# Step Bunches, Nanowires and Other Vicinal “Creatures”—Ehrlich–Schwoebel Effect by Cellular Automata

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

_{DS}diffusional steps are realized. One CA unit followed by one MC unit and the completion of the surface particles to their initial concentration c

_{0}represents one-time step of a simulation. This design allows for the study of large systems in long simulations.

_{0}. Periodic boundary conditions are imposed in the direction along the steps, while helical periodic boundary conditions preserving step differences assumed in the direction across steps are applied. The simulation procedure involves the MC part during which particles in the adatom layer diffuse along the surface and the realization of CA rules when particles can be built into the crystal structure. The time step of the vicCA procedure is completed by compensation of adatom concentration to its initial value c

_{0}, and then the next time step starts.

_{dES}, which is 1 for no barrier and 0 for an infinite barrier. All other cases in between are studied. Similarly, we set inverse Schwoebel barrier P

_{iES}, which is located at the bottom of step (Figure 1b) and prevents particles from jumping towards step from below or back. We combined these two cases and added one more parameter p

_{w}, namely, such one that decides about the energy of particle that stays at the bottom of the step. The site at the bottom of the step is at a particular position, because adatoms that occupy this site interact with particles that build crystal steps, and their energy is changed by these interactions. An assumption of different potential energy at this particular point causes that the model is more realistic. Such a particle, if its energy is larger than in other positions, jumps more easily over a left-hand or right-hand barrier (Figure 1c), while if its energy is lower, its jump is more difficult. The parameter p

_{w}changes from 0 (which means that the particle is blocked at its position) to the lower value of P

_{dES}

^{−1}or P

_{iES}

^{−1}Particle jumps out of the site at the bottom of the step with a probability of p

_{w}P

_{dES}or p

_{w}P

_{iES}, and for the maximal value of p

_{w}, one of them is equal to 1. The parameter causing the jumping over barriers becomes asymmetric but fulfills the detailed balance condition. As we see, together with the CA rules for incorporation at steps, it allows particles to form various surface patterns, depending on the choice of all three parameters p

_{w}, P

_{dES}, and P

_{iES}. During the diffusion process, all adatoms try to perform n

_{DS}diffusional jumps, but only those that point at an unoccupied neighboring lattice site are performed.

_{0}, thus setting equilibrium between the lattice gas and the ambience. The above procedure describes the sequence of a single time step and is repeated many times during each run of the simulation. Therefore, we measure the time in growth updates.

## 3. Pattern Formation

^{5}vicCA simulation time steps. We used the barrier of infinite height, given the probability of jumping across the steps P

_{dES}= 0. Each of the meandered steps are clearly seen in this plot. Together they form noticeable “fingers” in the direction vertical to the steps. Such fingers are quite often observed at the top of grown crystals.

_{iES}= 0, and we show the system formed after 10

^{5}vicCA simulation time steps. Three bunches parallel to the initial step direction are well seen in the figure. It can be noted that due to the introduced possibility of particle nucleation, islands appear at wider terraces. We observed that at given conditions, islands are attached to the moving steps and do not initiate further 3D growth. The height of bunches increases with the longer time of the simulation.

_{dES_asm}= 0.1. In the example illustrated in Figure 4, resulting surface patterns with well-formed nanowires can be clearly seen. The potential landscape, in this case, is presented at the right side of the plot. Because of the asymmetry of dES, which is not compensated by the asymmetry of iES like in Figure 1c, this case leads to aggregation of particles on the top of islands. In this respect, this case is different from all other systems presented in the current work. We can see that such conditions result in creation of long, thin nanowires. The positions of nanowires are random because we do not initiate nanowire growth at given places on the surface. In addition, the energy potential in Figure 4 works only when nanowire is formed at the surface.

_{w}. These patterns can be understood as a combination of the above-mentioned orderings or as a completely new system on the surface driven by a step potential.

_{iES}= 0.2, P

_{dES}= 0.6. When decreasing P

_{dES}= 0.4, which means that the height of direct Schwoebel barrier dES increases and more particles are trapped at the top of the step, we get a very interesting new pattern, referred to as nanopillars or nanocolumns (Figure 5b). It consists of cubic formations, much wider than the initial inter-step distance l

_{0}. They have smooth, straight-up walls, and they grow very tall when the simulation continues. Finally, the gaps between these formations become narrow and very deep.

_{dES}becomes smaller and is of the order of P

_{iES}, formations at the surface are closer to nanowires, as can be seen in Figure 5c, where the pattern obtained for P

_{dES}= 0.2 is shown. If these nanowires are compared to Figure 4, it can be seen that walls here are not so smooth, and nanowire diameter decreases with its height. Note, however, that these nanowires were created on setting the potential given in Figure 1c, not the one shown in the inset of Figure 4. Particles are not trapped at the top of the wire, as they are in the case of Figure 4. Nevertheless, without trapping, it was possible to build a structure with nanowires; this means no droplet to initiate and control nanowire growth is needed here. It should also be stressed here that we used low P

_{dES}, close to P

_{iES}, but not 0. It appears that when P

_{dES}is lowered to 0, we obtain another type of structure. In this case, the whole surface is covered by pyramids with similar shapes and sizes, as shown in Figure 5d. It is a very characteristic 3D formation, quite often generated at the surface of growing crystals. Such shapes change to classical meandered patterns shown in Figure 2 when the c

_{0}is lower, and the particles attach to steps before they stick together and nucleate. In addition, this means that meanders are formed in the case of very slow growth, while 3D growth in the form of pyramids will be present for a faster crystal growth process. It is worth noting that all the above-mentioned orderings reflect the underlying symmetry of the lattice. Therefore, the islands have square or rectangular shapes, and the formations shown in Figure 5 are also squares. A hexagonal lattice as the base would convert these shapes to triangles or hexagons.

_{iES}and P

_{dES}for one value of c

_{0}= 0.02. P

_{dES}and P

_{iES}were changed by 0.2, and as a result, in Figure 6, we can see a map of possible orderings in this case. Let us note that a large part of this plot is covered by a regular structure, which means that steps move evenly, with small fluctuations, forming perfect crystal structures. Apart from the regular structure, we have all patterns mentioned above, except meanders that would replace pyramids in this diagram for lower c

_{0}, which refers to the lower particle flux, determining the crystal growth rate. The point (0,0) corresponding to infinite direct and inverse barriers is very specific. It concerns the situation where the particle cannot diffuse to the step, neither from the bottom nor from the top. The only possibility is to land exactly at the step, and only then the particle can attach to it. Such events happen but are very rare; hence, we see very slow growth of rather straight steps for these parameters.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Table of Cellular Automaton Rules

Crystal seed Rule(1,1,1,1,0) = 1 Rule(1,1,0,1,1) = 1 Rule(1,1,1,1,1) = 1 Adsorption at step Rule(1,0,0,1,2) = 1 Rule(1,0,0,2,1) = 1 Rule(1,0,1,2,0) = 1 Rule(1,1,2,0,0) = 1 Rule(1,2,1,0,0) = 1 Rule(1,2,0,0,1) = 1 Rule(1,0,1,1,2) = 1 Rule(1,0,1,2,1) = 1 Rule(1,0,2,1,1) = 1 Rule(1,1,0,1,2) = 1 Rule(1,1,0,2,1) = 1 Rule(1,1,1,0,2) = 1 Rule(1,1,1,2,0) = 1 Rule(1,1,2,0,1) = 1 Rule(1,1,2,1,0) = 1 Rule(1,2,0,1,1) = 1 Rule(1,2,1,0,1) = 1 Rule(1,2,1,1,0) = 1 Rule(1,1,1,1,2) = 1 Rule(1,1,1,2,1) = 1 Rule(1,1,2,1,1) = 1 Rule(1,2,1,1,1) = 1 Rule(1,0,2,1,0) = 1 Rule(1,1,0,0,2) = 1 | Adsorption at kink Rule(1,0,0,2,2) = 1 Rule(1,0,2,2,0) = 1 Rule(1,2,0,0,2) = 1 Rule(1,2,2,0,0) = 1 Rule(1,0,1,2,2) = 1 Rule(1,0,2,1,2) = 1 Rule(1,0,2,2,1) = 1 Rule(1,1,0,2,2) = 1 Rule(1,1,2,0,2) = 1 Rule(1,1,2,2,0) = 1 Rule(1,2,0,1,2) = 1 Rule(1,2,0,2,1) = 1 Rule(1,2,1,0,2) = 1 Rule(1,2,1,2,0) = 1 Rule(1,2,2,0,1) = 1 Rule(1,2,2,1,0) = 1 Rule(1,0,2,2,2) = 1 Rule(1,2,0,2,2) = 1 Rule(1,2,2,0,2) = 1 Rule(1,2,2,2,0) = 1 Rule(1,1,1,2,2) = 1 Rule(1,1,2,1,2) = 1 Rule(1,1,2,2,1) = 1 Rule(1,2,1,1,2) = 1 Rule(1,2,1,2,1) = 1 Rule(1,2,2,1,1) = 1 Rule(1,1,2,2,2) = 1 Rule(1,2,1,2,2) = 1 Rule(1,2,2,1,2) = 1 Rule(1,2,2,2,1) = 1 Rule(1,2,2,2,2) = 1 | Filling voids Rule(0,1,1,1,1) = 1 No Adsorption Rule(1,0,0,0,0) = 0 Rule(1,0,0,0,1) = 0 Rule(1,0,0,1,0) = 0 Rule(1,0,1,0,0) = 0 Rule(1,1,0,0,0) = 0 Rule(1,0,0,1,1) = 0 Rule(1,0,1,0,1) = 0 Rule(1,0,1,1,0) = 0 Rule(1,0,0,1,1) = 0 Rule(1,1,0,1,0) = 0 Rule(1,1,0,0,1) = 0 Rule(1,1,1,0,1) = 0 Rule(1,0,1,1,1) = 0 Rule(1,0,0,0,2) = 0 Rule(1,0,0,2,0) = 0 Rule(1,0,2,0,0) = 0 Rule(1,2,0,0,0) = 0 Rule(1,0,1,0,2) = 0 Rule(1,0,2,0,1) = 0 Rule(1,1,0,2,0) = 0 Rule(1,2,0,1,0) = 0 Rule(1,0,2,0,2) = 0 Rule(1,2,0,2,0) = 0 Rule(0,*,*,*,*) = 0 |

## References

- Oreg, Y.; Refael, G.; von Oppen, F. Helical Liquids and Majorana Bound States in Quantum Wires. Phys. Rev. Lett.
**2010**, 105, 177002. [Google Scholar] [CrossRef] [Green Version] - Lutchyn, R.M.; Sau, J.D.; Das Sarma, S. Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures. Phys. Rev. Lett.
**2010**, 105, 077001. [Google Scholar] [CrossRef] [Green Version] - Mourik, V.; Zuo, K.; Frolov, S.M.; Plissard, S.R.; Bakkers, E.P.a.M.; Kouwenhoven, L.P. Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices. Science
**2012**, 336, 1003–1007. [Google Scholar] [CrossRef] [Green Version] - Grünberg, P.; Schreiber, R.; Pang, Y.; Brodsky, M.B.; Sowers, H. Layered Magnetic Structures: Evidence for Antiferromagnetic Coupling of Fe Layers across Cr Interlayers. Phys. Rev. Lett.
**1986**, 57, 2442. [Google Scholar] [CrossRef] [PubMed] - Baibich, M.N.; Broto, J.M.; Fert, A.; Nguyen, V.D.F.; Petroff, F.; Etienne, P.; Creuzet, G.; Friederich, A.; Chazelas, J. Giant Magnetoresistance of (001)Fe/(001)Cr Magnetic Superlattices. Phys. Rev. Lett.
**1988**, 61, 2472. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fermon, C.; Pannetier-Lecoeur, M. Noise in GMR and TMR Sensors. In Giant Magnetoresistance (GMR) Sensors. Smart Sensors, Measurement and Instrumentation; Reig, C., Cardoso, S., Mukhopadhyay, S.C., Eds.; Springer: Berlin/Heidelberg, Germany, 2013; Volume 6, pp. 47–70. [Google Scholar]
- Zheludev, N. The life and times of the LED—A 100-year history. Nat. Photonics
**2007**, 1, 189–192. [Google Scholar] [CrossRef] - Isobe, Y.; Iida, D.; Sakakibara, T.; Iwaya, M.; Takeuchi, T.; Kamiyama, S.; Akasaki, I.; Amano, H.; Imade, M.; Kitaoka, Y.; et al. Growth of AlGaN/GaN heterostructure on vicinal m-plane free-standing GaN substrates prepared by the Na flux method. Phys. Status Solidi (A) Appl. Mater. Sci.
**2011**, 208, 1191–1194. [Google Scholar] [CrossRef] - Amano, H.; Sawaki, N.; Akasaki, I. Metalorganic vapor phase epitaxial growth of a high quality GaN film using an AlN buffer layer. Appl. Phys. Lett.
**1986**, 48, 353. [Google Scholar] [CrossRef] [Green Version] - Yang, J.J.; Pickett, M.D.; Li, X.; Ohlberg, D.A.A.; Stewart, D.R.; Williams, R.S. Memristive switching mechanism for metal/oxide/metal nanodevices. Nat. Nanotechnol.
**2008**, 3, 429–433. [Google Scholar] [CrossRef] - Yang, J.J.; Strukov, D.B.; Stewart, D.R. Memristive Devices for Computing. Nat. Nanotechnol.
**2013**, 8, 13–24. [Google Scholar] [CrossRef] - Ehrlich, G.; Hudda, F.G. Atomic view of surface self-diffusion—Tungsten on tungsten. J. Chem. Phys.
**1966**, 44, 1039. [Google Scholar] [CrossRef] - Schwoebel, R.L.; Shipsey, E.J. Step motion on crystal surfaces. J. Appl. Phys.
**1966**, 37, 3682. [Google Scholar] [CrossRef] - Misbah, C.; Pierre-Louis, O.; Saito, Y. Crystal surfaces in and out of equilibrium: A modern view. Rev. Mod. Phys.
**2010**, 82, 981. [Google Scholar] [CrossRef] - Saúl, A.; Métois, J.-J.; Ranguis, A. Experimental evidence for an Ehrlich-Schwoebel effect on Si(111). Phys. Rev. B
**2001**, 65, 075409. [Google Scholar] [CrossRef] - Rogilo, D.I.; Fedina, L.I.; Kosolobov, S.S.; Ranguelov, B.S.; Latyshev, A.V. Critical Terrace Width for Two-Dimensional Nucleation during Si Growth on Si(111)-(7 × 7) Surface. Phys. Rev. Lett.
**2013**, 111, 036105. [Google Scholar] [CrossRef] - De Theije, F.K.; Schermer, J.J.; Van Enckevort, W.J.P. Effects of nitrogen impurities on the CVD growth of diamond: Step bunching in theory and experiment. Diam. Relat. Mater.
**2000**, 9, 1439–1449. [Google Scholar] [CrossRef] - Xie, M.H.; Cheung, S.H.; Zheng, L.X.; Ng, Y.F.; Wu, H.; Ohtani, N.; Tong, S.Y. Step bunching of vicinal GaN(0001) surfaces during molecular beam epitaxy. Phys. Rev. B Condens. Matter Mater. Phys.
**2000**, 61, 9983–9985. [Google Scholar] [CrossRef] [Green Version] - Zheng, H.; Xie, M.H.; Wu, H.S.; Xue, Q.K. Kinetic energy barriers on the GaN (0001) surface: A nucleation study by scanning tunneling microscopy. Phys. Rev. B
**2008**, 77, 045303. [Google Scholar] [CrossRef] - Gianfrancesco, A.G.; Tselev, A.; Baddorf, A.P.; Kalinin, S.V.; Vasudevan, R.K. The Ehrlich–Schwoebel barrier on an oxidesurface: A combined Monte-Carlo and in situ scanning tunneling microscopy approach. Nanotechnology
**2015**, 26, 455705. [Google Scholar] [CrossRef] - Sarma, D.S.; Punyindu, P.; Toroczkai, Z. Non-universal mound formation in non-equilibrium surface growth Z. Surf. Sci.
**2000**, 457, L369–L375. [Google Scholar] [CrossRef] [Green Version] - Leal, F.F.; Ferreira, S.C.; Ferreira, S.O. Modelling of epitaxial film growth with an Ehrlich–Schwoebel barrier dependent on the step height. J. Phys. Condens. Matter.
**2011**, 23, 292201. [Google Scholar] [CrossRef] [Green Version] - Palczynski, K.; Herrmann, P.; Heimel, G.; Dzubiella, J. Characterization of step-edge barrier crossing of para-sexiphenyl on the ZnO (101 [combining macron] 0) surface. J. Phys. Chem. Chem. Phys.
**2016**, 18, 25329. [Google Scholar] [CrossRef] - Xiang, S.K.; Huang, H. Ab initio determination of Ehrlich–Schwoebel barriers on Cu {111}. Appl. Phys. Lett.
**2008**, 92, 101923. [Google Scholar] [CrossRef] - Hao, J.; Zhang, L. Strongly reduced Ehrlich–Schwoebel barriers at the Cu (111) stepped surface with In and Pb surfactants. Surf. Sci.
**2018**, 667, 13–16. [Google Scholar] [CrossRef] - Xie, M.H.; Leung, S.Y.; Tong, S.Y. What causes step bunching-negative Ehrlich-Schwoebel barrier versus positive incorporation barrier. Surf. Sci.
**2002**, 515, L459–L463. [Google Scholar] [CrossRef] - Krzyżewski, F.; Załuska–Kotur, M.A. Coexistence of bunching and meandering instability in simulated growth of 4H-SiC (0001) surface. J. Appl. Phys.
**2014**, 115, 213517. [Google Scholar] [CrossRef] [Green Version] - Krzyżewski, F.; Załuska-Kotur, M.A. Stability diagrams for the surface patterns of GaN (0001¯) as a function of Schwoebel barrier height. J. Cryst. Growth
**2017**, 457, 80–84. [Google Scholar] [CrossRef] - Krasteva, A.; Popova, H.; Krzyżewski, F.; Załuska-Kotur, M.; Tonchev, V. Unstable vicinal crystal growth from cellular automata. AIP Conf. Proc.
**2016**, 1722, 220014. [Google Scholar] - Krzyżewski, F.; Załuska-Kotur, M.A.; Krasteva, A.; Popova, H.; Tonchev, V. Step bunching and macrostep formation in 1D atomistic scale model of unstable vicinal crystal growth. J. Cryst. Growth
**2017**, 474, 135–139. [Google Scholar] [CrossRef] [Green Version] - Krzyżewski, F.; Załuska-Kotur, M.A.; Krasteva, A.; Popova, H.; Tonchev, V. Scaling and Dynamic Stability of Model Vicinal Surfaces. Cryst. Growth Des.
**2019**, 19, 821–831. [Google Scholar] [CrossRef] - Toktarbaiuly, O.; Usov, V.O.; Coileáin, C.; Siewierska, K.; Krasnikov, S.; Norton, E.; Bozhko, S.I.; Semenov, V.N.; Chaika, A.N.; Murphy, B.E.; et al. Step bunching with both directions of the current: Vicinal W(110) surfaces versus atomistic-scale model. Phys. Rev. B Condens. Matter Mater. Phys.
**2018**, 97, 035436. [Google Scholar] [CrossRef] [Green Version] - Popova, H.; Krzyżewski, F.; Załuska-Kotur, M.A.; Tonchev, V. Quantifying the Effect of Step–Step Exclusion on Dynamically Unstable Vicinal Surfaces: Step Bunching without Macrostep Formation. Cryst. Growth Des.
**2020**, 20, 7246–7259. [Google Scholar] [CrossRef] - Turski, H.; Krzyżewski, F.; Feduniewicz-Żmuda, A.; Wolny, P.; Siekacz, M.; Muziol, G.; Cheze, C. Nowakowski-Szukudlarek Krzesimir, Xing Huili Grace, Jena Debdeep, Załuska-Kotur Magdalena, Skierbiszewski Czesław, Unusual step meandering due to Ehrlich-Schwoebel barrier in GaN epitaxy on the N-polar Surface. Appl. Surf. Sci.
**2019**, 484, 771–780. [Google Scholar] [CrossRef] [Green Version] - Sato, M.; Uwaha, M. Growth law of step bunches induced by the Ehrlich-Schwoebel effect in growth. Surf. Sci.
**2001**, 493, 494–498. [Google Scholar] [CrossRef]

**Figure 1.**Energy landscape for diffusing particles. (

**a**) Direct Ehrlich–Schwoebel (dES) barrier at the top of the step, with jump probability given by P

_{dES}. (

**b**) Inverse Ehrlich–Schwoebel (iES) barrier below the step with jump probability given by P

_{iES}. (

**c**) P

_{dES}, P

_{iES}and the changed depth of the potential well below. The jump rate out of the well is described by parameter p

_{w}.

**Figure 2.**Meanders, c

_{0}= 0.01, P

_{iES}= 1, P

_{dES}= 0, p

_{w}= 1, l

_{0}= 5, n

_{DS}= 10. System size 200 × 200.

**Figure 3.**Bunches, c

_{0}= 0.02, P

_{iES}= 0, P

_{dES}= 1, p

_{w}= 1, l

_{0}= 10, n

_{DS}= 10. System size 300 × 100.

**Figure 4.**Nanowires, c

_{0}= 0.02, P

_{iES}= 0, P

_{dES_asm}= 0.1, p

_{w}= 1, n

_{DS}= 10, l

_{0}= 100. System size 200 × 200. Inset: potential that realizes given pattern.

**Figure 5.**(

**a**) Antibunches, c

_{0}= 0.02, P

_{iES}= 0.2, P

_{dES}= 0.6, p

_{w}= 1.66, l

_{0}= 10, n

_{DS}= 10, time steps 3 × 10

^{6}. (

**b**) Nanopillars, c

_{0}= 0.02, P

_{iES}= 0.2, P

_{dES}= 0.4, p

_{w}= 2.5, l

_{0}= 10, n

_{DS}= 10, time steps 2 × 10

^{5}. (

**c**) Nanowires, c

_{0}= 0.02, P

_{iES}= 0.1, P

_{dES}= 0.2, p

_{w}= 5, l

_{0}= 10, n

_{DS}= 10, time steps 3 × 10

^{5}. (

**d**) Pyramids, c

_{0}= 0.02

_{,}P

_{iES}= 1.0, P

_{dES}= 0.0, p

_{w}= 1.0, l

_{0}= 5, n

_{DS}= 10, time steps 4 × 10

^{5}.

**Figure 6.**Diagram of pattern formation. Different symbols correspond to different patterns: describes regular step ordering, means bunches, bunches with antibunches, describe nanopillars, nanowires, pyramids, and is specific (0, 0) point.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Załuska-Kotur, M.; Popova, H.; Tonchev, V.
Step Bunches, Nanowires and Other Vicinal “Creatures”—Ehrlich–Schwoebel Effect by Cellular Automata. *Crystals* **2021**, *11*, 1135.
https://doi.org/10.3390/cryst11091135

**AMA Style**

Załuska-Kotur M, Popova H, Tonchev V.
Step Bunches, Nanowires and Other Vicinal “Creatures”—Ehrlich–Schwoebel Effect by Cellular Automata. *Crystals*. 2021; 11(9):1135.
https://doi.org/10.3390/cryst11091135

**Chicago/Turabian Style**

Załuska-Kotur, Magdalena, Hristina Popova, and Vesselin Tonchev.
2021. "Step Bunches, Nanowires and Other Vicinal “Creatures”—Ehrlich–Schwoebel Effect by Cellular Automata" *Crystals* 11, no. 9: 1135.
https://doi.org/10.3390/cryst11091135