Reasons for self ordering in multilayer quantum dots Part II: Interaction energy
Highlights
► The lateral interaction among QDs is very small since the misfit strains are relieved. ► Ordering is possible by the vertical interactions when surface protrusions are split. ► Vertical interactions are due to stresses set up by buried QDs in the covering layer.
Introduction
Growth of quantum dots (QDs) in multilayered semiconductors is observed to yield different types of arrays ranging from random arrays, vertically stacked arrays and vertically staggered arrays. The last arrays may be laterally random or ordered in FCC-like superlattices. Ordering has been shown to be the result of substrate mediated elastic interactions with dots in the underlying layer. The elastic fields associated with QDs have been analyzed by many researchers who used different methods [1], [2], [3], [4]: analytic continuum approach [e.g., [5], [6], [7], [8], [9], [10], [11], [12]], numeric continuum approach (finite difference [e.g., [13], [14]] or finite element [e.g., [15], [16], [17]] methods) and atomistic approach [e.g., [18], [19], [20], [21]]. In the analytical continuum approach, the QDs are treated as inclusions in a matrix. The elastic fields due to the lattice mismatch between the QDs and the matrix are obtained by integrating the Green's function over the interface of the inclusions and the matrix. Only limited geometries are solvable and homogeneity of the elastic constants is usually assumed. The finite difference and finite element approaches overcome these limitations. The atomistic approach describes the strain energy in terms of the potentials between the atoms and the strain fields are obtained by minimizing the potential energy. Due to the large number of atoms required in the analysis, this approach is computationally limited to a single QD. Comparison between the approaches shows that the continuum approaches reasonably approximate the atomistic calculations [18].
Following Tersoff et al. [5], Holy et al. [4], [7] and other authors considered a three layer situation consisting of a QD layer, a cupping layer, made of the material of the substrate and a wetting layer, made of the material of the dots. They calculated the density of the elastic energy in the wetting layer on the free surface and proposed that the favored sites for nucleation and growth of new dots in the Stransky Krastanov mode are those where the elastic energy density is minimal. Ignoring the lateral interactions they found that when the elastic anisotropy factor A ≡ 2C44/(C11–C12) > 1.6 (Cij are the elastic constants) and the substrate surface is the (001) crystallographic plane, an embedded dot creates favored sites that are expected to form a body centered tetragonal (BCT) superlattice. When A < 0.6 and the substrate surface is the (111) plane, trigonal or FCC-like superlattices are expected. In the other cases vertical stacking without lateral order is expected. A remarkable resemblance between the predictions and the experimental observations is found in different materials, except for the BCT superlattice that was not observed [1], [2], [3], [4].
Several groups have calculated the elastic energy density in the wetting layer [1], [4], [5] and in the cupping layer [13], [14], [15], [16]. Apparently only Liu et al. [15] admitted that the preferred sites for nucleation of new dots are those where the density of elastic energy in the cupping layer is maximal and those sites coincide with sites of minimum elastic energy density in the wetting layer. This difference can be explained in a straightforward way by the considerations in ref. [22]. A buried dot with lattice parameter larger than that of the substrate, generates tensile stresses in the cupping layer, made of the substrate material. The larger the tensile stresses in the cupping layer — the larger the density of the elastic energy there. On the other hand, on top of the cupping layer rests the wetting layer, made of the dot material. It is severely compressed in the lateral directions, due to the epitaxial relation with the substrate. Thus the minimum elastic energy density in the wetting layer is found when it is relatively relaxed from the compressive epitaxial stresses by the tensile stresses due to the underlying QD. These sites can be expected to be favored for nucleation of new QDs and it was indeed observed. While these are qualitative arguments, the actual energy gain of the new dot at a favored site should be directly calculated. This gain is defined as the interaction energy between the dots [23].
In the first part of the present work [22] we discussed qualitatively the elastic interactions among dots in multilayers. In the present part we rigorously prove two ways to calculate the elastic interactions in the framework of linear elasticity. From them we deduce a third simple estimate of the elastic interaction, obtainable from the solution of the elastic fields of a single dot and show that it is a better estimate than the elastic energy density in the cupping or the wetting layers. The three methods are applied and assessed numerically, to quantitatively determine the interaction energy in two types of systems where ordered arrays are observed: simple vertical stacking and vertically staggered stacking. Then the failure to form the expected BCT superlattice (in A > 1 materials with {001} free surface) is quantitatively compared with the FCC superlattice (in A < 1 materials with {111} free surface), which is experimentally observed. Finally the lateral interaction between dots in the same layer is calculated in order to quantitatively determine its role in the ordering of the QDs.
Section snippets
The elastic interaction energy
We follow the derivation of Eshelby in refs. [23], [24] of the interaction energy between two stress sources in infinite solid and adopt it to a solid with free surfaces. Suppose that in a body with volume Vo, enclosed by external surface Σο there are two systems of internal stresses that are due to misfit strains εijT: A whose sources lie entirely within a surface ΣΑ and B whose sources lie entirely outside ΣΑ (Fig. 1). Both may be embedded inside the body or attached to its surface.
Let EA and
Method of calculation
The calculations were performed by the finite element method using the MARC.MSC code [25]. Three approaches for the calculation of the interaction energy were mentioned in Section 2. They were represented by three types of models that are shown schematically in Fig. 2 (see also Fig. 2d,e in ref [22]).
- (a)
A model with two misfitting dots: one buried dot near a free surface and a second dot (with similar dimensions) on the free surface (Fig. 2a1). A similar model contained two misfitting dots on the
Assessment
Fig. 3 reproduced from ref. [22] shows maps of the hydrostatic stress component associated with a QD that is buried near the surface and a QD that is epitaxialy bonded on a {001} free surface of Fe and Nb substrates. The hydrostatic stress at the free surface was calculated by [8], [9], [10], [11], [13] who obtained similar results. As shown in Section 2, this stress component is proportional to the interaction energy between the existing QD and a newly formed one. In materials with anisotropy
Conclusions
Interaction energies were calculated in the framework of linear elasticity by two methods and found to be consistent with the hydrostatic pressure and to vary in a similar way. Thus the hydrostatic stress is a direct and simple representation of the local interaction energy and should serve as the potential for the elastic interaction. The vertical interaction energy is very small compared with the thermal energy (of the order of 0.004 kT for 1% of misfit), nevertheless it is just enough to
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