Scaling relation of domain competition on (2+1)-dimensional ballistic deposition model with surface diffusion

Kenyu Osada; Hiroyasu Katsuno; Toshiharu Irisawa; Yukio Saito

doi:10.1088/1674-4926/37/9/092001

J. Semicond. > 2016, Volume 37 > Issue 9 > 092001

SEMICONDUCTOR PHYSICS

Scaling relation of domain competition on (2+1)-dimensional ballistic deposition model with surface diffusion

Kenyu Osada¹, Hiroyasu Katsuno^{2, 3}, Toshiharu Irisawa² and Yukio Saito⁴

+ Author Affiliations

Corresponding author: Hiroyasu Katsuno, katsuno@fc.ritsumei.ac.jp

DOI: 10.1088/1674-4926/37/9/092001

Abstract: During heteroepitaxial overlayer growth multiple crystal domains nucleated on a substrate surface compete with each other in such a manner that a domain covered by neighboring ones stops growing. The number density of active domains ρ decreases as the height h increases. A simple scaling argument leads to a scaling law of ρ~h^-γ with a coarsening exponent γ=d/z, where d is the dimension of the substrate surface and z the dynamic exponent of a growth front. This scaling relation is confirmed by performing kinetic Monte Carlo simulations of the ballistic deposition model on a two-dimensional (d=2) surface, even when an isolated deposited particle diffuses on a crystal surface.

Key words: domain competition, ballistic deposition model, Kardar-Parisi-Zhang universality class, surface diffusion

References

[1]	Hayes W, Stoneham A M. Defects and defect processes in nonmetallic solids. New York: John Wiley & Sons, 1985
[2]	Amano H, Sawai N, Akasaki I, et al. Metalorganic vapor phase epitaxial growth of a high quality GaN film using an AlN buffer layer. Appl Phys Lett, 1986, 48: 353 doi: 10.1063/1.96549
[3]	Nakamura S. GaN growth using GaN buffer layer. Jpn J Appl Phys, 1991, 30: L1705 doi: 10.1143/JJAP.30.L1705
[4]	Zheleva T S, Nam O H, Bremser M D, et al. Dislocation density reduction via lateral epitaxy in selectively grown GaN structures. Appl Phys Lett, 1997, 71: 2472 doi: 10.1063/1.120091
[5]	Hersee S D, Sun X Y, Wang X, et al. Nanoheteroepitaxial growth of GaN on Si nanopillar arrays. J Appl Phys, 2005, 97: 124308 doi: 10.1063/1.1937468
[6]	Saito Y, Omura S. Domain competition during ballistic deposition: effect of surface diffusion and surface patterning. Phys Rev E, 2011, 84: 021601 http://cn.bing.com/academic/profile?id=2062908940&encoded=0&v=paper_preview&mkt=zh-cn
[7]	Family F, Vicsek T. Scaling of the active zone in the Eden process on percolation networks and the ballistic deposition model. J Phys A, 1985, 18(2): L75 doi: 10.1088/0305-4470/18/2/005
[8]	Barabási A L, Stanley H E. Fractal concepts in surface growth. Cambridge: Cambridge University Press, 1995
[9]	Kardar M, Parisi G, Zhang Y C. Dynamic scaling of growing interfaces. Phys Rev Lett, 1986, 56: 889 doi: 10.1103/PhysRevLett.56.889
[10]	Farnudi B, Vvdensky D D. Large-scale simulations of ballistic deposition: the approach to asymptotic scaling. Phys Rev E, 2011, 83: 020103(R) http://cn.bing.com/academic/profile?id=2070087653&encoded=0&v=paper_preview&mkt=zh-cn
[11]	Alves S G, Oliveira T J, Ferreira S C. Origins of scaling corrections in ballistic growth models. Phys Rev E, 2014, 90: 052405 doi: 10.1103/PhysRevE.90.052405
[12]	Ballestad A, Ruck B J, Admcyk M, et al. Evidence from the surface morphology for nonlinear growth of epitaxial GaAs films. Phys Rev Lett, 2001, 86: 2377 doi: 10.1103/PhysRevLett.86.2377
[13]	Halpin-Healy T, Palasantzas G. Universal correlators and distributions as experimental signatures of (2+1)-dimensional Kardar-Parisi-Zhang growth. Euro Phys Lett, 2014, 105: 50001 doi: 10.1209/0295-5075/105/50001
[14]	Michely T, Krug J. Island, mounds and atoms: patterns and processes in crystal growth far from equilibrium. Berlin: Springer-Verlag, 2004
[15]	Ehrlich G, Hudda F G. Atomic view of surface self-diffusion: tungsten on tungsten. J Chem Phys, 1966, 44: 1039 doi: 10.1063/1.1726787
[16]	Schwoebel R L, Shipsey E J. Step motion on crystal surfaces. J Appl Phys, 1966, 37: 3682 doi: 10.1063/1.1707904
[17]	Bortz A B, Kalos M H, Lebowitz J L. A new algorithm for Monte Carlo simulation of Ising spin system. J Comput Phys, 1975, 17: 10 doi: 10.1016/0021-9991(75)90060-1
[18]	Saberi A A, Dashti-Naserabadi H, Rouhani S. Classification of (2+1)-dimensional growing surfaces using Schramm-Loewner evolution. Phys Rev E, 2010, 82: 020101(R) http://cn.bing.com/academic/profile?id=2081862734&encoded=0&v=paper_preview&mkt=zh-cn
[19]	Oliveira T J, Alves S G, Ferreira S C. Kardar-Parisi-Zhang universality class in (2+1) dimensions: universal geometry-dependent distributions and finite-time corrections. Phys Rev E, 2013, 87: 040102(R) http://cn.bing.com/academic/profile?id=2012768116&encoded=0&v=paper_preview&mkt=zh-cn
[20]	Moser K, Wolf D E. Numerical solution of the Kardar-Parisi-Zhang equation in one, two and three dimensions. Physica A, 1991, 178: 215 doi: 10.1016/0378-4371(91)90017-7
[21]	Tamborenea P, Das Sarma S. Surface-diffusion-driven kinetic growth on one-dimensinoal substrate. Phys Rev E, 1993, 48: 2575 doi: 10.1103/PhysRevE.48.2575

Fig. 1. Surface diffusion of an isolated particle at a site O. It can move to either a site A or B. If an isolated particle moves to B,then it stops motion by forming a lateral bond.

DownLoad: Full-Size Img PowerPoint

Fig. 2. (Color online) Kinetic roughening of the growth front of the BD model without diffusion. (a) Log-log plot of growth front width W versus average height h for various system sizes L. (b) Log-log plot of asymptotic height-difference correlation function $G(r)$ versus distance r for various system sizes. (c) Sum of scaling exponents $\alpha+z$ versus system size L. The value asymptotically approaches a constant 2. Every data point is obtained by averaging 100 samples.

DownLoad: Full-Size Img PowerPoint

Fig. 3. (Color online) Kinetic roughening exponents, (a) α (circle) and β (triangle), and (b) z (circle) and α+ z (triangle) of the growth front versus surface diffusion rate $\tilde D$ . The system size is set L = 256.

DownLoad: Full-Size Img PowerPoint

Fig. 4. (Color online) Vertical cross sections of the domain coarsening during the 2D BD (a) without and (b) with surface diffusion with a surface diffusion rate $\tilde D$ = 1024. Colors differentiate domains. Crystals have a width L = 256 and grow up to the height h = 234 in this figure.

DownLoad: Full-Size Img PowerPoint

Fig. 5. (Color online) Vertical cross sections of the domain coarsening during the 2D BD (a) without and (b) with surface diffusion with a surface diffusion rate $\tilde D$ = 1024. Colors differentiate domains. Crystals have a width L = 256 and grow up to the height h = 234 in this figure.

DownLoad: Full-Size Img PowerPoint

Fig. 6. (Color online) The product $\gamma$ z with the coarsening exponent $\gamma$ for various diffusion rates $\tilde D$ . Error bars represent the standard deviation. The system size is L = 256. Solid line represents d = 2.

DownLoad: Full-Size Img PowerPoint

[1]	Hayes W, Stoneham A M. Defects and defect processes in nonmetallic solids. New York: John Wiley & Sons, 1985
[2]	Amano H, Sawai N, Akasaki I, et al. Metalorganic vapor phase epitaxial growth of a high quality GaN film using an AlN buffer layer. Appl Phys Lett, 1986, 48: 353 doi: 10.1063/1.96549
[3]	Nakamura S. GaN growth using GaN buffer layer. Jpn J Appl Phys, 1991, 30: L1705 doi: 10.1143/JJAP.30.L1705
[4]	Zheleva T S, Nam O H, Bremser M D, et al. Dislocation density reduction via lateral epitaxy in selectively grown GaN structures. Appl Phys Lett, 1997, 71: 2472 doi: 10.1063/1.120091
[5]	Hersee S D, Sun X Y, Wang X, et al. Nanoheteroepitaxial growth of GaN on Si nanopillar arrays. J Appl Phys, 2005, 97: 124308 doi: 10.1063/1.1937468
[6]	Saito Y, Omura S. Domain competition during ballistic deposition: effect of surface diffusion and surface patterning. Phys Rev E, 2011, 84: 021601 http://cn.bing.com/academic/profile?id=2062908940&encoded=0&v=paper_preview&mkt=zh-cn
[7]	Family F, Vicsek T. Scaling of the active zone in the Eden process on percolation networks and the ballistic deposition model. J Phys A, 1985, 18(2): L75 doi: 10.1088/0305-4470/18/2/005
[8]	Barabási A L, Stanley H E. Fractal concepts in surface growth. Cambridge: Cambridge University Press, 1995
[9]	Kardar M, Parisi G, Zhang Y C. Dynamic scaling of growing interfaces. Phys Rev Lett, 1986, 56: 889 doi: 10.1103/PhysRevLett.56.889
[10]	Farnudi B, Vvdensky D D. Large-scale simulations of ballistic deposition: the approach to asymptotic scaling. Phys Rev E, 2011, 83: 020103(R) http://cn.bing.com/academic/profile?id=2070087653&encoded=0&v=paper_preview&mkt=zh-cn
[11]	Alves S G, Oliveira T J, Ferreira S C. Origins of scaling corrections in ballistic growth models. Phys Rev E, 2014, 90: 052405 doi: 10.1103/PhysRevE.90.052405
[12]	Ballestad A, Ruck B J, Admcyk M, et al. Evidence from the surface morphology for nonlinear growth of epitaxial GaAs films. Phys Rev Lett, 2001, 86: 2377 doi: 10.1103/PhysRevLett.86.2377
[13]	Halpin-Healy T, Palasantzas G. Universal correlators and distributions as experimental signatures of (2+1)-dimensional Kardar-Parisi-Zhang growth. Euro Phys Lett, 2014, 105: 50001 doi: 10.1209/0295-5075/105/50001
[14]	Michely T, Krug J. Island, mounds and atoms: patterns and processes in crystal growth far from equilibrium. Berlin: Springer-Verlag, 2004
[15]	Ehrlich G, Hudda F G. Atomic view of surface self-diffusion: tungsten on tungsten. J Chem Phys, 1966, 44: 1039 doi: 10.1063/1.1726787
[16]	Schwoebel R L, Shipsey E J. Step motion on crystal surfaces. J Appl Phys, 1966, 37: 3682 doi: 10.1063/1.1707904
[17]	Bortz A B, Kalos M H, Lebowitz J L. A new algorithm for Monte Carlo simulation of Ising spin system. J Comput Phys, 1975, 17: 10 doi: 10.1016/0021-9991(75)90060-1
[18]	Saberi A A, Dashti-Naserabadi H, Rouhani S. Classification of (2+1)-dimensional growing surfaces using Schramm-Loewner evolution. Phys Rev E, 2010, 82: 020101(R) http://cn.bing.com/academic/profile?id=2081862734&encoded=0&v=paper_preview&mkt=zh-cn
[19]	Oliveira T J, Alves S G, Ferreira S C. Kardar-Parisi-Zhang universality class in (2+1) dimensions: universal geometry-dependent distributions and finite-time corrections. Phys Rev E, 2013, 87: 040102(R) http://cn.bing.com/academic/profile?id=2012768116&encoded=0&v=paper_preview&mkt=zh-cn
[20]	Moser K, Wolf D E. Numerical solution of the Kardar-Parisi-Zhang equation in one, two and three dimensions. Physica A, 1991, 178: 215 doi: 10.1016/0378-4371(91)90017-7
[21]	Tamborenea P, Das Sarma S. Surface-diffusion-driven kinetic growth on one-dimensinoal substrate. Phys Rev E, 1993, 48: 2575 doi: 10.1103/PhysRevE.48.2575

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Received: 16 March 2016 Revised: 19 April 2016 Online: Published: 01 September 2016

Article Navigation > Journal of Semiconductors > 2016 > 37(9): 092001

Kenyu Osada, Hiroyasu Katsuno, Toshiharu Irisawa, Yukio Saito. Scaling relation of domain competition on (2+1)-dimensional ballistic deposition model with surface diffusion[J]. Journal of Semiconductors, 2016, 37(9): 092001. doi: 10.1088/1674-4926/37/9/092001 ****K. Osada, H Katsuno, T Irisawa, Y Saito. Scaling relation of domain competition on (2+1)-dimensional ballistic deposition model with surface diffusion[J]. J. Semicond., 2016, 37(9): 092001. doi: 10.1088/1674-4926/37/9/092001.

Citation:

K. Osada, H Katsuno, T Irisawa, Y Saito. Scaling relation of domain competition on (2+1)-dimensional ballistic deposition model with surface diffusion[J]. J. Semicond., 2016, 37(9): 092001. doi: 10.1088/1674-4926/37/9/092001.

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Scaling relation of domain competition on (2+1)-dimensional ballistic deposition model with surface diffusion

DOI: 10.1088/1674-4926/37/9/092001

1.
Department of Physics, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan
2.
Computer Centre, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan
3.
Department of Physical Science, Ritsumeikan University, 1-1-1 Noji-higashi, Kusatsu, Shiga 525-8577, Japan
4.
Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan

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Corresponding author: Hiroyasu Katsuno, katsuno@fc.ritsumei.ac.jp
Received Date: 2016-03-16
Revised Date: 2016-04-19
Published Date: 2016-09-01

Abstract

Abstract

During heteroepitaxial overlayer growth multiple crystal domains nucleated on a substrate surface compete with each other in such a manner that a domain covered by neighboring ones stops growing. The number density of active domains ρ decreases as the height h increases. A simple scaling argument leads to a scaling law of ρ~h^-γ with a coarsening exponent γ=d/z, where d is the dimension of the substrate surface and z the dynamic exponent of a growth front. This scaling relation is confirmed by performing kinetic Monte Carlo simulations of the ballistic deposition model on a two-dimensional (d=2) surface, even when an isolated deposited particle diffuses on a crystal surface.
- domain competition,
- ballistic deposition model,
- Kardar-Parisi-Zhang universality class,
- surface diffusion

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References(21)

References

[1]	Hayes W, Stoneham A M. Defects and defect processes in nonmetallic solids. New York: John Wiley & Sons, 1985
[2]	Amano H, Sawai N, Akasaki I, et al. Metalorganic vapor phase epitaxial growth of a high quality GaN film using an AlN buffer layer. Appl Phys Lett, 1986, 48: 353 doi: 10.1063/1.96549
[3]	Nakamura S. GaN growth using GaN buffer layer. Jpn J Appl Phys, 1991, 30: L1705 doi: 10.1143/JJAP.30.L1705
[4]	Zheleva T S, Nam O H, Bremser M D, et al. Dislocation density reduction via lateral epitaxy in selectively grown GaN structures. Appl Phys Lett, 1997, 71: 2472 doi: 10.1063/1.120091
[5]	Hersee S D, Sun X Y, Wang X, et al. Nanoheteroepitaxial growth of GaN on Si nanopillar arrays. J Appl Phys, 2005, 97: 124308 doi: 10.1063/1.1937468
[6]	Saito Y, Omura S. Domain competition during ballistic deposition: effect of surface diffusion and surface patterning. Phys Rev E, 2011, 84: 021601 http://cn.bing.com/academic/profile?id=2062908940&encoded=0&v=paper_preview&mkt=zh-cn
[7]	Family F, Vicsek T. Scaling of the active zone in the Eden process on percolation networks and the ballistic deposition model. J Phys A, 1985, 18(2): L75 doi: 10.1088/0305-4470/18/2/005
[8]	Barabási A L, Stanley H E. Fractal concepts in surface growth. Cambridge: Cambridge University Press, 1995
[9]	Kardar M, Parisi G, Zhang Y C. Dynamic scaling of growing interfaces. Phys Rev Lett, 1986, 56: 889 doi: 10.1103/PhysRevLett.56.889
[10]	Farnudi B, Vvdensky D D. Large-scale simulations of ballistic deposition: the approach to asymptotic scaling. Phys Rev E, 2011, 83: 020103(R) http://cn.bing.com/academic/profile?id=2070087653&encoded=0&v=paper_preview&mkt=zh-cn
[11]	Alves S G, Oliveira T J, Ferreira S C. Origins of scaling corrections in ballistic growth models. Phys Rev E, 2014, 90: 052405 doi: 10.1103/PhysRevE.90.052405
[12]	Ballestad A, Ruck B J, Admcyk M, et al. Evidence from the surface morphology for nonlinear growth of epitaxial GaAs films. Phys Rev Lett, 2001, 86: 2377 doi: 10.1103/PhysRevLett.86.2377
[13]	Halpin-Healy T, Palasantzas G. Universal correlators and distributions as experimental signatures of (2+1)-dimensional Kardar-Parisi-Zhang growth. Euro Phys Lett, 2014, 105: 50001 doi: 10.1209/0295-5075/105/50001
[14]	Michely T, Krug J. Island, mounds and atoms: patterns and processes in crystal growth far from equilibrium. Berlin: Springer-Verlag, 2004
[15]	Ehrlich G, Hudda F G. Atomic view of surface self-diffusion: tungsten on tungsten. J Chem Phys, 1966, 44: 1039 doi: 10.1063/1.1726787
[16]	Schwoebel R L, Shipsey E J. Step motion on crystal surfaces. J Appl Phys, 1966, 37: 3682 doi: 10.1063/1.1707904
[17]	Bortz A B, Kalos M H, Lebowitz J L. A new algorithm for Monte Carlo simulation of Ising spin system. J Comput Phys, 1975, 17: 10 doi: 10.1016/0021-9991(75)90060-1
[18]	Saberi A A, Dashti-Naserabadi H, Rouhani S. Classification of (2+1)-dimensional growing surfaces using Schramm-Loewner evolution. Phys Rev E, 2010, 82: 020101(R) http://cn.bing.com/academic/profile?id=2081862734&encoded=0&v=paper_preview&mkt=zh-cn
[19]	Oliveira T J, Alves S G, Ferreira S C. Kardar-Parisi-Zhang universality class in (2+1) dimensions: universal geometry-dependent distributions and finite-time corrections. Phys Rev E, 2013, 87: 040102(R) http://cn.bing.com/academic/profile?id=2012768116&encoded=0&v=paper_preview&mkt=zh-cn
[20]	Moser K, Wolf D E. Numerical solution of the Kardar-Parisi-Zhang equation in one, two and three dimensions. Physica A, 1991, 178: 215 doi: 10.1016/0378-4371(91)90017-7
[21]	Tamborenea P, Das Sarma S. Surface-diffusion-driven kinetic growth on one-dimensinoal substrate. Phys Rev E, 1993, 48: 2575 doi: 10.1103/PhysRevE.48.2575

Scaling relation of domain competition on (2+1)-dimensional ballistic deposition model with surface diffusion

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