Epitaxy coherency strain

A primary focus of our work has been to understand the ferroelectric phase transition in thin epitaxial films of PbTiOs. It is expected that epitaxial strain effects are important in such films because of the large, anisotropic strain associated with the phase transition. Figure 8.3 shows the phase diagram for PbTiOs as a function of epitaxial strain and temperature calculated using Landau-Ginzburg-Devonshire (lgd) theory [9], Here epitaxial strain is defined as the in-plane strain imposed by the substrate, experienced by the cubic (paraelectric) phase of PbTiOs. The dashed line shows that a coherent PbTiOs film on a SrTiOs substrate experiences somewhat more than 1 % compressive epitaxial strain. Such compressive strain favors the ferroelectric PbTiOs phase having the c domain orientation, i.e. with the c (polar) axis normal to the film. From Figure 8.3 one can see that the paraelectric-ferroelectric transition temperature Tc for coherently-strained PbTiOs films on SrTiOs is predicted to be elevated by 260°C above that of... 

Calculate the strains and that would be applied if the lattice parameters in the interface plane of the layer were forced to conform to the substrate (full coherent epitaxy). Multiply these by (1-i ) where R is the (fractional) relaxation of the layer. (See the discussion of measurement of relaxation in Chapters.)... 

Early observations of elastic strain relaxation during growth of epitaxial layers led to paradoxical results. An attempt to interpret the observations on the basis of the critical thickness theory in its most elementary form suggested that, once the thickness of a film exceeded the critical thickness, the final elastic strain of the film should be determined by the thickness of the film alone, independent of the original, or fuUy coherent, mismatch strain. This is implied by the result in (6.27), which states that that the mean elastic strain predicted by the equilibrium condition G(/if) = 0 is completely determined by hf beyond critical thickness, no matter what the value of Cni- However, it was found that the post-growth elastic strain as measured by x-ray diffraction methods did indeed vary with the initial elastic mismatch strain, and it did so in different ways for different film thicknesses (Bean et al. 1984). As a consequence, the critical thickness theory came under question, and various alternate models were proposed to replace it. However, further study of the problem has revealed the relaxation process to be much richer in physical phenomena than anticipated, with the critical thickness theory revealing only part of the story. 

Johnson, H. T. and Freund, L. B. (1997), The mechanics of coherent and dislocated island morphologies in strained epitaxial material syste. Journal of Applied Physics 81, 6081-6090. 

The contribution of the surface/interface can be neglected in most cases. However, it plays an important role when the bulk volume of the crystalline phase is small. This is the case in the initial phase of crystal formation or when a thin (epitaxial) crystalline layer is formed. In the latter case besides the energy of free or unstrained interfaces the elastic strain imposed by coherent intergrowth with the substrate of different lattice constant also provides a contribution to the total energy balance that can not be neglected. 

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