Surface morphology and a dynamic scaling analysis in epitaxial growth: A kinetic Monte Carlo study

Physica A 524 (2019) 112–120

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Surface morphology and a dynamic scaling analysis in epitaxial growth: A kinetic Monte Carlo study ∗

Sonia Blel a , , Ajmi B.H. Hamouda a,b a b

Quantum and Statistical Physics Laboratory, Faculty of Science, University of Monastir, Monastir 5019, Tunisia Department of Physics, University of Maryland, College Park, MD 20742-4111, USA

highlights • • • •

The homo-epitaxial growth Cu (001), Co (001), Ge (001) and GaAs (001) are investigated. The kinetic Monte-Carlo simulation is used. The materials are classified into two sets based on the resultant growth morphology. Growth temperature effect on the critical exponents is studied.

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Article history: Received 20 December 2017 Received in revised form 20 January 2019 Available online 14 March 2019 Keywords: kinetic Monte-Carlo simulations Vicinal surfaces Surface morphology Roughening Dynamic scaling

a b s t r a c t We use Kinetic Monte-Carlo (KMC) simulations based on the solid-on-solid model to see the evolution of the surface morphology for different materials. We have performed a qualitative and quantitative study of the homo-epitaxial growth for (Cu, Co, Ge, and GaAs) over a large range of temperature, varying between 300 K and 700 K. Based on the growth morphology, we found two sets of materials. This classification is also confirmed by the analysis of their dynamic scaling properties (measurements of the critical exponents derived from the height–height correlation functions). Our results were compared to the available experimental and theoretical results and seem advantageous for the understanding of the growth dynamics. Different atomistic mechanisms may intervene in favoring adatoms attachment to surface steps. Here we have focused on the effect of the Ehrlich–Schwoebel barrier. However, we discussed how the magnitude of this barrier affects the scaling exponents as well as the surface morphology. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Recently, theoretical and experimental studies have confirmed the existence of scaling laws during the epitaxial growth. In particular, the growth of non-equilibrium surfaces is generally described by a Langevin-type equation and atomistic models which are called the universality class of MBE. We have studied the roughness and eventually determined the scaling invariance laws corresponding to these surfaces. In the last three decades, much attention has been focused on the problem of kinetic surface roughening associated with the non-equilibrium dynamics of growing interfaces [1]. Various models of epitaxial growth have been proposed and studied analytically or numerically, revealing a rich variety of interesting phenomena [2]. Particularly, the molecular beam epitaxy (MBE) has been described by these models and has been widely studied for many decades due to its technological applications and scientific interest. ∗ Corresponding author. E-mail address: [email protected] (S. Blel). https://doi.org/10.1016/j.physa.2019.03.021 0378-4371/© 2019 Elsevier B.V. All rights reserved.

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Experimental studies have confirmed the existence of scaling laws during the epitaxial growth [3]. Theoretical researches have proposed a scaling behavior for a growing rough surface [4–7]. For these reasons, it becomes important to identify the universality class of each material by measuring the scaling exponents. Indeed, many models, which are believed to be related to molecular beam-epitaxial (MBE) growth, have been extensively studied [8,9]. Most of these models are characterized by conservative processes where the number of particles is conserved after being deposited. Among the MBE growth models, the Wolf–Villain (WV) model [10,11] has been studied intensively because of its complex crossover behavior [12–15]. In contrast, several models wherein the evaporation of deposited particles is considered have been proposed for more realistic vapor deposition phenomena [8,9,16]. Obviously, the main goal of the dynamic scaling approach is to classify growth processes into a few universality classes, in order to develop general rules to ascertain their asymptotic behavior. Each university class is characterized by a set of critical exponents that seems to be specific for some dynamical features belonging to that class. In order to better understand the details of the growth processes, and from the conviction suggesting a close link between the grown morphology and the surface dynamics, we have analyzed the scaling behavior of several materials. These scaling exponents can be obtained by computing the height–height correlation function G, defined as the mean square of the height difference between two surfaces site-positions separated by a lateral distance r. At a given time t, we write:

⟨

G(X , t) = [h(x + X , t) − h(x, t)]2

⟩

(1)

where the upper bar denotes a spatial average and the angular brackets denote statistical ensemble-average [17–19]. It was computed in the direction parallel to the step-edge (the x direction) with X = n.a, where n is an integer, a is the lattice parameter and t is the time expressed in monolayer. A similar expression can be written for the correlation function G (Y , t) for separation Y in the direction perpendicular to the steps. Each surface, at a given time t, is characterized by specific values of scaling exponents; particularly the roughness exponent α and the growth exponent β . From the dynamic scaling, the correlation function has a similar dependence to the expression of interface width [20,21], and it is scaled according to the Family Vicsek ansatz [22]: G(r , t) = t 2β g(r /ξ )

(2) 2α

where the scaling function g(u) ∼ u for u ≪ 1 and g(u) ∼ constant for u ≫ 1 [23,24]. We consider the case of (2 + 1)D surface with ξ ∼ t 1/Z is the correlation length. The roughness exponent α and the dynamic exponent z describes the asymptotic behaviors of the growing interface at short length scale and long-time scale respectively. The ratio β = α /z is called the growth exponent and characterizes the short-time behavior of the surface. According to the values of these characteristic exponents, it is possible to classify the growth processes into different universality classes [3,23]. This paper is organized as follows. In Section 2, we have described the computational details of our kinetic Monte Carlo (kMC) simulations. In Section 3, first we have devoted to a comparative study, based on a dynamic scaling analysis, between the different materials like: Cu, Co, Ge, and GaAs, encountered in Ref [17–19]. Second, we have studied the effect of the Ehrlich–Schwoebel barrier (EES ) on the surface morphology and the scaling exponents. Conclusions remarks are given in Section 4. 2. Computational details The atomistic simulations are useful for the investigation of different fundamental physical phenomena such as the growth instabilities and the formation of nanostructures. However, it is a powerful tool for the modeling of many microscopic mechanisms such as: the diffusion anisotropy, the Ehrlich–Schwoebel (ES) barrier [25,26], the kink ES effect (KESE) [27–29]. At epitaxial surfaces, the incident particles are deposited on substrates for specific values of the temperature and the deposition flux, and then diffuse under chemical-bonding environment. Monte Carlo methods allow one to go in either direction, to find a description of nature that is as simple as possible, or to be as accurate as desired by including sufficiently many details in the simulations. Especially, KMC simulations allow a detailed understanding of all mechanisms induced in the simulations and a control of various processes during MBE-like growth at the microscopic scale. In this work, we simulated the epitaxial growth using the KMC method based on the SOS model of a simple cubic substrate which is essentially governed by two fundamental processes: the deposition and the diffusion on the substrate surface and no desorption is allowed [30]. This model is similar to that used by Kotrla et al. [31] and extensively tested by one of us [17–19]. More details are reported in Refs. [17–20]. The surface is characterized by an integer height h(r); at each point r on a square lattice of dimensions (Lx × Ly ). We have used periodic boundary conditions in both directions. The initial surface is a vicinal surface, with dimensions (400a × 400a) and terrace width L = Lx /N = 5a (expressed in lattice unit, a = 1), where N is the number of steps. When adsorbed on the surface, the adatoms can diffuse into four possible neighboring surface cells with a jump rate given by the Arrhenius form:

Γ = ν0 exp(−

Ed kB T

)

(3)

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S. Blel and A.B.H. Hamouda / Physica A 524 (2019) 112–120 Table 1 A summary of the energetic barriers: diffusion barriers (Ed ), bonding energies (E1 ) between nearest-neighboring site for different materials. Barrier Material

Ed (eV)

E1 (eV)

Cu/Cu [18] Co/Co [18,33] Ge/Ge [34–36] GaAs/GaAs [37]

0.564 0.891 1.3 1.2

0.350 0.224 0.370 0.3

Table 2 Scaling exponents versus the growth temperature T after deposition of 1000 ML for: (a) GaAs, (b) Ge, (c) Co, (d) Cu. In the last line, κ is determined from Eq. (7) for x = 1, that should be compared to (α − αloc )/z, see Ref. [16] for more details. (a) T (K)

300

350

400

450

500

600

700

αloc β 1/z α =β ∗z β − αloc /z κ

0 0.501

0 0.500

0.018 0.492

∞ ∞

∞ ∞

∞ ∞

0.501 0.501

0.500 0.500

0.492 0.487

0.185 0.390 0.144 2.708 0.363 0.355

0.274 0.274 0.241 1.136 0.207 0.179

0.321 0.172 0.297 0.579 0.076 0.035

0.324 0.117 0.365 0.320 0.001 0.004

(b) T (K)

300

350

400

450

500

600

700

αloc β 1/z α =β ∗z β − αloc /z κ

0 0.501

0 0.501

0.050 0.459

∞ ∞

∞ ∞

0.501 0.500

0.501 0.501

0 0.500 0.139 3.597 0.499 0.500

0.459 0.443

0.078 0.458 0.256 1.789 0.438 0.442

0.278 0.180 0.296 0.608 0.097 0.078

0.314 0.149 0.290 0.513 0.057 0.012

(c) T (K)

300

350

400

450

500

600

700

αloc β 1/z α =β ∗z β − αloc /z κ

0.010 0.489

0.249 0.307

∞ ∞

∞ ∞

0.489 0.483

0.307 0.241

0.304 0.221 0.274 0.806 0.137 0.107

0.328 0.165 0.287 0.574 0.070 0.022

0.351 0.153 0.295 0.517 0.049 0.007

0.381 0.108 0.247 0.437 0.013 0.004

0.391 0.040 0.102 0.392 0.000 0.000

(d) T (K)

300

350

400

450

500

600

700

αloc β 1/z α =β ∗z β − αloc /z κ

0.232 0.150 0.307 0.488 0.078 0.035

0.234 0.132 0.345 0.382 0.051 0.019

0.319 0.137 0.381 0.359 0.015 0.005

0.376 0.130 0.354 0.367 −0.003 0.004

0.420 0.132 0.246 0.357 0.028 0.011

0.426 0.013 0.039 0.333 0.003 0.008

0.442 0.013 0.042 0.309 −0.005 0.001

∞ ∞

where ν0 = 1013 Hz is a typical adatoms vibration frequency, Ed is the diffusion barrier for free adatoms, T is the growth temperature, and kB is Boltzmann’s constant. Furthermore, our model takes also into account the possible diffusion anisotropy of the incorporated adatoms at the step edges, the so-called ES barrier (EES ); that is introduced in the second part of Section 3. On the substrate surface, the adatoms-hopping barrier E takes the form: E = Ed + n1 E1 + EES

(4)

where Ed is the diffusion barrier, n1 is the number of nearest-neighbor adatoms before hopping (n1 = 0, 1, 2, 3, 4), E1 is the lateral bonding energy, and EES is an additional energy barrier added for ascending or descending moves in order to model the presence of a step-edge (ES) barrier [32]. The energies of diffusion barrier and the lateral bonding energy are given in Table 1. Therefore, for a free adatoms on the terrace, the corresponding energy is called Ed . However, this energy is (Ed + EES ) when it reaches the step-edge from the upper terrace or (Ed + E1 + EES ) from the lower terrace, i.e. for a step-edge position. In our simulations, we have fixed the flux at 1 ML/s, where ML denotes monolayers, varying the temperature between 300 K and 700 K, and for vanishing Ehrlich–Schwoebel barrier EES = 0 eV. The growth is conducted up-to 103 monolayers (ML)-thickness; at this point we find that the surface has reached a stationary state, characterized by nearly constant critical exponents. All of these exponents are extracted from the corresponding asymptotic behavior of the height–height correlation function Eq. (1) (see Ref. [17] for more details). On the other hand, we have included the ES barrier of 0.1 eV, The growth temperature T is fixed at 700 K and at fixed flux F = 1 ML/s.

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Fig. 1. KMC simulations of the surface morphology and height profile-plot in the direction perpendicular to the steps (y). For all materials, the initial surface is vicinal with (400a × 400a) sites with terrace widths L = 5a sites, at the substrate temperature T = 700 K and at fixed deposition flux F = 1 ML/s. The images were recorded after deposition of 1000 ML.

3. Results and discussion In this report, we have made a comparative study between different materials (semiconductor and metallic) based on the surface morphology and the scaling exponents for the stable (EES = 0 eV) and unstable growth (EES = 0.1 eV). 3.1. Qualitative description Experimental data obtained from Cu(001) [38], Ge(001) [39] and GaAs(001) [37,40] have revealed that a completely different morphological scenario can occur for homo-epitaxial growth on high quality single crystal surfaces [41]. Based on KMC simulation of the epitaxial growth and the available energetic barriers (diffusion, bonding energy) computed using VASP [18], the surface morphology resulting from our simulations for different materials, are shown in Fig. 1. The Fig. 1 shows the images of the surfaces morphology after growth of 1000 ML at fixed temperature of T = 700 K and deposition flux F = 1 ML/s. As it is shown, we can distinguish two sets of materials having qualitatively very similar surface morphology: we call (Cu, Co) as set-1 and (Ge, GaAs) as set-2. The surface morphology of set-1 is very similar to the results of Ref. [17] in the case of Si(001). These surfaces exhibit relatively rough morphology and very clearly distinguishable stepped structure along the step-edge; where the local roughness exponent αloc of order 0.41 in the studied temperature range. Set-2 exhibits, however, a more smooth morphology with a weaker local roughness of order 0.31. Indeed, from the height–height correlation function (plotted in Fig. 2) in the direction parallel to the step-edge G (X,t) (x-direction) and in the direction perpendicular to the steps G (Y,t) (y-direction), the critical exponents (α , β , z) and the correlation length ξ , at a given time, could be extracted (see Table 2). Besides the difference in the critical exponent, the correlation functions plotted in Fig. 2, confirm our previous classification of these materials in two set based on their morphology. To summarize, we can conclude that materials of set-2 are characterized by stronger correlations between the heights of atomic sites, with smaller local roughness αloc exponents; but higher global roughness α than those of set-1. We argue that the local roughness is a signature of short-scale correlations; however, the global roughness is a signature of large-scale correlations. This means that the local order is governed by a high αloc ; but the global order is governed by a high α . Then, since 1D-structures were formed step-by-step (i.e. from neighbor-to-neighbor); we assume they are

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Fig. 2. height–height correlation function G (x) (at time t corresponding to 1000 ML, F = 1 ML/s and T = 700 K) as a function of separation x in the direction parallel to the steps edge (left) and in the direction perpendicular to the steps (right), for different materials. The local roughness exponent αloc is shown for each material.

more sensitive to the local than the global roughness. Consequently, it is obvious to think that, materials of set-1 should be more convenient for growing 1D-nano-structures (like nanowires along step-edges [32,42,43]); whereas, materials of set-2, should be more convenient to grow 3D-nanostructures [41,44]. We can also note a higher local roughness is being linked to a very small anomalous exponent κ ; we believe that is the key factor to form long 1D-structures. Indeed, we found an exponent κ < 1/10 for set-1 (see Table 2). 3.2. Quantitative description In this section, we discuss the scaling dynamics with particular focus on two phenomena: anomalous scaling and super-roughness. However, the growth of non-equilibrium surfaces is generally described by a Langevin-type equation and atomistic models which are called the universality class of MBE [23,45]. The anomalous scaling is observed for many theoretical models particularly the model of Wolf–Villain (WV) [10], Kim–Das Sarma (KD) [46] and Das Sarma– Tamborenea (DT) [46,47]. In this study, we are interested in the equation of Lai–Das Sarma–Villain (LDV) (Eqs. (5)) [47], because it allows us to make comparison with our simulation results for the scaling exponents. This equation reads:

∂ h(⃗r , t) ⃗ 2 [∇ ⃗ h(⃗r , t)]2 + η(⃗r , t) ⃗ 4 h(⃗r , t) + λ22 ∇ = −υ4 ∇ ∂t where critical exponents are given by: α = 2/3, β = 1/5 and z = 10/3 [48,49].

(5)

Furthermore, Pang et al. [50] have proposed a space–time correlated noise to describe the super-roughening (where

α > 1) using a linear growth equation in (1 + 1) dimensions: ∂ h(x, t) = (−1)m+1 υ∇ 2m h(x, t) + η(x, t) (6) ∂t The critical exponents are given by: α = 2m/3, β = 1/3, z = 2m and κ = (m − 1)/3m [51]. It is interesting to note that for m = 1, Eq. (6) with white noise denotes the Edwards–Wilkinson equation [23,52] and Mullins–Wolf–Villain for m = 2 [10]. By using a simple scaling analysis (power counting [23]), it is easy to obtain the analytical expressions of the critical exponents (α , β , z), for both of Eqs. (5) and (6). Now we will summarize the basic knowledge about the autocorrelation function G (x, t). A self-affine surface is characterized by a scaling exponent κ = 0 and a roughness exponent (α = αloc ). Indeed, theoretical and experimental studies of self-affine kinetic roughening have discovered a rich variety of novel features [3]. Especially, the existence of anomalous roughening has received much attention [53]. Hence, it is worthwhile to emphasize that the local and global surface fluctuations are governed by distinctly different scaling exponents. Also, the anomalous roughening has been reported in many experimental studies including MBE of Si/Si (111) [53]. In surface roughening, the anomalous scaling is found for κ ̸ = 0 with local roughness exponent αloc ̸ = α = β × z [54]. Particularly, the height–height correlation function G (x, t) at short times writes: G(x, t) = t 2k x2αloc

(7)

where k is the anomalous scaling exponent and αloc is the local roughness exponent. On the other hand, the correlation function G obeys the anomalous scaling law: G(x, t) = t 2β gA (x/t 1/z ), where gA (u) ≈ u2αloc for u ≪ 1

(8)

Eventually, we can write: G(x, t) ≈ t 2(β−αloc /z) x2αloc for x ≪ t 1/z , where κ = (α − αloc )/z and β = α/z

(9)

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Fig. 3. height–height correlation function G (x) (at time t corresponding to 1000 ML, F = 1 ML/s and T = 500 K) as a function of separation x in the direction parallel to the steps (left) and in the direction perpendicular to the steps (right), for different materials. The value of local roughness exponent αloc is also indicated for each material. Table 3 Critical exponents after deposition of 1000 ML, T = 500 K for different materials. Unlike Set-1, Set-2 exhibits super-roughness (α > 1, k ̸ = 0) and small correlation length. Materials

Set-1

Exponents

Cu

Co

GaAs

Ge

αloc β α =β ∗z

0.420 0.132 0.357 0.011

0.351 0.153 0.517 0.007

0.274 0.274 1.136 0.179

0.078 0.458 1.789 0.442

k

ξ

Set-2

100 → 200 2 → 6

G ∗ 102

30 → 50 15 → 80

Table 4 Scaling exponents for different materials at fixed Ehrlich–Schwoebel barrier EES = 0.1 eV. The growth temperature is T = 700 K, after deposition of 1000 ML. Materials exponents

GaAs

Ge

Cu

Co

αloc β 1/z α =β ∗z β − αloc /z κ

0.826 0.714 0.134 5.328 0.603 0.478

0.661 0.598 0.148 4.040 0.500 0.449

0.430 0.039 0.059 0.661 0.013 0.011

0.358 0.040 0.075 0.533 0.013 0.010

In our simulations, the critical exponents (reported in Table 2 as a function of the growth temperature), were extracted from the asymptotic behavior of the height–height correlation function G (x, t) (Eqs. (2), (7), (8), (9)). In the temperature ranging from 300 K to 700 K, different growth regimes have been observed for all the studied materials. A general analysis of Table 3, show the existence of a super-roughening in set-2 (Ge, GaAs) rather than in set-1 (Cu, Co). Their correlation functions plotted in Fig. 3 in the direction parallel to the steps and in the direction perpendicular to the steps. For GaAs, the super-roughness is found at 450 K and 500 K. The same phenomenon is also observed in Ge(001) at 400 K and 500 K. For all materials, the global roughness undergoes a monotonic decay with increasing temperature, with crossover at the critical temperatures for materials presenting super-roughening. On the other hand, the local roughness rises almost monotonically with temperature, but its value is much higher in the case of set-2 materials. Since the overall behaviors of the critical exponents are almost the same for each set, only GaAs and Co materials will be described in detail and compared with the theoretical-predictions [55] and the experimental available data. The material GaAs (001) (Table 2.(a)) is characterized by a similar surface morphology obtained with the material Ge, but it is different from the critical exponents. Below T = 450 K, the surface exhibits a random deposition morphology. This surface is characterized by a growth exponent β = 1/2 and α = z = ∞. Above T = 450 K, contrary to previous observations, we have obtained that the roughness exponent α decreases with the increasing temperature and the same behavior for the growth exponent β . This study can give an idea for the experiments in the choice of the substrate temperature for the metal growth. Hence, for m = 2 (α = 4/3 β = 1/3, z = 4, κ = 1/6), the scaling exponents predicted in Eq. (6) are in good agreement with our simulation results at T = 500 K. At T = 600 K, our results (α = 0.58, β = 29, z = 3.36) are close to those predicted by the LDV equation in 2D (α = 2/3, β = 1/5, z = 10/3) [42]. Note that the difference between the global and local roughness exponent implies that these surfaces are not self-affine. Furthermore,

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Fig. 4. Monte Carlo simulation images of the surface morphology for different materials at fixed parameters (T = 700 K, EES = 0.1 eV).

the roughness exponent α > 1 implying the growth of the unstable surface has been investigated in recent works including the growth instability [34,35] and anomalous scaling behavior [26] The result of set-1 material is different from our previous observations of the set-2. At low temperature (T < 375 K), in the case of the Co, we have found a random deposition model (RD) (β = 1/2, α = z =∝); where there is insufficient surface diffusion (see Table 2(c)). Moreover, at T = 500 K, the critical exponent are comparable to those predicted by the LDV equation in 2D (α = 2/3, β = 1/5, z = 10/3) [27]. At higher temperature (T ≥ 600 K), the surface is self-affine, characterized by α = αloc and an anomalous scaling κ ≈ 0. In the intermediate temperature (400 K ≤ T ≤ 450 K), the computed results show a crossover regime. Their exponents are found between the model of random deposition and the equation of Lai–Das Sarma–villain (LDV). We note a monotonic decay in the global roughness exponent α with the increasing temperature. The same behavior is also obtained for the growth exponent β . In summary, our results for the set-1 as well as the set-2 are comparable to some theoretical and experimental results found in the literature at specific temperatures. Indeed, the two sets present a great difference in the computed critical exponents and hence, in their surface morphology. In the case of set-1, except for Cu, a very weak anomalous scaling is found (κ < 0.1) at moderate temperature that vanishes at relatively high temperature (T > 400 K). However this phenomena, is more pronounced (κ ≈ 1/2) in set-2 until T = 400 K. This result indicates that the super-roughening as well as the anomalous scaling are being a transient behavior and tend to disappear (κ → 0, asymptotically) at high temperature for all materials. For set-1 materials, the asymptotic behavior of the correlation function is reached much more quickly than for set-2 materials. Moreover, the super-roughening behavior manifests in set-2, but not in set-1. 3.3. Effect of Ehrlich–Schwoebel (ES) barrier In this work, our SOS model is essentially governed by the deposition rate, the surface diffusion and aggregation. In present section, we have included the Ehrlich–Schwoebel (ES) barrier, which is added at step-edges. However, we will discuss their effect on the scaling exponents and the evolution of the growth morphology. We have used a fixed temperature of T = 700 K and an ES barrier of EES = 0.1 eV; a value of this barrier allows to produce anisotropic surface diffusion of adatoms along the direction perpendicular to the step-edges. During the diffusion process, if an atom approaches to the step from the top side, directly it meets an extra barrier which can be even greater than the diffusion barrier on the terrace. This adatoms occupies temporarily an unfavorable position. The Fig. 4 shows the images of the surfaces morphology after growth of 1000 ML at fixed temperature of T = 700 K and ES barrier of EES = 0.1 eV. Set-2 (Ge, GaAs) exhibits a more rough morphology with the formation of islands on the substrate surface. In contrast, Set-1 (Co, Cu) show a more smooth morphology with a weaker local roughness. The height–height correlation function in the direction parallel to the step-edge G (X,t) and in the direction perpendicular to the steps G (Y,t), after deposition of 1000 ML, are presented in Fig. 5. Also, the scaling exponents are reported in Table 4 after deposition of 1000 ML. Table 4 shows super-roughness is obtain only in the Set-2 (α > 1) and a weaker anomalous exponent (κ ≈ 0) for Set-1. An important point is that the materials of set-2, should be more convenient to grow 3D-nanostructures, but the materials of set-1 should be more convenient for growing 1D-nano-structures in presence of the ES barrier. 4. Conclusions In this work, several materials (semi-conductors and metallic) were grown and studied from a morphological and dynamic-scaling point of view. By varying the temperature from 300 K to 700 K, these materials are found belonging to two different morphologies that can be classified into two sets from the analysis of their surface morphology and roughness. Our growth model shows different dynamical regimes characterized by super-roughness and scaling anomaly. According, we conclude that the super-roughness leads necessarily to an anomalous scaling. We also show that the set of materials (Cu, Co) are very similar to the results of Hamouda et al. [17] in case of the Si (001), but the other sets (Ge and GaAs) are different from the surface morphology and the scaling exponents. We argue that set-1 provides materials

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Fig. 5. height–height correlation function G (x) (at time t corresponding to 1000 ML, F = 1 ML/s and T = 700 K) as a function of separation x in the direction parallel to the steps (left) and in the direction perpendicular to the steps (right), for different materials at fixed Ehrlich–Schwoebel barrier EES = 0.1 eV.

that should be more appropriate for growing one-dimensional structures 1D (like nanowires). However materials of set-2 should be more appropriate for growing three-dimensional structures 3D (like islands and aggregates). Therefore, even though the temperature-roughening dependence of the different materials is dominated by the same kinetic processes, the specific parameters of each material (diffusion barrier, bonding energy), lead to quantitatively different behaviors in the critical exponents (as a function of the temperature) and different surface morphologies. The flux dependence of the surface morphology is beyond the scope of this paper (more details can be found in Ref. [16] and recent results will appear elsewhere). This classification of different materials into two sets, according to their morphologies, proves two interesting conclusions: first, the same condition of growth does not necessarily imply the same growth mode; and second, the same surface morphology does not necessarily imply the same universality class as it is believed. Hence, the theory of universality classes must be questioned. The extension of this study will focus on the understanding of the link between the scaling exponent and the surface morphology. Also, for a better understanding of the optimum growth conditions and precisely the roughness role in directing either 1D, 2D or 3D structures, it is worth interesting to know what should be the most suitable material for that purpose. Acknowledgment We gratefully acknowledge the financial support from the ‘‘Direction Générale de la Recherche Scientifique et de la Technologie’’ (DGRST), Tunisia. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

M. Kardar, G. Parisi, Y.-C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986) 889. J.M. Kim, S.D. Sarma, Discrete models for conserved growth equations, Phys. Rev. Lett. 72 (1994) 2903. J. Krug, Origins of scale invariance in growth processes, Adv. Phys. 46 (1997) 139. P. Meakin, The growth of rough surfaces and interfaces, Phys. Rep. 235 (1993) 189–289. J. Krug, H. Spohn, in: C. Godrèche (Ed.), Solids far from Equilibrium: Growth, Morphology and Defects, Cambridge University Press, Cambridge, 1992. S.D. Sarma, P. Punyindu, Z. Toroczkai, Non-universal mound formation in non-equilibrium surface growth, Surf. Sci. 457 (2000) L369 – L375. P.P. Chatraphorn, Z. Toroczkai, S. Das Sarma, Epitaxial mounding in limited-mobility models of surface growth, Phys. Rev. B 64 (2001) 205407. J. Villain, Continuum models of crystal growth from atomic beams with and without desorption, J. Phys. I 1 (1991) 19–42. J.M. Kim, S. Das Sarma, Dynamical universality of the nonlinear conserved current equation for growing interfaces, Phys. Rev. E. 51 (1995) 1889. D.E. Wolf, J. Villain, Growth with surface diffusion, J. Eur. Phys. Lett. 13 (1990) 1350. L. Changhan, L.S. Bub, Surface growth on diluted lattices by a restricted solid-on-solid model, Phys. Rev. E 80 (2009) 021134. J. Krug, in: A.J. McKane, M. Droz, J. Vannimenus, D.E. Wolf (Eds.), In Scale Invariance Interfaces and Non-Equilibrium Dynamics, Plenum Press, New York, 1995. K. Park, B. Kahng, S.S. Kim, Surface dynamics of the Wolf-Villain model for epitaxial growth in 1 + 1 dimensions, Phys. A 210 (1994) 146–154. P. Šmilauer, M. Kotrla, Crossover effects in the Wolf-Villain model of epitaxial growth in 1+1 and 2+1 dimensions, Phys. Rev. B 49 (1994) 57–69. M. Kotrla, P. Šmilauer, Nonuniversality in models of epitaxial growth, Phys. Rev. B 53 (1996) 13777. S.H. Yook, Y. Kim, Dynamical universality classes of vapor deposition models with evaporation, Phys. Rev. E 60 (1999) 3837. A.B.H. Hamouda, A. Pimpinelli, R.J. Phaneuf, Anomalous scaling in epitaxial growth on vicinal surfaces: meandering and mounding instabilities in a linear growth equation with spatiotemporally correlated noise, Surf. Sci. 602 (2008) 2819–2827. A.B.H. Hamouda, R. Sathiyanarayanan, A. Pimpinelli, T.L. Einstein, Role of codeposited impurities during growth. I. Explaining distinctive experimental morphology on Cu(0 0 1), Phys. Rev. B 83 (2011) 035423.

120

S. Blel and A.B.H. Hamouda / Physica A 524 (2019) 112–120

[19] A.B. Hamouda, A. Pimpinelli, F. Nita, Linear growth equation with spatiotemporally correlated noise to describe the meandering instability, IOP Conf. Ser. 100 (2008) 072009; A. Ben-Hamouda, N. Absi, P.E. Hoggan, A. Pimpinelli, Growth instabilities and adsorbed impurities: a case study, Phys. Rev. B 77 (2008) 245430. [20] K.D. Ho, K.J. Min, A model of random deposition with relaxation on fractal substrates, J. Stat. Mech. Theory Exp. 08 (2010) 08008. [21] A.A. Masoudi, M. Zahedifar, S.V. Farahani, Scaling properties of fractional continuous growth equations, Phys. B 410 (2013) 177–181. [22] F. Family, T. Vicsek, Scaling of the active zone in the Eden process on percolation networks and the ballistic deposition model, J. Phys. A: Math. Gen. 18 (1985) L75. [23] A.-L. Barabási, H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, 1995. [24] P. Meakin, Fractals, Scaling and Growth Far from Equilibrium, Cambridge University Press, New York, Cambridge, U.K.; New York, 1998. [25] R.L. Schwoebel, E.J. Shipsey, Step motion on crystal surfaces, J. Appl. Phys. 37 (1966) 3682; R.L. Schwoebel, J. Appl. Phys. 40 (1969) 614. [26] G. Ehrlich, F.G. Hudda, Atomic view of surface self-diffusion: tungsten on tungsten, J. Chem. Phys. 44 (1966) 1039. [27] O. Pierre-Louis, M.R. D’Orsogna, T.L. Einstein, Edge diffusion during growth: the Kink Ehrlich-Schwoebel effect and resulting instabilities, Phys. Rev. Lett. 82 (1999) 3661. [28] M.V. Ramana Murty, B.H. Cooper, Instability in molecular beam epitaxy due to fast edge diffusion and corner diffusion barriers, Phys. Rev. Lett. 83 (1999) 352. [29] P. Politi, J. Krug, Crystal symmetry, step-edge diffusion, and unstable growth, Surf. Sci. 446 (2000) 89–97. [30] S. Blel, A.B.H. Hamouda, B. Mahjoub, T.L. Einstein, Competing growth processes induced by next-nearest-neighbor interactions: effects on meandering wavelength and stiffness, Phy. Rev. B. 95 (2017) 085404. [31] M. Kotrla, J. Krug, P. Šmilauer, Submonolayer epitaxy with impurities: kinetic Monte Carlo simulations and rate-equation analysis, Phys. Rev. B 62 (2000) 2889. [32] S. Blel, A.B.H. Hamouda, Formation of Ag and Co nanowires at the step of the Cu(100) vicinal surfaces: A kinetic Monte Carlo study, Vacuum 151 (2018) 133–139. [33] H. Garbouj, M. Said, F. Picaud, C. Ramseyer, D. Spanjaard, M.C. Desjonquères, Growth of perfect and smooth Ag and Co monatomic wires on Pt vicinal surfaces: A kinetic Monte Carlo study, Surface Sci. 603 (2009) 22–26. [34] L. Huang, F. Liu, X. Gong, Strain effect on adatom binding and diffusion in homo- and heteroepitaxies of Si and Ge on (001) surfaces, Phys. Rev. B 70 (2004) 155320. [35] J.-N. Aqua, T. Frisch, Elastic interactions and kinetics during reversible submonolayer growth: Monte Carlo simulations, Phys. Rev. B 78 (2008) 121305 (R). [36] A. Ben Hadj Hamouda, Thése de Doctorat Morphologie et stabilité des surfaces cristallines nanostructurées, dynamiques des instabilités : Théorie et modélisation, Université Blaise Pascal, 2007. [37] C.-F. Lin, A.B.H. Hammouda, H.-C. Kan, N.C. Bartelt, R.J. Phaneuf, Directing self-assembly of nanostructures kinetically: patterning and the Ehrlich-Schwoebel barrier, Phys. Rev. B 85 (2012) 085421. [38] H.-J. Ernst, F. Fabre, R. Folkerts, J. Lapujoulade, Observation of a growth instability during low temperature molecular beam epitaxy, Phys. Rev. Lett. 72 (1994) 112; H.-J. Ernst, F. Fabre, R. Folkerts, J. Lapujoulade, Kinetics of growth of Cu on Cu(001), J. Vac. Sci. Technol. A 12 (1994) 1. [39] J.E. Van Nostrand, S. Jay Chey, M.-A. Hasan, D.G. Cahill, J. Greene, Surface morphology during multilayer epitaxial growth of Ge(001), Phys. Rev. Lett. 74 (1995) 1127. [40] C.-F. Lin, H.-C. Kan, S. Kanakaraju, C. Richardson, R. Phaneuf, Directed kinetic self-assembly of mounds on patterned GaAs (001): tunable arrangement, pattern amplification and self-limiting growth, Nanomaterials 4 (2014) 344–354. [41] J.A. Stroscio, D.T. Pierce, M.D. Stiles, A. Zangwill, L.M. Sander, Coarsening of unstable surface features during Fe(001) homoepitaxy, Rev. Lett. 75 (1995) (1995) 4246. [42] H. Garbouj, M. Said, C. Ramseyer, F. Picaud, Modelling the Cu mono-atomic wire formation on Pt vicinal surfaces using kinetic Monte Carlo simulations, Modell. Simul. Mater. Sci. Eng. 18 (2010) 085009, 14pp. [43] N.N. Negulyaev, V.S. Stepanyuk, W. Hergert, P. Bruno, J. Kirschner, Atomic-scale self-organization of Fe nanostripes on stepped Cu(111) surfaces: molecular dynamics and kinetic Monte Carlo simulations, Phys. Rev. B 77 (2008) 085430. [44] G. Apostolopoulos, N. Boukos, J. Herfort, A. Travlos, K.H. Ploog, Surface morphology of low temperature grown GaAs on singular and vicinal substrates, Mater. Sci. Eng. B88 (2002) 205–208. [45] A. Pimpinelli, J. Villain, Physics of Crystal Growth, Cambridge University Press, Cambridge, 1998. [46] S. Das Sarma, C. Lanczycki, R. Kotlyar, S.V. Ghaisas, Scale invariance and dynamical correlations in growth models of molecular beam epitaxy, Phys. Rev. E 53 (1996) 359; S. Das Sarma, P. Tamborenea, A new universality class for kinetic growth: one- dimensional molecular-beam epitaxy, Phys. Rev. Lett. 66 (1991) 325. [47] P.I. Tamborenea, S. Das Sarma, Surface-diffusion-driven kinetic growth on one-dimensional substrates, Phys. Rev. E 48 (1993) 2575. [48] Y. Shim, D.P. Landau, Dynamic finite-size scaling of the normalized height distribution in kinetic surface roughening, Phys. Rev. E 64 (2001) 036110. [49] Z.-W. Lai, S. Das Sarma, Kinetic growth with surface relaxation: continuum versus atomistic models, Phy. Rev. Lett. 66 (1991) 2348. [50] N.-N. Pang, W.-J. Tzeng, Superroughening by linear growth equations with spatiotemporally correlated noise, Phys. Rev. E 70 (2004) 011105. [51] S. Blel, A.B.H. Hamouda, B. Mahjoub, P. Hoggan, B. Oujia, Scaling analysis of self-assembled structures and related morphological information in epitaxial growth, Superlattices Microstruct. 102 (2017) 155–165. [52] S.F. Edwards, D.R. Wilkinson, The surface statistics of a granular aggregate, Proc. Roy. Soc. London A 381 (1982) 17–31. [53] H.N. Yang, G.C. Wang, T.M. Lu, Instability in low-temperature molecular-beam epitaxy growth of Si/Si(111), Phys. Rev. Lett. 73 (1994) 2348. [54] J.M. López, Scaling approach to calculate critical exponents in anomalous surface roughening, Phys. Rev. Lett. 83 (1999) 4594; Juan M. López, Mario Castro, Rafael Gallego, Scaling of local slopes, conservation laws, and anomalous roughening in surface growth, Rev. Lett. 94 (2005) 166103. [55] M. Constantin, C. Dasgupta, P.P. Chatraphorn, S.N. Majumdar, S.D. Sarma, Persistence in nonequilibrium surface growth, Phys. Rev. E 69 (2004) 061608.