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# KAIST Internship Program

Fractal Concepts in Surface Growth
by

## Kirstie Raquel Natalia Toreh

on 27 November 2014

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#### Transcript of KAIST Internship Program

Dynamics of Growing Fractal Surfaces
Kirstie Raquel Natalia Toreh
Department of Physics
Institut Teknologi Bandung (ITB)

Motivation
Random Deposition
Ballistic Deposition
Random Deposition with Diffusion (Relaxation)
Molecular Beam Epitaxy (MBE)
Conclusion
Outline
“Most of our life takes place on the surface of something”
Motivation
Every point at the paper have different speed of burning, which influence the height of every point and the global width of the rough surface
The modern concepts used to study various roughening process is scaling we shall see that many measurable quantites obey simple scaling relations.
Define universality classes
The other examples that can be explained are:
(liquid flow in a tissue that is suspended in water, snow on the ground surface, bacterial colonies (growth), surface growth in thin film technology, roughening in MBE)  roughening process
Experiments
Discrete models
Continuum equations
The Mandelbrot Set
What is Fractal?
"Fractals is a geometry which have the same shape from near as from far"
Prof. Jeong, Hawoong
Kihong Chung & Daniel Kim
Department of Physics-Complex System and Statistical Physics Lab (CSSPL)
Korea Advanced Institute of Science and Technology (KAIST)

Fractal
Why Fractal??
So many surfaces in nature are rough
Many rough surfaces has fractal dimension
Fractal can be used to describe the morphology of surface growth
Random Deposition
Ballistic Deposition
Random Deposition with Surface Diffusion (Relaxation)
Surface Growth Model
“From a randomly chosen site over the surface, a particle falls vertically until it reaches the top of the column under it, whereupon it is deposited’
Random Deposition
will give an interface
Simulation and Results of Random Deposition
Fig. The time evolution of the surface width for Random Deposition
Fig. The interface of Random Deposition with L=100
The very interesting thing is, every different system size (L) will give the same growth exponent  why?? exact solution (discrete model) and stochastic growth equations
Random Deposition
Random deposition described either by the discrete model or by the stochastic growth equation defines the simplest growth process and forms a distinct universality class and growth phenomena.
Its most important property lies in the uncorrelated nature of the interface.
Random Deposition
“Particles from randomly chosen position above the surface, located at a distance larger than the maximum height of the interface. The particles follow a straight vertical trajectory until it reaches the surface, whereupon it sticks.”
Ballistic Deposition
will give an interface
Java animation
Simulation and Results of Ballistic Deposition
Dynamic exponent (z)
Roughness exponent (αalpha)
Growth exponent (betha)
Scaling exponents
b. The interface width (characterizes the roughness of the interface)
Roughening
a. The mean height of the surface is defined by
Ballistic Deposition
alpha = 0,4401
z = alpha/betha
= 1,6
Rescaled Graph
The BD simulations resclaed according to the equation . The obtained curve is in fact the scaling function f(u), with the properties of scaling exponents.
Random Deposition with Diffusion (Surface Relaxation)
“The freshly-deposited atoms do not stick irreversibly to the site they fall on, but rather they can ‘relax’ to a nearest neighbor if it has a lower height.”
Results of Random Deposition with Diffusion (Surface Relaxation)
Rescaling
Comparison RDSR with RD
Random deposition with diffusion (surface relaxation) interface
Random deposition without diffusion
Edwards-Wilkinson (EW) Equation
Random deposition with diffusion (surface relaxation) can be described by Edwards-Wilkinson equation --> which we also can find the scaling exponent by this equation
If we solve the equation, then we will get
Erdwards-Wilkinson Equation
Kardar-Parisi-Zhang (KPZ) equation
z=3/2
,
KPZ can explain Ballistic Deposition model
Molecular Beam Epitaxy (MBE)
The morphology of this interface is determined by the interplay between deposition, desorption, and surface diffusion
At low temperature (T), surface diffusion can be practically neglected. In this limit, only deposition determines the growth, and the material becomes amorphous with a very rough surface.
At intermediate T, surface diffusion becomes relevant and the interface becomes rough.
At higher T, the diffusion length becomes larger and the atoms can search the entire system for an energetically favorable position to stick. As the result, the interface becomes smooth.
MBE
MBE
Conclusion
A. –L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, New York, 1995).
B. Farnudi and D. D. Vvedensky, Phys. Rev. E 83, 020103(R) (2011).
S. Das Sarma and P. Tamborenea, Phys. Rev. Lett. 66, 3 (1991).
S. Das Sarma, C. J. Lanczycki, R. Kotlyar, and S. V. Ghaisas, Phys. Rev. E 53, 1(1996).
References
A simple surface growth model can be described by random deposition, ballistic deposition, and random deposition with diffusion (surface relaxation)
Every growth model obey simple scaling relations (alpha, betha, z) and can be described by experiments, dicrete models, and continuum equation.
Random deposition with diffusion (surface relaxation) can be described by Edwards-Wilkinson (EW) equation and Ballistic deposition can be described by Kardar-Parisi-Zhang (KPZ) equation.
In Molecular Beam Epitaxy (MBE), higher temperature of system will make a smoother surface (interface), on the other hand lower temperature will give a rough surface (interface)
THANK YOU
,
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Edwards-Wilkinson (EW) Equation
Kardar-Parisi-Zhang (KPZ) Equation
h
L
Surface plot L=100
and so on
0,5
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