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Volumn 57, Issue 6, 1998, Pages 6851-6858

Simple model for anisotropic step growth

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EID: 0000476665     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.57.6851     Document Type: Article
Times cited : (13)

References (53)
  • 2
    • 85036263867 scopus 로고    scopus 로고
    • A.-L. Barabási and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, City, 1995)
    • A.-L. Barabási and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, City, 1995).
  • 26
    • 85036290164 scopus 로고    scopus 로고
    • the case of one-particle diffusion, the drift term is not present strictly speaking since the step does not move. In our model [formula presented] may be considered as a finite-density correction to this limit
    • In the case of one-particle diffusion, the drift term is not present strictly speaking since the step does not move. In our model ld may be considered as a finite-density correction to this limit.
  • 42
    • 85036408841 scopus 로고    scopus 로고
    • On a Si(001) terrace, diffusion along the dimer rows is much faster than across them. For [formula presented], however, the step morphology is controlled by the drift term, and thus the exact value of the ratio [formula presented] is not important
    • On a Si(001) terrace, diffusion along the dimer rows is much faster than across them. For ld>0, however, the step morphology is controlled by the drift term, and thus the exact value of the ratio Dxx/Dyy is not important.
  • 44
    • 0031097281 scopus 로고    scopus 로고
    • J. Krug, Adv. Phys. 46, 139 (1997).ADPHAH
    • (1997) Adv. Phys. , vol.46 , pp. 139
    • Krug, J.1
  • 45
    • 85036258304 scopus 로고    scopus 로고
    • The zero drift limit ld→0 is rather subtle in the present model. For systems with finite Lx, we find that asymptotically w(t)∼t. However, the behavior of the finger coarsening is somewhat different from the case of ld>0. We do find an intermediate time regime where r0(t)∼t0.33, but for larger systems there seems to be a late-time crossover to a larger effective coarsening exponent. In this regime, our model becomes equivalent to the “Laplacian needle model A” of Krug et al. 25. In the needle model, the average distance between the needles grows like ha/ln (ha), where ha is the height of “active” needles 26. This would mean that r0(t)∼t/ln(t), which is qualitatively supported by our simulation results for ld=0
    • The zero drift limit ld→0 is rather subtle in the present model. For systems with finite Lx, we find that asymptotically w(t)∼t. However, the behavior of the finger coarsening is somewhat different from the case of ld>0. We do find an intermediate time regime where r0(t)∼t0.33, but for larger systems there seems to be a late-time crossover to a larger effective coarsening exponent. In this regime, our model becomes equivalent to the “Laplacian needle model A” of Krug et al. 25. In the needle model, the average distance between the needles grows like ha/ln (ha), where ha is the height of “active” needles 26. This would mean that r0(t)∼t/ln(t), which is qualitatively supported by our simulation results for ld=0.
  • 50
    • 85036207117 scopus 로고    scopus 로고
    • As defined in Eq. (13), the scaling function [formula presented] depends on the drift [formula presented] We can also write the scaling form as [formula presented], in which case the scaling plots for the new function [formula presented] collapse within the accuracy of our data for the values of the drift studied, namely [formula presented], 1/4, 1/2, and 1
    • As defined in Eq. (13), the scaling function gld(x) depends on the drift ld. We can also write the scaling form as G(r,t)=w2(ld,t)g(r/r0(ld,t)), in which case the scaling plots for the new function g(x) collapse within the accuracy of our data for the values of the drift studied, namely ld=1/8, 1/4, 1/2, and 1.
  • 52
    • 85036274475 scopus 로고    scopus 로고
    • the long-time limit, for an infinite system (Lx=Ly=∞) the aspect ratio r0/w approaches zero, and the fingers become needlelike even for finite values of ld. In the needle model, the penetration probability of the diffusing particles between needles decays exponentially as exp(-πr/ξ) 25, where the distance r is measured from the top of the needles and ξ is the distance between two needles. In our model with a finite drift parameter ld, the corresponding result can be shown to be exp-(1/2Dyy)[4DxxDyyπ2/ξ2+vd2(ld)+vd(ld)]r}, where vd(l), Dxx, and Dyy are defined in the text. For zero drift (where vd=0, and Dyy=Dxx), this correctly reduces to the previous result
    • In the long-time limit, for an infinite system (Lx=Ly=∞) the aspect ratio r0/w approaches zero, and the fingers become needlelike even for finite values of ld. In the needle model, the penetration probability of the diffusing particles between needles decays exponentially as exp(-πr/ξ) 25, where the distance r is measured from the top of the needles and ξ is the distance between two needles. In our model with a finite drift parameter ld, the corresponding result can be shown to be exp-(1/2Dyy)[4DxxDyyπ2/ξ2+vd2(ld)+vd(ld)]r}, where vd(l), Dxx, and Dyy are defined in the text. For zero drift (where vd=0, and Dyy=Dxx), this correctly reduces to the previous result.


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