Abstract
An efficient method for the simulation of strained heteroepitaxial growth with intermixing using kinetic Monte Carlo is presented. The model used is based on a solid-on-solid bond counting formulation in which elastic effects are incorporated using a ball and spring model. While idealized, this model nevertheless captures many aspects of heteroepitaxial growth, including nucleation, surface diffusion, and long-range effects due to elastic interaction. The algorithm combines a fast evaluation of the elastic displacement field with an efficient implementation of a rejection-reduced kinetic Monte Carlo based on using upper bounds for the rates. The former is achieved by using a multigrid method for global updates of the displacement field and an expanding box method for local updates. The simulations show the importance of intermixing on the growth of a strained film. Further, the method is used to simulate the growth of self-assembled stacked quantum dots.
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Asaro R.J., Tiller W.A.: Interface morphology development during stress corrosion cracking: Part I. Via surface diffusion. Metall. Trans. B. 3, 1789–1796 (1972)
Blue J.L., Beichl I., Sullivan F.: Faster Monte Carlo simulations. Phys. Rev. E. 51, 867–868 (1995)
Briggs, M.W.L.: A Multigrid Tutorial. SIAM, Philadelphia (1987)
Caflsich R.E., Lee Y.J., Shu S., Xiao Y.X., Xu J.: An application of multigrid methods for a discrete elastic model for epitaxial systems. J. Comp. Phys. 219, 697–714 (2006)
Cullis A.G., Norris D.J., Walther T., Migliorato M.A., Hopkinson M.: Stranski–Krastanow transition and epitaxial island growth. Phys. Rev. B. 66, 081305 (2002)
Grinfeld M.A.: The stress driven instability in elastic crystals: mathematical models and physical manifestations. J. Nonlinear Sci. 3, 35–83 (1993)
Lam, C.H., Lee, C.K., Sander, L.M.: Competing roughening mechanisms in strained heteroepitaxy: a fast kinetic Monte Carlo study. Phys. Rev. Lett. 89, 16102 (1–4) (2002)
Lee S., Caflsich R.E., Lee Y.J.: Exact artifical boundary conditions for continuum and discrete elasticity. SIAM J. Appl. Math. 66, 1749–1775 (2006)
Lita B., Goldman R.S., Phillips J.D., Bhattacharya P.K.: Nanometer-scale studies of vertical organization and evolution of stacked self-assembled InAs/GaAs quantum dots. Appl. Phys. Lett. 74, 2824–2826 (1999)
Millunchick J.M., Twesten R.D., Follstaedt D.M., Lee S.R., Jones E.D., Zhang Y., Ahrenkiel S.P., Mascarenhas A.: Lateral composition modulation in AlAs/InAs short period superlattices grown on InP(001). Appl. Phys. Lett. 70, 1402–1404 (1997)
Niu X., Vardavas R., Caflisch R.E., Ratsch C.: Level set simulation of directed self-assembly during epitaxial growth. Phys. Rev. B. 74, 193403 (2006)
Orr B.G., Kessler D.A., Snyder C.W., Sander L.M.: A model for strain-induced roughening and coherent island growth. Europhys. Lett. 19, 33–38 (1992)
Politi P., Grenet G., Marty A., Ponchet A., Villain J.: Instabilities in crystal growth by atomic or molecular beams. Phys. Rep. 324, 271 (2000)
Quek S.S., Liu G.R.: Simulation of surface evolution of quantum dot using meshfree approximation. Thin Solid Films 479, 297–309 (2005)
Russo G., Smereka P.: Computation of strained epitaxial growth in three dimensions by kinetic Monte Carlo. J. Comp. Phys. 214, 809–828 (2005)
Russo G., Smereka P.: A multigrid-Fourier method for the computation of elastic fields with application to heteroepitaxy. Multiscl. Model. Simul. 5, 130–148 (2006)
Schulze, T.P., Smereka, P.: An energy localization principle and its application to fast kinetic Monte Carlo simulation of heteroepitaxial growth. J. Mech. Phys. Solids (in press) (2009)
Shchukin V.A., Bimberg D.: Spontaneous ordering of nanostructure on crystal surfaces. Rev. Mod. Phys. 71, 1125 (1999)
Smilauer P., Vvedensky D.D.: Coarsening and slope evolution during unstable epitaxial-growth. Phys. Rev. E. 52, 14263–14272 (1995)
Spencer B.J., Voorhees P.W., Davis S.H.: Morphological instability in epitaxially strained disclocation-free solid films. Phys. Rev. Lett. 67, 3696 (1991)
Spencer B.J., Voorhees P.W., Tersoff J.: Morphological instability theory for strained alloy film growth: the effect of compositional stresses and species-dependent surface mobilities on ripple formation during epitaxial film deposition. Phys. Rev. B. 64, 235318 (2001)
Tu Y., Tersoff J.: Origin of apparent critical thickness for island formation in heteroepitaxy. Phys. Rev. Lett. 93, 216101 (2004)
Tu, Y., Tersoff, J.: Coarsening, mixing, and motion: the complex evolution of epitaxial islands. Phys. Rev. Lett. 98, Art. No. 096103 (2007)
Walther T., Hopkinson M., Cullis A.G.: Observation of vertical and lateral Ge segregation in thin undulating SiGe layers on Si by electron energy-loss spectroscopy. Appl. Phys. Lett. 71, 809–811 (1997)
Walther T., Cullis A.G., Norris D.J., Hopkinson M.: Nature of the Stranski–Krastanow transition during epitaxy of InGa on GaAs. Phys. Rev. Lett. 86, 2381–2384 (2001)
Wise S.M., Lowengrub J.S., Kim J.S., Johnson W.C.: Efficient phase-field simulation of quantum dot formation in a strained heteroepitaxial film. Superlattices Microstruct. 36, 293–304 (2004)
Wise, S.M., Lowengrub, J.S., Kim, J.S., Thornton, K., Voorhees, P.W., Johnson, W.C.: Quantum dot formation on a strain-patterned epitaxial thin film. Appl. Phys. Lett. 87, Art. No. 133102 (2005)
Acknowledgments
We thank Len Sander and Tim Schulze for helpful conversations. This study was supported in part by the following grants from the National Science Foundation, DMS-0553487, DMS-0509124, and DMS-0810113.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Baskaran, A., Devita, J. & Smereka, P. Kinetic Monte Carlo simulation of strained heteroepitaxial growth with intermixing. Continuum Mech. Thermodyn. 22, 1–26 (2010). https://doi.org/10.1007/s00161-009-0118-0
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DOI: https://doi.org/10.1007/s00161-009-0118-0