Interplay between misfit strain relief and Stranski–Krastanov growth in fcc (111)/bcc (110) ultrathin film epitaxy: Part 1. Analytical approach

The main objective of this investigation is to study the impact of misfit (homogeneous) strain (MS) relief by misfit dislocations (MDs) — or a misfit vernier where the relaxation into oscillatory strains characteristic of MDs becomes insignificant — on the Stranski–Krastanov (SK) growth mode, the perception being that the driving force for the transition is dominated by short range proximity effects of the vacuum and a strongly bonding substrate. The influence of misfit strain relief with increasing thickness on the transition from two-dimensional (2D) to three-dimensional (3D) growth, characteristic for SK growth, is investigated. Here, the dominating parameters are identified and the mathematical formulae for the governing relations, needed for quantification in Part 2, using embedded atom methods (EAM) potentials, are derived: (i) the dependence of substrate coverage on epilayer misfit (f) and deformation (ē); (ii) the structure of Fourier series to model fcc (111) epilayer–bcc (110) substrate interaction potentials V(x, y) for a Kurdjumov–Sachs (KS) orientation, including optimum Fourier coefficients Vhk for low order truncations; (iii) the harmonic intra-epilayer interaction suitable to describe MDs and to model misfit vernier accommodation of mismatch related to homogeneous Poisson strain; (iv) the average in-plane strain energy suitable (a) to handle the large repeat periods involved in homogenous epilayer deformation within the short wavelength periodic epilayer–substrate interaction potentials and (b) to define stiffness constants for epilayer deformation within such a field; (v) stability criteria for MS relief of 2D coherent and one-dimensional (1D) KS coherent — registry of two opposing sets of parallel closest packed atom rows on either side of the interface — epilayers in Nishiyama–Wassermann and KS orientations, respectively, applying continuum and discrete approaches; and (vi) criteria for growth mode realization, emphasizing transitions to the SK mode and its dependence on proximity effects. The analytical considerations reveal (a) that the transition from a 2D coherent misfit accommodation mode to a 1D KS mode is favored by a large excess strain energy ε2D−ε1D and large substrate surface free energy, but opposed by strong epilayer–substrate bonding and (b) the MS energy ε makes no contribution to the growth mode discriminant if proximity effects of the interface and free surface are absent.