Theoretical Physics Discrete simulation models of surface growth Martin Bjork¨ Erik Deng [email protected] [email protected] SA104X Degree Project in Engineering Physics, First Level Department of Theoretical Physics Royal Institute of Technology (KTH) Supervisor: Jack Lidmar May 22, 2014 Abstract In this thesis the time evolution and scaling properties of different discrete models of surface growth using computer simulation is studied. The models are mainly using random deposition at a perpendicular angle to the substrates to model the adsorption process, and both one dimensional and two dimensional surfaces are considered. The Edwards-Wilkinson, Kardar-Parisi-Zhang and Mullins equations are also studied as analytical methods to describe the growth of surfaces. The scaling exponents derived from these equations are used as reference when analysing the exponents calculated from the simulation models studied in this thesis. We have found that the simulation models do not correspond perfectly with the analytical models for surface growth, suggesting possible flaws in our models or definitions. Despite the possible flaws, the models prove to be powerful tools for analysing the time evolution of surface growth. Furthermore, we have shown that most of the simulation models exhibit the expected scaling properties, which indicates that the surfaces do have the self-affine structure they are presumed to have. Sammanfattning I denna avhandling har tidsutvecklingen och skalningsegenskaperna hos olika diskreta modeller for¨ yttillvaxt¨ studerats med hjalp¨ av datorsimuleringar. De flesta modellerna anvander¨ slumpmassig¨ deposition vinkelratt¨ mot substraten for¨ att modellera adsorption, och bade˚ en- och tvadimensionella˚ modeller har studerats. Edward-Wilkingson-, Kardar-Parisi-Zhang- och Mullinsekvationen har aven¨ studerats som analytiska modeller for¨ att beskriva yttillvaxt.¨ Skalningsexponenterna som har erhallits˚ fran˚ dessa ekvationer har anvants¨ som referenser vid analysen av exponenterna som har raknats¨ ut fran˚ de simuleringsmodeller som har studerats i denna avhandling. Vi har kommit fram till att simuleringsmodellerna inte stammer¨ overrens¨ perfekt med de analytiska modellerna, vilket tyder pam˚ ojliga¨ brister i vara˚ modeller eller definitioner. Trots de mojliga¨ bristerna har modellerna visat sig vara kraftfulla verktyg vid analys av tidsutvecklingen av yttillvaxt.¨ Vidare har vi visat att de flesta av simuleringsmodellerna uppvisar de forv¨ antade¨ skalningsegenskaperna, vilket ar¨ ett tecken pa˚ att ytorna har den sjalvaffina¨ struktur de antas ha. 1 Contents Introduction 3 Background theory 3 Stochastic growth equations . 5 Random deposition . 5 Edwards-Wilkinson equation . 5 Kardar-Parisi-Zhang equation . 6 Mullins equation . 6 Models of surface growth 7 Random deposition . 7 Ballistic deposition . 8 Simple surface diffusion . 9 Kinetic Monte Carlo . 9 Results and discussion 12 Random deposition . 12 Ballistic deposition . 12 Simple surface diffusion . 14 Kinetic Monte Carlo . 14 Summary and conclusions 18 References 19 2 Introduction Background theory Surface growth is often associated with the accretion of In this section, we present the background theory of the a physical surface, such as growing crystals and metals. subject at hand. This theory is presented in its entirety in [9] This is also largely the focus of the area, and there are and [10], but is reiterated here for convenience. We begin many applications of this approach, both in academics by introducing the concept of fractals and self-affinity, and industries [1], including crystal growth [2], biological followed by a presentation of important topics, such as growth [3] and growing snow layers [4]. However, there are roughness and scaling exponents. We conclude this section also other areas where the same general ideas and concepts by presenting different analytical methods for describing are applicable, such as fluid flows [5], fire fronts [6] and surface growth, i.e. the Edwards-Wilkinson equation (EW), bacterial growth [7][8]. the Kardar-Parisi-Zhang equation (KPZ) and the Mullins diffusion equation. The results derived from these equations Generally, surface growth can be considered to be the will be used as references when comparing with the models time evolution of an interface - the interface representing we have used. the growing surface. It has been observed that these interfaces share some common properties in a wide range of When observed at different scales, the morphology of applications, for example the time evolution of the interface fractals does not change, meaning that they look the same roughness and the self-affine structure of the interface [9]. regardless of the scale at which they are being observed. These common properties makes this a very interesting For fractals, the scaling factor is the same in all directions, subject to study because of the wide range of possible but there are also objects that share a lot of properties with applications of the results. fractals, the sole difference being that they have different scaling factors in each direction. Objects with this property In this thesis, we have chosen to study different discrete are called self-affine. Surfaces are an example of self-affine simulation models of surface growth. The main focus has objects; when scaled correctly it is virtually impossible been the implementation of the different models, but we to tell at which magnification the surface is observed. have also investigated the time evolution of the roughness of However, they are not self-affine in the classical sense since each model, as well as their scaling properties. Furthermore, they do not have identical patterns repeating on different we have analysed a number of analytical stochastic growth scales, but they are statistically self-affine; the statistical equations as a comparison to the results obtained from the properties of the surface are conserved when scaling - an simulation models. The equations we have analysed are example of this is shown in 1. This self-affinity can be used the Edward-Wilkinson equation, the Kardar-Parisi-Zhang to determine the scaling properties of the surface. equation and the Mullins equation - each having a different approach accounting for the processes occurring in surface growth. The scaling properties of each equation are com- pared to the simulation models to determine the validity of the different models. We begin by presenting the background theory of the subject, introducing the most important concepts of surface growth as well as definitions of the properties we study in this thesis and analyses of the stochastic growth equa- tions. We continue by describing the simulation models we have considered and our implementations of them, including a short summary of the strengths and weaknesses of each model. Subsequently, we present our results and a discus- sion, giving a full analysis of each model and their relation with the stochastic growth equations. Lastly, a summary of our findings is presented, concluding this thesis. Figure 1: An example of a statistically self-affine structure; without a reference it is virtually impossible to tell which image is shown at a higher magnification. Image adapted from [11]. 3 The surface roughness, also sometimes called the inter- single value, and by plotting the roughness as a function face width, is defined as the standard deviation of the time of t/t , the systems will be scaled to saturate at the same ⇥ dependent height, as time, as can be seen in figure 3. d 1 L w(L, t) [h(i, t) h¯(t)]2 , (1) ⌘ vLd − u i=1 u X where w is the surfacet roughness, L is the system size, d is the dimension, h(i, t) is the height of the surface at lattice site i and time t, and h¯(t) is the mean height of the surface at time t. For a general surface the roughness increases as a power of time up to a crossover time t , sometimes called ⇥ the saturation time, w(L, t) tβ , [t t ] . (2) ⇠ ⌧ ⇥ The exponent β is called the growth exponent which de- scribes the time-dependent roughening dynamics. After the crossover time is reached the roughening saturates, giving the saturation value, wsat. The saturation value increases with increased system size L and the dependence also fol- lows a power law, ↵ wsat(L) L , [t t ] , (3) ⇠ ⇥ where ↵ is the roughening exponent that describes the roughening after system saturation. The crossover time also Figure 3: Roughness curves for different system sizes, before and depends on a power law after scaling. The roughness is scaled using the saturation value ↵ wsat L and the time is scaled using the crossover time t z ⇠ ⇥ ⇠ t L , (4) Lz. ⇥ ⇠ where z is called the dynamic exponent. One easy way of estimating t is shown in figure 2. ⇥ This suggests that w(L, t)/wsat(L) is only dependent on t/t , thus giving the general scaling relation of the rough- ⇥ ness, also called the Family-Vicsek scaling relation [12] t w(L, t) w (L)f ⇠ sat t ✓ ⇥ ◆ t w(L, t) L↵f . (5) ⇠ Lz ✓ ◆ Here the function f(u), with u = t/t , is a scaling function ⇥ satisfying f(u) uβ , [u 1] ⇠ ⌧ f(u)=const , [u 1] . (6) The exponents are also related to each other. Con- sider the fitted straight lines in figure 2; Approaching the Figure 2: Two fitted lines, one for the time dependent growth, the β other for the saturated value, are used to estimate the crossover crossover point from the left we get w(t ) t , while ap- ⇥ ⇠ ↵⇥ time, t , at their intersection point. proaching from the right gives us w(t ) L , according ⇥ ⇥ ⇠ to (2) and (3). Thus tβ L↵ and by using (4) we get that ⇥ ⇠ ↵ z = . (7) For different system sizes, by plotting w(L, t)/wsat(L) β as a function of time, the saturation values are scaled to a 4 This relation between the exponents holds for any Taking the mean of the square of (11) we get growth process that obeys the scaling relation (5).