Accommodation of misfit in epitaxial interfaces: discrete Frenkel-Kontorova model with real interatomic forces

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Abstract
The accommodation of misfit in epitaxial interfaces by homogeneous strain and by misfit dislocations (MD) is studied by means of the 1D model of Frank and van der Merwe in which the elastic interactions are replaced by a real pair-wise potential. A relevant feature of the real potential is its inflection point beyond which the potential is not convex. The latter leads to a significant difference in the physical behaviour in comparison with the system with a convex potential. An alternation of long, weak and short, strong bonds occurs in expanded epilayers. This transition from undistorted to distorted structures is found to be of second-order with respect to both the misfit and the substrate modulation and the critical exponents are equal to 1/2. The commensurate-incommensurate transition is continuous in compressed chains. In expanded chains it is also continuous as long as all the atoms in the ground state sample the convex part of the potential. Above a certain value of the interfacial bonding this condition is violated and the commensurate-incommensurate transition changes into a first-order transition. The energy of MD interaction is considerably smaller in expanded than in compressed chains. It is suppressed additionally by the distortion of the chemical bonds between the MD. The energy of a single MD depends strongly on the absolute value of the misfit; it grows in compressed and decreases in expanded epilayers. At zero value of the misfit the energy of a positive MD (empty potential trough) is smaller than the energy of a negative MD (two atoms in a trough).

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