İtici ve Çekici Etkileşimlerin Basamaklı Yüzeyler Üzerinde Oluşan Basamak Gruplaşmaları Üzerindeki Etkileri
Abstract
İki boyutta eş merkezli
dairesel monoatomik basamaklardan oluşan
pürüzlülük sıcaklığı altında olan bir yüzey tartışılmıştır. Yüzey üzerindeki
basamaklar arasında çekici ve itici etkileşimlerin olduğu varsayılmıştır.
Çekici ve itici etkileşimlerin sırasıyla ve şeklinde
değiştikleri düşünülmüştür. Buradaki monoatomik basamaklar arasındaki teras genişliğini
göstermektedir. Difüzyon denkleminin çözümü kutupsal koordinatlar kullanılarak iki
boyutta elde edilmiştir. İlk yüzey gelişirken, yerel kütle transferinin sadece
yüzey difüzyonu nedeniyle basamak akış modeli altında gerçekleştiği farz
edilmiştir. Çalışma içerisinde biçimine
sahip zarf fonksiyonları ile sınırlandırılan ilk yüzeyler ele alınmıştır. Basamaklar
arasında sadece çekici etkileşimin olduğu durumda yüzeyler düzgün şekilde gelişmiştir.
Basamaklar arasında çekici ve itici etkileşimlerin her ikisinin olduğu kabul
edildiği zaman yüzey üzerinde geniş düz teraslarla ayrılan basamak
gruplaşmaları oluşmuştur. Difüzyon sınırlı düzende (DL) bütün yüzey yapıları
için, zaman içerisindeki yüzey morfolojisi ve yüzey yükseklik gelişimi incelenirken,
basamak gruplaşmasının olduğu ve basamak gruplaşmasının olmadığı bölgelere ait
bir parametre uzayı elde edilmiştir.
References
- [1]. W. K. Burton, N. Cabrera, F. C. Frank, The Growth of Crystals and the Equilibrium Structure of Their Surfaces, Philos. Trans.R. Soc. London A 243 (1951) 299.
- [2]. J. Villain, Healing of a Rough Surface at Low Temperature, Europhys. Lett. 2 (1986) 532.
- [3]. M. Uwaha, Relaxation of Crystal Shapes Caused by Step Motion, J. Phys. Soc. Jpn 57 (1988) 1681.
- [4]. N. Israeli, D. Kandel, Profile Scaling in Decay of Nanostructures, Phys. Rev. B 80 (1998) 3300.
- [5]. N. Israeli, D. Kandel, Profile of Decaying Crystalline Cone, Phys. Rev. B 60 (1999) 5946.
- [6]. M. Uwaha, K. Watanabe, Decay of an Island on a Facet via Surface Diffusion, J. Phys. Soc. Jpn 69 (2000) 497.
- [7]. A. Ichimiya, K. Hayashi, E.D. Williams, T.L. Einstein, M. Uwaha, K. Watanabe, Decay of Silicon Mounds: Scaling Laws and Description with Continuum Step Parameters, Appl. Surf. Sci. 175-176 (2001) 33.
- [8]. S. Kodambaka, N. Israeli, J. Bareno, W. Swiech, K.Ohmori, I. Petrov, J.E. Greene, Low-Energy Electron Microscopy Studies of Interlayer Mass Transport Kinetics on TiN(1 1 1), Surf. Sci. 560 (2004) 53.
- [9]. M. Esen, A.T. Tüzemen, M. Ozdemir, Equilibration of a Cone: KMC Simulation Results, European Physical Journal B 85 (2012) 117.
- [10]. A. F. Andreev, A.Y. Kosevich, Capillary Phenomena in the Theory of Elasticity, Sov. Phys. JETP 54 (1981) 761.
- [11]. E. M. Pearson, T. Halicioglu, W. A. Tiller, Long-range Ledge-ledge Interactions on Si(1 1 1) Surfaces: I. No Kinks or Surface Point Defects, Surf. Sci., 184, (1987) 401.
- [12]. A. F. Andreev, Faceting Phase Transitions of Crystals, Sov. Phys. JETP 53 (1981) 1063.
- [13]. E. E. Gruber, W.W. Mullins, On the Theory of Anisotropy of Crystalline Surface Tension, J. Chem. Solids 28 (1967) 875.
- [14]. J. J. Saenz, N. Garcia, Classical Critical Behaviour in Crystal Surfaces Near Smooth and Sharp Edges, Surf. Sci., 155, (1985) 24.
- [15]. J. C. Heyraud, J. J. Metois, Establishment of the Equilibrium Shape of Metal Crystallites on a Foreign Substrate: Gold on Graphite, J. Cryst. Growth, 50, (1980) 571.
- [16]. J. J. Metois, J. C. Heyraud, Analysis of the Critical Behaviour of Curved Regions in Equilibrium Shapes of in Crystals, Surf. Sci., 180, (1987) 647.
- [17]. J. Frohn, M. Giesen, M. Poensgen, J. F. Wolf, H. Ibach, Attractive Interaction Between Steps, Phys. Rev. Lett., 67, (1991), 3543.
- [18]. K. Sudoh, H. Iwasaki, E. D. Williams, Facet Growth Due to Attractive Step-Step Interactions on Vicinal Si(1 1 3), Surf. Sci., 452 (2000) 287-292.
- [19]. V. B. Shenoy, S. Zhang, W. F. Saam, Step Bunching Transitions on Vicinal Surfaces with Attractive Step Interactions, Surf. Sci., 467 (2000) 58-84.
- [20]. W. Kossel, Zur Theorie des Kristallwachstums, Nachr. Ges. Wiss. Göttingen, Math-Phys. Kl., (1927) 135.
- [21]. I. N. Stranski, Zur Theorie des Kristallwachstums, Z. Phys. Chem. (Leipzig), 136, (1928) 259.
- [22]. M. Ozdemir, The Morphology of Crystalline Surfaces in the Presence of Attractive Step Interactions, J. Phys.: Condens. Matter 11, (1999) 1915.
- [23]. M. Ozdemir, Epitaxial Growth to Non-Planar Substrates by Step-Flow, App. Surf. Sci., 152 (1999) 200-212.
- [24]. A. T. Tüzemen, M. Esen, M. Ozdemir, Scaling Properties of Equilibrating Semiconductor Mounds of Various Initial Shapes, J. Crystal Growth, 470 (2017) 94-98.
- [25]. S. Tanaka, N. C. Bartelt, C. C. Umbach, R. M. Tromp, J. M. Blakely, Step Permeability and the Relaxation of Biperiodic Gratings on Si(0 0 1), Phys. Rev. Lett. 78 (1997) 3342.
- [26]. H. C. Jeong, E. D. Williams, Steps on Surfaces: Experiment and Theory, Surface Science Reports 34 (1999) 171-294.
Abstract
A surface which consists concentric circular monoatomic steps in two
dimensions and below its roughening temperature is discussed. Repulsive and
attractive interactions between steps on the surface are considered. It is
supposed that repulsive and attractive interactions vary as and respectively. Here indicates the terrace width
between monoatomic steps. The solution of diffusion equation is achieved in
two- dimension by using polar coordinates. While the initial surface evolves,
it is supposed that the local mass transfer exists because of the surface
diffusion only under the step-flow model. In the study initial surfaces bounded
by envelope functions which have the form of are dealt. In the case of only
repulsive interaction between steps surfaces evolve properly. When both
repulsive and attractive interactions between steps are accepted step bunchings
separated by large flat terraces occur on the surface. While the surface
morphology and the evolution of the height of surface in time are investigated
for all surface structures in Diffusion-Limited (DL) regime, a parameter space
of bunching and no bunching regions is derived.
References
- [1]. W. K. Burton, N. Cabrera, F. C. Frank, The Growth of Crystals and the Equilibrium Structure of Their Surfaces, Philos. Trans.R. Soc. London A 243 (1951) 299.
- [2]. J. Villain, Healing of a Rough Surface at Low Temperature, Europhys. Lett. 2 (1986) 532.
- [3]. M. Uwaha, Relaxation of Crystal Shapes Caused by Step Motion, J. Phys. Soc. Jpn 57 (1988) 1681.
- [4]. N. Israeli, D. Kandel, Profile Scaling in Decay of Nanostructures, Phys. Rev. B 80 (1998) 3300.
- [5]. N. Israeli, D. Kandel, Profile of Decaying Crystalline Cone, Phys. Rev. B 60 (1999) 5946.
- [6]. M. Uwaha, K. Watanabe, Decay of an Island on a Facet via Surface Diffusion, J. Phys. Soc. Jpn 69 (2000) 497.
- [7]. A. Ichimiya, K. Hayashi, E.D. Williams, T.L. Einstein, M. Uwaha, K. Watanabe, Decay of Silicon Mounds: Scaling Laws and Description with Continuum Step Parameters, Appl. Surf. Sci. 175-176 (2001) 33.
- [8]. S. Kodambaka, N. Israeli, J. Bareno, W. Swiech, K.Ohmori, I. Petrov, J.E. Greene, Low-Energy Electron Microscopy Studies of Interlayer Mass Transport Kinetics on TiN(1 1 1), Surf. Sci. 560 (2004) 53.
- [9]. M. Esen, A.T. Tüzemen, M. Ozdemir, Equilibration of a Cone: KMC Simulation Results, European Physical Journal B 85 (2012) 117.
- [10]. A. F. Andreev, A.Y. Kosevich, Capillary Phenomena in the Theory of Elasticity, Sov. Phys. JETP 54 (1981) 761.
- [11]. E. M. Pearson, T. Halicioglu, W. A. Tiller, Long-range Ledge-ledge Interactions on Si(1 1 1) Surfaces: I. No Kinks or Surface Point Defects, Surf. Sci., 184, (1987) 401.
- [12]. A. F. Andreev, Faceting Phase Transitions of Crystals, Sov. Phys. JETP 53 (1981) 1063.
- [13]. E. E. Gruber, W.W. Mullins, On the Theory of Anisotropy of Crystalline Surface Tension, J. Chem. Solids 28 (1967) 875.
- [14]. J. J. Saenz, N. Garcia, Classical Critical Behaviour in Crystal Surfaces Near Smooth and Sharp Edges, Surf. Sci., 155, (1985) 24.
- [15]. J. C. Heyraud, J. J. Metois, Establishment of the Equilibrium Shape of Metal Crystallites on a Foreign Substrate: Gold on Graphite, J. Cryst. Growth, 50, (1980) 571.
- [16]. J. J. Metois, J. C. Heyraud, Analysis of the Critical Behaviour of Curved Regions in Equilibrium Shapes of in Crystals, Surf. Sci., 180, (1987) 647.
- [17]. J. Frohn, M. Giesen, M. Poensgen, J. F. Wolf, H. Ibach, Attractive Interaction Between Steps, Phys. Rev. Lett., 67, (1991), 3543.
- [18]. K. Sudoh, H. Iwasaki, E. D. Williams, Facet Growth Due to Attractive Step-Step Interactions on Vicinal Si(1 1 3), Surf. Sci., 452 (2000) 287-292.
- [19]. V. B. Shenoy, S. Zhang, W. F. Saam, Step Bunching Transitions on Vicinal Surfaces with Attractive Step Interactions, Surf. Sci., 467 (2000) 58-84.
- [20]. W. Kossel, Zur Theorie des Kristallwachstums, Nachr. Ges. Wiss. Göttingen, Math-Phys. Kl., (1927) 135.
- [21]. I. N. Stranski, Zur Theorie des Kristallwachstums, Z. Phys. Chem. (Leipzig), 136, (1928) 259.
- [22]. M. Ozdemir, The Morphology of Crystalline Surfaces in the Presence of Attractive Step Interactions, J. Phys.: Condens. Matter 11, (1999) 1915.
- [23]. M. Ozdemir, Epitaxial Growth to Non-Planar Substrates by Step-Flow, App. Surf. Sci., 152 (1999) 200-212.
- [24]. A. T. Tüzemen, M. Esen, M. Ozdemir, Scaling Properties of Equilibrating Semiconductor Mounds of Various Initial Shapes, J. Crystal Growth, 470 (2017) 94-98.
- [25]. S. Tanaka, N. C. Bartelt, C. C. Umbach, R. M. Tromp, J. M. Blakely, Step Permeability and the Relaxation of Biperiodic Gratings on Si(0 0 1), Phys. Rev. Lett. 78 (1997) 3342.
- [26]. H. C. Jeong, E. D. Williams, Steps on Surfaces: Experiment and Theory, Surface Science Reports 34 (1999) 171-294.
Details
| Primary Language | English |
|---|---|
| Journal Section | Natural Sciences |
| Authors | |
| Publication Date | September 30, 2018 |
| Submission Date | March 28, 2018 |
| Acceptance Date | May 24, 2018 |
| Published in Issue | Year 2018Volume: 39 Issue: 3 |