Nano-patterning of surfaces by ion sputtering: Numerical study of the Kuramoto-Sivashinsky Equation

Transcription

1 Nano-patterning of surfaces by ion sputtering: Numerical study of the Kuramoto-Sivashinsky Equation Eduardo Vitral Freigedo Rodrigues Supervisors: Gustavo Rabello dos Anjos José Pontes GESAR - UERJ Mechanical Engineering Graduate Program eduardo.vitral@gmail.com April 5, 25 Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25

2 Overview Sputtering 2 Experimental results 3 Theoretical approaches Kadar-Parisi-Zhang equation Kuramoto-Sivashinsky equation 4 Numerical scheme Governing equation Semi-implicit scheme Internal iterations The splitting scheme 5 Linear stability analysis 6 Numerical results Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25 2

3 Sputtering Physics of Sputtering Applications: thin film deposition, micromachining, pattern etching for the fabrication of integrated circuits and chemical analysis Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25 3

4 Sputtering Ion Beam x Glow Discharge Sputtering Sputtering yields: E < KeV : S = 3α π 2 M M 2 (M + M 2 ) 2 E i E L M E > KeV : S (Z Z 2 ) S n (E) M + M 2 Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25 4

5 Experimental results Two main classes of surface morphology by ion sputtering according to experimental results: Periodic ripples on the surface Rough eroded surfaces Experimental data and parameters: i) Ripple formation Angle of incidence Temperature dependence Ion energy System chemistry ii) Kinetic roughening Temperature dependece Ion energy Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25 5

6 Experimental results Rippled patterns formed by ion sputtering have close relatives in nature: (a) Ar + sputtering on Si (b) Namib desert (c) Cloudy sky (d) Ripples on the water Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25 6

7 Experimental results Bradley and Harper theory (HP): theoretical approach describing the process of ripple formation on amorphous substrates Ripples are a result of a surface instability caused by the curvature dependence of the sputter yield. Wavelength and orientation prediction in agreement with numerous experimental results Cannot account for surface roughening A nonlinear theory is required to model the time evolution of ion-sputtered surfaces and predict their morphology: Short time scales: describe the development of a periodic ripple structure Large time scales: the predicted surface morphology should be rough or dominated by new ripples Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25 7

8 Experimental results Experimental data and imaging (AFM): 3D image of hexagonally ordered nanoholes on a Ge surface (Ga + FIB bomdardment) obtained from AFM data AFM image of a nanodot pattern created by Ar + sputtering on GaSb() Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25 8

9 Theoretical approaches Kardar-Parisi-Zhang equation (KPZ): t h = ν 2 h + λ 2 ( h)2 + η Nonlinear stochastic partial differential equation Describe the time evolution of a nonequilibrium interface η ( x, t): noise, reflects the random fluctuations in the process EW equation, λ = : equilibrium fluctuations of an interface which tries to minimize its area under the influence of external noise Scaling properties may change drastically due to anisotropy (AKPZ) Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25 9

10 Theoretical approaches Kuramoto-Sivashinsky equation (KS): t h = ν 2 h K 4 h + λ 2 ( h)2 Deterministic equation, originally proposed to describe chemical waves and flame fronts Chaotic solution rises from its unstable and highly nonlinear character D: for long time and length scales, the surface described by the KS equation is similar to the one described by the KPZ equation; for short time scale solution, the morphology is reminiscent of ripples 2D: contradictory computer simulations Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25

11 Numerical scheme Governing equation: An anisotropic damped Kuramoto-Sivashinsky equation was adopted for the present study One simplified dimensionless form of such equation can be described as follows: h τ = ᾱ h + µ 2 h ( ) 2 h h 2 ( ) h 2 c2 X 2 Y 2 + ν x c 3 4 h D XX X Y X 4 4 h ( 4 h +D XY X 2 + c2 Y 2 Y 4 K 4 h X h X 2 Y h ) Y 4 Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25

12 Numerical scheme Semi-implicit scheme: We propose the following second order in time Cranck-Nicolson semi-implicit scheme for solving the target equation with a µ = 4, high temperatures and θ < 65.3 : h n+ h n τ = Λ X h n+ + h n 2 + Λ Y h n+ + h n 2 + f n+/2 Including the factor /2 into the operators Λ X and Λ Y, they may be defined as follows: Λ X = [ ] ᾱ 2 2 (D XX + K) 4 X 4 Λ Y = [ ] ᾱ 2 2 K 4 Y 4 Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25 2

13 Numerical scheme Internal iterations: Since the operators Λ n+/2 X, Λ n+/2 Y and the function f n+/2 contain terms in the new stage we do internal iterations at each time step according to: h n,m+ h n τ = Λ X ( h n,m+ + h n) + Λ Y ( h n,m+ + h n) + f n+/2 where the superscript (n, m + ) identifies the new iteration. The iterations proceed until the following criteria is satisfied: max h n,m+ h n,m max h n,m+ < δ Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25 3

14 Numerical scheme The splitting scheme: The splitting of the target equation is made according to the Douglas second scheme (also known as scheme of stabilizing correction ). It is designed as follows: h h n τ h n,m+ h τ = Λ X h + ΛY hn + f n+/2 + (Λ X + Λ Y ) h n = Λ ( hn,m+ Y h n) Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25 4

15 Linear stability analysis In Fourier transform, for small perturbations, the governing equation may be written as: h p ( q) = h o e i(qx X +qy Y ) e στ σ τ hp ( q) = [ ᾱ + ( µq 2 x) + ( νq 2 y ) D XX q 4 x + D XY q 2 xq 2 y +D YY q 4 y K(q 2 x + q 2 y ) 2] hp ( q) The anisotropic coefficients D XX, D XY and D YY can be hidden for an easier manipulation of the equation. Besides, we may consider q 2 = q 2 x + q 2 y. σ τ = ɛ K(q 2 q 2 c ) 2 ( µ ν )q 2 y Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25 5

16 Random initial case L norm 3 4 hn max τ τ Y 2.4 Y X X Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25 6

17 Initial pattern: monomode qo~x.8.6 h n max L norm τ τ h n max Internal iterations Eduardo Vitral 2 (GESAR) 3 τ UERJ Mechanical Eng. Graduate Program 8,2 τ,6 2, 2,4 April 5, 25 7

18 Initial pattern: monomode qo~x Y 2 2 Y X X Y.2 Y X Eduardo Vitral (GESAR) X UERJ Mechanical Eng. Graduate Program April 5, 25 8

19 Initial pattern: monomode q o y L norm 3 4 hn max τ τ 4 6 Internal iterations τ 4 Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25 9

20 Initial pattern: monomode qo~y Y Y X X Y Y X Eduardo Vitral (GESAR) X UERJ Mechanical Eng. Graduate Program April 5, 25 2

21 References [] Materials Science of Thin Films: Deposition & Structure Milton Ohring [2] Fractal Concepts in Surface Growth A.-L. Barabási, H.E. Stanley [3] M. A. Makeev, R. Cuerno, A.-L. Barabási (22) Morphology of ion-sputtered surfaces Nuclear Instruments and Methods in Physics Research B [4] F. Family (99) Dynamic scaling and phase transitions in interface growth (99) Physica A: Statistical Mechanics and its Applications Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25 2

22 References [5] theory-and-operation-of-magnetron-sputter-deposition-coating-process (Retrieved: 5//24) [6] (Retrieved: 4//24) [7] (Retrieved: 4//24) [8] (Retrieved: 5//24) Eduardo Vitral (GESAR) UERJ Mechanical Eng. Graduate Program April 5, 25 22