003. Tabe et al., 2002 & 2003. Yang, Liu and Lagally, 2004.

3. Wrinkle Patterns Bowden et al., 1998 & 1999. Stafford et al., 2004. Cahill et al., 2002. Muller-Wiegand et al., 2002.

4. Part I: Surface Diffusion-Controlled Patterns

5. Competition between surface energy and strain energy leads to a critical wavelength: (~300nm) Chemical potential on surface: Surface evolution: Linear analysis: Surface Instability of Stressed Solid Asaro and Tiller (1972); Grinfeld (1986); Srolovitz (1989)…… Nonlinear analysis: develop crack-like grooves or cusps.

6. Instability of Epitaxial Films Spencer, Voorhees and Davis (1991); Freund and Jonsdottir (1993); Gao (1993)…… The film is stressed due to lattice mismatch between the film and the substrate (e.g., Ge on Si). Stress relaxation leads to formation of dislocations and/or surface roughening. Linear analysis: similar to that of stressed solids Nonlinear analysis: self-assembly of quantum dots How to control the size and order of quantum dots?

7. The Base Model Surface chemical potential: Surface flux: Equation of surface evolution: Nonlinear terms arise from wavy surface as the boundary condition for the stress field and from the wetting effect.

8. First-order: (B.C.) (In the order of ) Boundary condition on the surface: Zeroth-order: Stress Analysis

9. Linear Evolution Equation Length scale: Fourier transform Time scale: Critical wavelength: Fastest growing wavelength:

10. Nonlinear Stresses Second-order: (B.C.) Nonlinear evolution equation:

11. Spectral Method Fourier transform of the nonlinear equation: Semi-implicit integration: Numerical simulations: Calculate spatial differentiation in the Fourier space Calculate nonlinear terms in the physical space Communicate between physical and Fourier spaces via FFT and its inverse

12. 1D Simulations Linear equation Nonlinear equation Consideration of nonlinear stress leads to unstable evolution and formation of deep grooves.

13. 2D Simulations t = 0 t = 50 t = 20 t = 85 Downward blow-up instability: nanopits?

14. Effect of Wetting Transition of surface energy (Spencer, 1999): Linear evolution equation with wetting:

15. Linear Analysis Thick films: no effect; Very thin films: stabilized. Critical film thickness: Typical values:

16. 1D Simulations Stable growth Coarsening Blow-up instability

17. 2D Simulations t = 0 t = 50 t = 200 t = 259 t = 260 Upward blow-up instability: nano whiskers?

18. Nonlinear Stress + Wetting: 1D Simulation Stable growth Coarsening No blow-up instability!

19. Nonlinear Stress + Wetting: 2D t = 0 t = 50 t = 200 t = 250 t = 500 t = 1000

20. Part I: Summary Nonlinear analysis of surface diffusion-controlled pattern evolution in strained epitaxial films: • Nonlinear stress field leads to downward blowup instability. • Wetting effect leads to upward blowup instability. • Combination of nonlinear stress and wetting stabilizes the evolution.

21. Part II: Compression-Induced Wrinkle Patterns

22. (Lee at al., 2004)

23. Other equilibrium states: energetically unfavorable Freestanding film: Euler buckling Critical load: • Buckling relaxes compressive stress • Bending energy favors long wavelength

24. Elastic substrate On elastic substrates • Deformation of the substrate disfavors wrinkling of long wavelengths and competes with bending to select an intermediate wavelength Wrinkling: short wavelength, on soft substrates, no delamination Buckling: long wavelength, on hard substrates, with delamination

25. Critical Condition for Wrinkling Thick substrate (hs >> hf): The critical strain decreases as the substrate stiffness decreases. In general, the critical strain depends on the thickness ratio and Poisson’s ratios too. In addition, the interface must be well bonded.

26. Equilibrium Wrinkle Wavelength Thick substrate (hs >> hf): Measure wavelength to determine film stiffness The wrinkle wavelength is independentof compressive strain. The wavelength increases as the substrate stiffness decreases. In general, the wavelength depends on thickness ratio and Poisson’s ratios too.

27. Equilibrium Wrinkle Amplitude Thick substrate (hs >> hf): Measure amplitude to determine film stress/strain. The wrinkle amplitude increases as the compressive strain increases. For large deformation, however, nonlinear elastic behavior must be considered.

28. Equilibrium Wrinkle Patterns In an elastic system, the equilibrium state minimizes the total strain energy. However, it is extremely difficult to find such a state for large film areas. More practically, one compares the energy of several possible patterns to determine the preferred pattern. How does the pattern emerge? How to control wrinkle patterns?

29. Wrinkle Amplitude Rubbery State Glassy State 0 Compressive Strain Wrinkling on Viscoelastic Substrates Cross-linked polymers • Evolution of wrinkles: • Viscous to Rubbery • Glassy to Rubbery

30. Wrinkling Kinetics I: Wrinkles of intermediate wavelengths grow exponentially; The fastest growing mode dominates the initial growth. GrowthRate Fastest mode 0 m  For hs >> hf : The kinetically selected wavelength is independent of substrate!

31. Wrinkling Kinetics II: Instantaneous wrinkle at the glassy state: Kinetic growth at the initial stage: Long-term evolution:

32. Metal film Polymer Substrate Evolution Equations

33. Numerical Simulation t = 0 Growing wavelengths t = 1104 Coarsening t = 1105 Equilibrium wavelength t = 1107

34. Evolution of Wrinkle Wavelength Initial stage: kinetically selected wavelengths Intermediate stage: coarsening of wavelength Final stage: equilibrium wavelength at the rubbery state

35. Evolution of Wrinkle Amplitude Initial stage: exponential growth Intermediate stage: slow growth Final stage: saturating

36. 2D Wrinkle Patterns I t = 0 t = 104 t = 105 t = 107 t = 106

37. 2D Wrinkle Patterns II t = 0 t = 105 t = 106 t = 5X106 t = 2X107

38. 2D Wrinkle Patterns III t = 5X105 t = 0 t = 104 t = 107 t = 106

39. On a Patterned Substrate t = 0 t = 104 t = 105 t = 107 t = 106

40. Circular Perturbation t = 0 t = 104 t = 105 t = 107 t = 5105 t = 106

41. Evolution of Wrinkle Patterns • Symmetry breaking in isotropic system: • from spherical caps to elongated ridges • from labyrinth to herringbone. • Symmetry breaking due to anisotropic strain • from labyrinth to parallel stripes • Controlling the wrinkle patterns • On patterned substrates • By introducing initial defects

42. Large-Cell Simulation t=1X104 t=3X104 t=5X104 t=8X104 t=1X105 t=1X106 t=1X107 t=3X107

43. Acknowledgments • Co-workers: Se Hyuk Im, Yaoyu Pang, Hai Liu, S.K. Banerjee, H.H. Lee, C.M. Stafford • Funding: NSF, ATP, Texas AMRC Thank you !