**Dynamics of Growing Fractal Surfaces**

**Kirstie Raquel Natalia Toreh**

Department of Physics

Institut Teknologi Bandung (ITB)

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)

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

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