A critical thickness estimate

These two conditions determine the elastic strain in each layer prior to formation of dislocations as 

Note that this result is consistent with the more familiar result for a thick substrate, that is, Cg — 0 and Cf — Cm as /if//ig — 0. 

The self-energy of the dislocation at the interface is then the total work which must be done in overcoming this image force as the dislocation is moved from the surface of the substrate to the interface, which is 

The external work done in creating the dislocation in the bi-layer initially strained according to (6.65) is This work is positive if 

Several noteworthy features of (6.68) are immediately evident. For one thing, it is S5rmmetric in hg and hf, a feature which might have been anticipated in advance. Because there is no fundamental distinction between the 61m and the substrate, their roles should be interchangeable. A second observation is that, if the condition is examined in the limit of a thin film on a much thicker substrate, then the result obtained in Section 6.2 is recovered, namely, 

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