Universality class change due to angle of deposition of thin-film growth by random particles aggregation

In this work we study numerically the effects of the angle of deposition of particles in the growth process of a thin-film generated by aggregation of particles added at random. The particles are aggregated in a random position of an initially flat surface and with a given angle distribution. This process gives rise to a rough interface after some time of deposition. We performed Monte Carlo simulations and, by changing the angle of deposition, we observed a transition from the random deposition (RD) universality class to the Kardar-Parisi-Zhang (KPZ) universality class. We measured the usual scaling exponents, namely, the roughness ($\alpha$), the growth ($\beta$) and the dynamic ($z$) exponents. Our results show that the particles added non-perpendicularly to the substrate, can change the universality class in a discrete atomistic random deposition model. When particles are deposited with an angle of $45^{\circ}$ in relation to the surface, the same values of the Ballistic Deposition model are observed in the Random Deposition model. We also propose an analytic approach, using a differential stochastic equation to analyze the growth process evolution, and our theoretical results corroborate the computer simulations.


I. INTRODUCTION
The understanding of physical process that take place at surfaces and interfaces has attracted interest of researchers from different fields 1,2 . Specially motivated by the technological applications developed from thin-films 3,4 , the investigation on the formation of structures due to the deposition of atoms or particles has been the subject of large number of recent studies both experimental and theoretical [5][6][7][8] .
By controlling the surface processes, one can control physical, chemical, optical and mechanical properties of the material, leading to the development of new devices with practical purposes 9 . Theoretical and computational models represent a powerful tool to study the growth of thin-films and interfaces, where the physicists can apply the well-known methods of statistical physics to describe these non-equilibrium phenomena.
New advances in recent years allowed a better understanding of the fundamental phenomena which govern the deposition of particles forming a thin film at nanoscale. Atomistic models have been largely applied in this field of study, using different forms of particles 5,10,11 . Although many models are quite simpler, one can use them as a good starting point to study more sophisticated processes that are directly related to the experimental growth process and techniques.
New experimental techniques, such as sputtering or Molecular Beam Epitaxy (MBE), can provide suitable materials for a large range of applications, such as medicine 12 , in which thin and ultra-thin film coatings for stent devices are, perhaps, one of most remarkable examples of nanostructured biomaterials. In another field, the electronic and nano electronic industry have an increasing demand for devices at nanoscale, in which the surface morphology play a very important role for applications in solar cells 13 , information storages 14 and carbon nanotubes 15 .
In this paper we introduce a modification in the Random Deposition (RD) model, considering that the particles can be aggregated with different angles in relation to the initially flat substrate. Due to the simplicity of the model, random deposition is widely studied, but the uncorrelated interface resulting from this process is not realistic. In order to make the model more realistic and applicable to the surfaces science, we modify the RD model and introduce a natural correlation on the surface by adding particles obliquely. We study our model by means of computer simulations and stochastic growth equations and we show that, by adding particles obliquely to the surface, the RD model can produce the same scaling exponents of the Ballistic Deposition (BD) model when the angle of deposition is 45°in relation to the initially flat substrate. This paper is organized as follows: In Sec II we analyze the standard methods for theoretical analysis of thin-films. In Sec. III we present our model, the deposition rules and simulations details. The discussion of the results of the numerical simulations and the theoretical analysis of the stochastic equations is presented in Sec. IV, and the main conclusions are presented in Sec. V.

THEORETICAL ANALYSIS
In the field of theoretical surface growth there are a few standard tools developed for the analysis of surfaces and interfaces. One method of analysis of surface growth is through scaling concepts. There are some characteristics of surfaces and interfaces which obey some scaling relations. Studying these relations and their corresponding exponents, one can define a few universality classes in which different processes share the same scaling behavior 1,2 .
Another possible way to study theoretically these processes is through continuum growth equations. Stochastic differential equations describe the interface at large length scales. One can associate a specific growth process with an equation which classify them into the proper universality class. The Random Deposition process is the simplest discrete atomistic model and can be described by the equation where F represents the average number of particles per unit time that are added to the substrate at given position and η( r, t) represents the random fluctuation of this process, a noise that does not show spatial correlation in the substrate.
On the other hand, one of the most important continuum growth equations, related to the BD model of particles, represents a wide variety of processes of surface growth and nonequilibrium interfaces, such as those related to the formation of porous surfaces, corrosion processes of metallic surfaces and dissolution of a crystalline solid in a liquid medium 16,17 .
The equation 2, called KPZ, has the form, and includes a nonlinear term λ 2 (▽h) 2 , that takes into account lateral growth, describing the aggregation of particles parallel to the surface. Unfortunately, in many cases, continuum growth equations do not have and exact solutions for a specific class of problems or a specific spatial dimension, restricting their use.
For the class of problems previously mentioned, one can analyze them by means of discrete growth models, where the deposition process is simulated by a computer algorithm and the surface morphology is reproduced. Simulations are an essential link between theory and experiments and can provide some morphological details that are usually neglected by the equations but revealed by experimental techniques 18 .
In order to study numerically the morphology of a surface, one can calculate the interface width (surface roughness), w(L, t), a function of time and the linear size of the substrate.
In order to calculate w(L, t), we determine the vertical height of the surface relative to the  In both cases, by scaling concepts, one can study and characterize a growth model that represents in some sense a real surface growth process. The surface roughness increases as a power of time initially, w(L, t) ∼ t β , and, after some time of deposition, t x , the roughness saturates, w sat (L) ∼ L α . The time necessary to saturation depends on the system size, t x ∼ L z . These exponents are not independent and they are related in the form z = α/β 19 .
In theoretical studies of surface growth, one is interested in the calculation of these scaling exponents.

III. A MODEL FOR ANGULAR PARTICLE AGGREGATION
In the present study, particles with size of one lattice unit are randomly dropped over the

IV. RESULTS AND DISCUSSION
Our Monte Carlo simulations were performed on squared lattices with linear size ranging from L = 128 to 4096 and with different angles of deposition between 0°≤ θ ≤ 45°. At initial time steps, the growth is close to a surface generated by a RD model. However, as the time goes by, the lateral growth take place and the surface morphology change drastically.
A cross section of a surface generated by our simulation in shown in the figure 3, as one can see that the lateral aggregation of particles as the growth process evolve.
We found the best estimative for the roughness exponent was α ≈ 0.157 ± 0.001. This With the appropriated chosen of the constants and using γ = 2/3, the function S(f ) became where t is time and C and D are constants. By integration of Φ(( r, t)) from 0 to t,one can write h( r, t) , h 2 and h 2 as which leads to Using the definition of the surface roughness, and the growth exponent β = 1/3, the expected value for the Ballistic Deposition. The roughness exponent α cannot be obtained analytically by using the equation 1, since the simple equation used in this work does not depend on the height h. Only through Monte Carlo simulations, for our model, we can obtain the exponent α.

V. CONCLUSIONS AND FURTHER REMARKS
We studied the surface growth due to the deposition of particles dropped at random with different angles of deposition over a liner square lattice using computer simulations and stochastic differential equations. Our model is a modification of the simple RD model in an attempt to make it closer to real deposition processes.
We showed that the surface roughness evolves in time with different behavior, even when θ = 0. At initial times, the roughness behaves as in the RD model and β = 0.5. At long deposition times, the surface roughness grows slowly and the exponent β ≈ 0.33, as observed in the BD model. From our Monte Carlo simulations, we changed the angle of deposition and we showed that when θ = 45°and the size of the system is large enough, the growth exponent β = 1/3 is the same of the Ballistic Deposition model.
From our calculations, one can see that the same result from the simulations was obtained by a stochastic equation, using a proper noise -the pink noise -to describe the interface growth and its evolution.