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SPECIAL SECTION ON NANOMATERIALS AND NANOMECHANICS

Growth and Ordering of Si-Ge Quantum Dots on Strain Patterned Substrates

[+] Author and Article Information
A. Ramasubramaniam, V. B. Shenoy

Division of Engineering, Brown University, Providence, RI 02912shenoyv@engin.brown.edu

Strictly speaking, the strains prescribed here hold only at the film-substrate interface. The actual strains on the surface of the film must contain higher order corrections in the film thickness Δh. For the limiting case of a thin film, these corrections may be neglected—the strains on the surface of the film are thus the same as the misfit strains at the film-substrate interface.

J. Eng. Mater. Technol 127(4), 434-443 (Jan 30, 2005) (10 pages) doi:10.1115/1.1924559 History: Received September 21, 2004; Revised January 30, 2005

Manipulating the strain distribution along the surface of a substrate has been shown experimentally to promote spatial ordering of self-assembled nanostructures in heteroepitaxial film growth without having to resort to expensive nanolithographic techniques. We present here numerical studies of three-dimensional modeling of self-assembly in Si-Ge systems with the aim of understanding the effect of spatially varying mismatch strain-fields on the growth and ordering of quantum dots. We use a continuum model based on the underlying physics of crystallographic surface steps in our calculations. Using appropriate parameters from atomistic studies, the (100) orientation is found to be unstable under compressive strain; the surface energy now develops a new minimum at an orientation that may be interpreted as the (105) facet observed in SiGeSi systems. This form of surface energy allows for the nucleationless growth of quantum dots which start off via a surface instability as shallow stepped mounds whose sidewalls evolve continuously toward their low-energy orientations. The interaction of the surface instability with one- and two-dimensional strain modulations is considered in detail as a function of the growth rate. One-dimensional strain modulations lead to the formation of rows of dots in regions of low mismatch—there is some ordering within these rows owing to elastic interactions between dots but this is found to depend strongly upon the kinetics of the growth process. Two-dimensional strain modulations are found to provide excellent ordering within the island array, the growth kinetics being less influential in this case. For purposes of comparison, we also consider self-assembly of dots for an isotropic surface energy. While the results do not differ significantly from those for the anisotropic surface energy with the two-dimensional strain variation, the one-dimensional strain variation produces profoundly different behavior. The surface instability is seen to start off initially as stripes in regions of low mismatch. However, since stripes are less effective at relaxing the mismatch strain they eventually break up into islands. The spacing of these islands is determined by the wavelength of the fastest growing mode of the Asaro-Tiller-Grinfeld instability. However, the fact that such a growth mode is not observed experimentally indicates the importance of accounting for surface energy anisotropy in growth models.

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Copyright © 2005 by American Society of Mechanical Engineers
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Figures

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Figure 10

Evolution sequence of quantum dots grown on a substrate with an equi-biaxial strain modulation shown in Fig. 9 at a deposition flux f̃=2. Starting off from a random perturbation (a) mounds are seen to self-assemble in a square array in regions of low mismatch strain (b, c) and grow ultimately to their optimal orientations (d). The island sizes and their spatial distribution is quite uniform in both the Cartesian directions.

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Figure 11

Evolution sequence of quantum dots grown on a substrate with an equi-biaxial strain modulation shown in Fig. 9 at a deposition flux f̃=4. Once again, starting off from a random perturbation (a) mounds are seen to self-assemble in a square array in regions of low mismatch strain (b, c) and grow ultimately to their optimal orientations (d). [Islands with red caps in (d) are at their optimal orientation while some of the others are still mounds at lower sidewall orientations.] Kinetic limitations at a higher flux lead to some nonuniformity in the array with pairs of islands being obtained at some sites where previously (Fig. 1) only a single island was obtained. The islands that are in pairs are also relatively smaller than the single islands in the array. Some random nucleation events in regions of higher mismatch may also be observed.

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Figure 12

Fourier spectra for the island arrays in Figs.  1011. Distinct peaks are obtained at frequencies that correlate spatially to the periodicity of the strain modulation.

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Figure 9

Contour plot of the two-dimensional mismatch strain distribution ϵ110(x̃1,x̃2)=ϵ220(x̃1,x̃2)=ϵ0(1−0.5exp[−4sin2(x̃1)−4sin2(x̃2)]) with ϵ0=−1%

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Figure 8

Evolution sequence of quantum dots grown on a substrate with a one-dimensional strain modulation shown in Fig. 4 for an isotropic surface energy at a deposition flux f̃=4. Starting off from a random perturbation (a), the film organizes into stripes in regions of low mismatch. On continued deposition, the stripes eventually break up into three-dimensional islands (c, d) since 3D islands are better at relaxing the mismatch strain as compared to 2D ones. The periodicity of the islands in the x2 direction is determined by the fastest growing wavelength of the ATG instability (λmax=54.22nm) for the parameters chosen here. The growth of the initial instability and subsequent island formation is thus very different for isotropic surface energies as compared to the anisotropic case.

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Figure 7

Fourier spectra for the island arrays in Figs.  56. Distinct peaks are obtained at frequencies that correlate spatially to the strain-field in the modulated direction while a more diffuse distribution of frequencies is obtained in the unmodulated direction. The diffuse frequencies are, however, not entirely random, thereby suggesting some ordering (due to elastic interactions) in the x2 direction as well.

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Figure 6

Evolution sequence of quantum dots grown on a substrate with a one-dimensional strain modulation shown in Fig. 4 at a deposition flux f̃=4. Once again, starting off from a random perturbation (a) mounds are seen to self-assemble in rows in four regions of low mismatch strain (b, c) and grow ultimately to their optimal orientations (d). [Dots with red caps in (d) are at their optimal orientations while the others are still mounds at lower sidewall slopes.] The spacing of dots is less uniform in the x2 direction as compared to that in Fig. 5 owing to kinetic limitations (see discussion in text); the island sizes are also comparatively smaller than those in that case. Additionally, some random nucleation events are also observable in less preferred regions of higher mismatch owing to kinetic limitations.

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Figure 5

Evolution sequence of quantum dots grown on a substrate with a one-dimensional strain modulation shown in Fig. 4 at a deposition flux f̃=2. Starting off from a random perturbation (a) mounds are seen to self-assemble in rows in regions of low mismatch strain (b, c) and grow ultimately to their optimal orientations (d). Although elastic interactions lead to reasonably uniform spacing along the x2 direction in each row, there is no overall ordering in this direction owing to ϵ22 being constant.

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Figure 4

Plot of the one-dimensional strain modulation ϵ110(x̃1,x̃2)=ϵ0(1−0.5exp[−4sin2(x̃1)]) where ϵ0=−1%

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Figure 3

Schematic illustration of a substrate with buried SiGe islands. The strain distribution ϵ0(x) is the mismatch strain experienced by the film when deposited on this substrate. Since the film material (Si1−xGex or Ge) has a larger lattice parameter than the substrate, the film experiences a compressive mismatch strain. The magnitude of this mismatch, as indicated, is lesser in regions above the buried islands since these “inclusions” cause stretching of bonds on the substrate surface in their vicinity.

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Figure 2

Plot of the surface energy, γ=γ0+β̂1∣∇h∣+(β3∕3)∣∇h∣3, for a compressively strained film (β̂1<0). The minimum is at ∣∇h∣*=−β̂1∕β3. h1 and h2 denote the slopes in the two Cartesian directions.

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Figure 1

Schematic of a thin film on a lattice mismatched substrate. Spatial variations in the mismatch lead to a position-dependent biaxial strain ϵij0(x)—the corresponding stress state is σij0(x). The combined volume of the film and substrate is represented by V and the traction-free film surface is represented by S.

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