Self Organized Array of Quantum Nanostructures Via a Strain Induced Morphological Instability
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Self organized array of quantum nanostructures via a strain induced morphological instability
David Montiel[1], Judith M¨ uller[2] and Eugenia Corvera Poir´ e[1]. [1] Departamento de F´ısica y Qu´ımica Te´ orica, Facultad de Qu´ımica, UNAM. Ciudad Universitaria. M´ exico, D.F. 04510, MEXICO [2] Instituut-Lorentz, Universiteit Leiden. Postbus 9506, 2300 RA Leiden, THE NETHERLANDS Abstract Motivated by the work of Li et al. [1], we have studied the strain induced morphological instability at the submonolayer coverage stage of heteroepitaxial growth on a vicinal substrate with regularly spaced steps. We have performed a linear stability analysis and determined for which conditions of coverage a flat front is unstable and for which conditions it is stable. For low coverages the instability will cause the front to break in an array of islands. Assuming that the fastest growing mode of the instability determines the properties of the array, we make an estimation of the islands sizes and aspect ratios as well as an estimation of the separation length between islands of the array formed when the dominant mechanism for transport of matter is diffusion of particles along the growing front. These estimations are given as functions of the terrace width and coverage. Since these ones are experimentally controllable parameters, our results could be used to tailor the spontaneous formation of quantum nanostructures. Growing material Growth
Substrate (vicinal surface)
Interstep
W
FIG. 1: Step flow on a vicinal surface.
In a recent letter [2] we have considered the stability of a flat front during heteroepitaxial growth on a vicinal surface. Since misfit strain is present, the deposited material causes a force distribution on the substrate. By using theory of continuum media and by assuming a relaxational mechanism consisting on diffusion of particles along the growing fronts we found a dispersion relation of the form
ω=
2Es (1 − σ)Dπ cot(πθ)k2 − D γk4 W
AA3.3.1/N2.3.1
(1)
ω
0
k∗
k
FIG. 2: Dispersion relation for coverages θ < 0.5. 1+σ 2 Here D and D are proportional to the diffusion constant, Es = 2πE F0 is a unit strain energy introduced in reference [3]. E and σ are the Young’s modulus and Poisson’s ratio of the substrate respectively. F0 is the magnitude of the force per unit length. The terrace width W is shown in figure 1 and the coverage θ, is the fraction of the terrace width W covered by the growing film. This dispersion relation gives the growth rate of a perturbation of the form y(x, t) = y0 + δeωt cos(kx), where y0 is the position of a flat unperturbed growing front and δeωt is the amplitude of the Fourier mode with wavenumber k. t stands for time and x and y for spatial coordinates. It is worth to emphasize that a general perturbation can be expressed as a superposition of Fourier modes and that for a linear stability analysis all modes decouple in such a way that one can treat each of them individually. Our results indicate that a straight stripe is unstable for coverages θ < 0.5 and stable for coverages θ > 0.
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