An analytical model for the growth of quantum dots on ultrathin substrates

S. P. A. Gill
2011 Applied Physics Letters  
The self-assembly of heteroepitaxial quantum dots on ultrathin substrates is analyzed within the context of small perturbation theory. Analytical expressions are derived for the dependence of the quantum dot separation on the substrate thickness. It is shown that the substrate thickness is critical in determining this separation when it is below the intrinsic material length scale of the system. The model is extended to simultaneous dot growth on both sides of the substrate. It is shown that
more » ... It is shown that vertically anticorrelated structures are preferred with an increase in the dot separation of 15% above that found in the one-sided case. The instability of elastically strained heteroepitaxial thin films leads to the self-assembly of quantum dot ͑QD͒ nanostructures. Known as the Asaro-Tiller-Grinfeld ͑ATG͒ instability, 1 the size and spacing of the QDs is determined by the competition between the energetic driving forces ͑strain and surface energy͒ and the kinetics of the material transport process ͑surface diffusion͒. Recently, it has been shown that some control of the QD size and spacing can be achieved through varying the thickness of the substrate. 2,3 The energetics of the system are strongly affected when the thickness of the substrate is comparable with the QD length scale. For ultrathin substrates, such as nanomembranes, 2,4 significant stress relief can arise due to bending of the substrate. The local strain field beneath the QDs is also modified due to the proximity of the lower free surface. If QDs are simultaneously deposited on both sides they can interact through the substrate to create a vertically anticorrelated QD structure. 2 In this paper the ATG instability model is extended to consider growth of QDs on ultrathin substrates to quantify how the substrate thickness affects the QD size and spacing and their spatial correlation. The problem is formulated within the context of a kinetic variational principle, 5 whereby has contributions from the dissipation potential, ⌿, which represents the work done in material transport, and the rate of change in the Gibb's free energy of the system, Ġ , which provides the driving force for the evolution. The optimal kinetic field is that which render the variational functional stationary, ␦⌸ =0. First we consider a single planar epitaxial film of thickness h 0 ͑see Ref. 6 for morphological effects͒. This is subject to a perturbation of amplitude A͑t͒ and wavelength , such that h͑x , t͒ = h 0 + A͑t͒sin͑2x / ͒, as shown in Fig. 1 . Mass conservation requires that ͓d͑j s ͒ / dx͔ + v n = 0, where j s is the material surface flux and the normal velocity of the surface v n Ϸ ḣ for small slopes ͑A Ӷ͒. The dissipation potential ͑per unit wavelength͒ is then where D s is the surface diffusion coefficient. The Gibb's free energy has surface energy and elastic strain energy contributions. For an isotropic surface energy density, ␥ 0 , and a surface elastic strain energy density, w͑x͒, one has 5 where ͑x͒Ϸ͑d 2 h / dx 2 ͒ is the surface curvature. The elasticity problem consists of two parts: global bending/stretching of the substrate by the initially planar film; and a local sinusoidal contribution from the thin film perturbation, as shown in Fig. 1 . These are solved separately and combined through linear superposition. The film in Fig. 1 is subject to a ͑compressive͒ mismatch strain ⑀ m Ͻ 0 which is relaxed by bending of the assembly. Let the thickness of the substrate be 2c and assume the variation in the normal strain in the x-direction through the thickness is linear such a͒ Electronic mail: spg3@le.ac.uk. y x c c + FIG. 1. ͑Color online͒ Geometry of the film-substrate assembly ͑top͒. The epitaxial film experiences a mismatch strain ⑀ m and a sinusoidal surface perturbation of amplitude A and wavelength . The assembly can relieve the strain by a combination of global elongation and bending ͑middle͒ with the addition of a sinusoidal component due to the local surface waviness ͑bot-tom͒. Contours show magnitude of stress x in the substrate. APPLIED PHYSICS LETTERS 98, 161910 ͑2011͒
doi:10.1063/1.3583447 fatcat:silqx6wiunfwrpcmkdqzr4qdzi