Morphologies of Self-Assembled Quantum Dots: A Variational Approach
- PDF / 120,131 Bytes
- 6 Pages / 595 x 842 pts (A4) Page_size
- 8 Downloads / 170 Views
Morphologies of Self-Assembled Quantum Dots: A Variational Approach R. Arief Budiman and Harry E. Ruda University of Toronto, Toronto, Ontario M5S 3E3, Canada Abstract We construct a 3D model for coherent island formation by (i) using a novel 3D strain tensor to account for bulk strains and (ii) representing adatom diffusion as an external field that perturbs an otherwise flat strained layer. Equilibrium shapes of coherent islands and wetting layer thickness are obtained. Coherently compressed layers are typically unstable, but become stable in tension. Comparisons with Si1−x Gex /Si(001) and Si0.5 Ge0.5 /Si1−x Gex (001) layers are discussed.
Introduction Nanometer-size 3D coherent islands form spontaneously during strained layer deposition of several monolayers (ML) thick. Applications of these self-assembled quantum dots hinge on our ability to predict and obtain a uniform island size distribution [1]. For this reason, several fundamental issues on the nature of coherent islanding transition need to be clarified. Two such issues revolve around the physics of wetting layer thickness. What governs its thickness? Why does it coexist with coherent islands? In this paper, a simple answer to the first issue is offered by our 3D model: in the limit of zero shear strains, flat wetting layer thickness is determined by the force balance between surface and interface tension, and the surface stress from (vertical) tetragonal strain. The basic idea of our single-component model is that shape transition that gives rise to the 3D islands must be accompanied by the emergence of bulk shear strains. Several experiments [2, 3, 4, 5] have consistently verified this notion. Our formulation allows for the tetragonal strain to change so that plane strain condition (uzz = 0, or hz = 0 as later defined) is no longer needed. The removal of plane strain assumption is crucial, since the resulting force equilibrium that includes the tetragonal strain yields the wetting layer thickness. In addition, shapes of the 3D islands are predicted from the accumulation of shear and tetragonal strains throughout the layer thickness. The model also addresses three additional issues. First, the coexistence of a wetting layer and coherent islands is explained by coupled nonlocal shear and longitudinal surface force equilibria along the growth direction. The nonlocality arises from the accumulation of bulk stresses with layer thickness, which defines the elastic surface stresses. Second, the layer stability against shear strains, thus islanding, depends on the sign of biaxial misfit strain. Coherently compressed layers are typically unstable above their wetting layer thickness, but become stable in tension. Third, the nanometer-size property is shown to arise from the exponential decay length ≈ (γs /µ) of bulk shear strains with thickness, where γs and µ are surface tension and shear modulus, respectively. The exponential decay of shear strains also explains shape metastability of coherent islands.
3D Model For sufficiently thin strained layers, a 3D deformation map:
Data Loading...
