Enhanced elastic interactions between conical quantum dots

An analytical model for the elastic energy of a system of conical heteroepitaxial quantum dots of finite slope is presented. An expression for the surface tractions at the dot-substrate interface is proposed. This includes a singularity in the stress field at the perimeter of the dot. The strength of this singularity increases as the slope of the dot increases. This dramatically enhances the elastic interaction between dots and the metastability of a quantum dot array. This could help explain the stability of bimodal island size distributions observed in some quantum dot systems. © 2006 American Institute of Physics. DOI: 10.1063/1.2390651

The elastic strain energy of a system of heteroepitaxial quantum dots is critical in determining their ability to selforganize and achieve a metastable state. 1,24][5][6][7][8] These models represent the effect of the dots' presence on the substrate by a distribution of point forces on an elastic half-space.This is a small slope approximation based on the assumption that the energy change in the island-substrate system due to the relaxation of the mismatch strain is small compared to the energy of the unrelaxed island. 9In this case, the tractions are simply proportional to the slope of the surface profile.This is a valid approximation for small slopes if the tractions on the halfspace are continuous. 10,11For the case of quantum dots this is not the case, as there is a discontinuity in the dot profile at the edge of the dot.This generates a singularity in the stress field of the dot at its outer perimeter.These singularities are expected to interact strongly over relatively large distances.These singularities have not been treated explicitly in previous analyses, although fitted models for the interaction energy between dots based on finite element calculations have been proposed. 2onsider a conical dot of height h and base radius b, such that the slope = tan = h / b is constant.The elastic mismatch strain is T and E s and s are Young's modulus and Poisson's ratio.The subscripts s and d are used to denote substrate and dot properties, respectively.The small slope approximation for the radial surface traction distributed over the circular footprint of the dot is 10 for 0 ഛ r ഛ b and zero otherwise.The predictions of ͑1͒ are compared with results calculated using the finite element package FEMLAB in Fig. 1 for E s = E d and s = d = 0.3.There is reasonable agreement for the very small slope of = 0.01 ͑0.6°͒ but this is not the case for even the modest slope of = 0.1 ͑5.7°͒.Even Ge/ Si͑001͒ dots, which have very low slopes, have Ϸ 0.2 ͑11°͒ and dots with much higher slopes are not uncommon. 12The surface tractions in Fig. 1 are singular at the dot perimeter. 13The radial surface traction at the dot-substrate interface is fitted to where ␣ is the strength of the singularity at the island edge and the second term ensures that f r ͑0͒ = 0.The two exponents are well fitted by ␣ = 1 2 tanh͑1.41͒and ␦ = 0.114␣ −1 + 0.638 for E s = E d and s = d = 0.3.The magnitude of the surface traction, c, is that which minimizes the elastic strain energy ͑see below͒.In the small slope limit ͑ → 0͒, ͑2͒ is equivalent to ͑1͒. Figure 1 shows that ͑2͒ is a good representation of the numerical results.Shchukin et al. 5 analyzed the change in strain energy of an elastic half-space ͑substrate͒ due to radial surface tractions distributed over a number of circular patches p of radius b p .Their result is for the constant radial surface traction model ͑1͒, but their elegant derivation is applicable for any radial surface traction model.To first order, the change in strain energy is given by where ⌬E pp is the self-relaxation energy of patch p and ⌬E pq is the interaction energy between patches p and q.The selfrelaxation energy is 14 a͒ Electronic mail: spg3@le.ac.uk where c p is the magnitude of the surface traction due to the pth dot and J͑␣͒ = 1.059+ 41.25␣ 3 is used as a good approximation to the exact expression. 14The elastic interaction energy between two conical dots p and q with center-to-center separation where the interaction function is ⌫͑z͒ is the gamma function and

͑7͒
Note that ͑4͒ and ͑5͒ reduce to the small slope result 5 for ␣ = 0. Figure 2 shows the interaction function for the two identical interacting dots.The interaction increases dramatically with the dot slope due to the long-range elastic field of the perimeter singularity.The total energy for a regular array of identical dots was derived by Gill and Cocks 12 for the traction model ͑1͒.The result for the more realistic traction model ͑2͒ is 14 where V p is the volume of dot p, ␤ ˆis an interfacial energy, 15 w d = E d T 2 / ͑1− d ͒ is a strain energy density, the relaxation factor for conical dots is g = 1.115− 0.814 + 0.136 2 + 0.0594 3 This captures the effects of dot shape, elastic property mismatch, and interaction effects in a simple form and includes relaxation of strain in the dots due to elastic interactions.Note that this reproduces the energy expression of Shchukin et al. 15 if we take E s = E d and s = d ͑similar materials͒ and linearize the strain energy term such that it is the sum of a self-relaxation energy and an interaction energy, such that ͑1+I͒ / ͑1+I + ␤g͒ −1Ϸ −␤g / ͑1+I͒Ϸ−␤g + ␤gI, where the first step assumes ␤g Ӷ 1 ͑small slopes͒ and the second assumes I Ӷ 1 ͑small interactions͒.
The energy model of ͑8͒ is now used to investigate the stability of dot arrays to coarsening.This follows the approach of Shchukin et al. 15 who demonstrated that the most unstable array configuration to coarsening is the hexagonal array ͑the K point on the Brillouin zone͒.A stability phase diagram for this configuration is shown in Fig. 3, in which the boundary between metastable and unstable regions is plotted as a function of surface coverage q s and island size b / b cr where b cr = ␤ ˆ͑1+I + ␤g͒ 2 / w d ␤g 2 is a critical radius.The solution of Shchukin et al. 15 ͑for small slopes, small interactions, and similar materials͒ is the same as Fig. 2 in that reference.It predicts that the system is always unstable below coverages of 88.8%.Allowing for large interactions increases the region of metastability to 86.5% for the small slope model ͑ =0͒.Including large slope effects as well also dramatically enhances the size of the region of metastability.The critical coverage for =8°͑␣ = 0.1͒ is 83.1%, for = 17°͑␣ = 0.2͒ is 80.6%, for = 28°͑␣ = 0.3͒ is 78.4%, and for = 45°͑␣ = 0.4͒ it is as low as 76.7%.Recall that this case of an ideal hexagonal array is the least stable dot configuration.The stability of a random dot array would be expected to be enhanced by an equal order of magnitude.
In conclusion, a model for realistic surface tractions at the dot-substrate interface has been proposed.There is a singularity in the stress field at the perimeter of a dot.The strength of this singularity increases as the slope of the dot increases.This greatly enhances the self-relaxation energy of a single dot and the interaction energy between dots.A 15͒ is small.This is greatly increased by the inclusion of large slope and large interaction effects.The maximum surface coverage, q s = 0.9068, is indicated by the vertical line.

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Simon P. A. Gill Appl.Phys.Lett.89, 203115 ͑2006͒ model for the elastic energy of a conical dot system with realistic slopes, interactions, and material properties has been proposed.The stability of this model to coarsening has been investigated.A system of highly sloped dots is predicted to be metastable at much lower coverages than previously predicted.This could help explain the stability of bimodal island size distributions observed in some quantum dot systems. 16