The critical thickness criterion

If the inner products implied by the integrand are formed, and if the fact that the mismatch stress is zero throughout the substrate is taken into account, then 

The identity rai csca = —lfor0 a 7r has been used in obtaining the result. Note that kkm(O) = 0, that kkm( ) varies linearly with rj if the dislocation is formed in the film, and that it is independent of r/ if the dislocation is formed in the substrate. The corresponding configurational force acting on the dislocation in the direction of increasing rj is then 

This force tends to move a strain-relieving dislocation in the film away from the free surface and toward the film-substrate interface. The force is discontinuous at 7/ = /if and is everywhere zero in the substrate. 

On physical grounds, only strain relieving dislocations are possible. This notion is reflected in (6.16) by the requirement that Cm and must have the same algebraic sign for positive roots of her to exist. 

In cases for which ficr b, which is the situation for many film-substrate systems of practical interest, the left side of (6.16) is dominated by the logarithmic term. As a consequence, the critical thickness condition 

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Fig. 7.7. Curve in the plane of film thickness hi h/b) versus mismatch strain 7m showing the criterion described by (7.17) for blocking of a threading dislocation due to a misfit dislocation in its path. Also shown by the dashed curve is the critical thickness criterion for the same system as given in (7.13). Blocking is predicted for states below and to the left of the solid curve, whereas the blocking mechanism should not be effective for states above and to the right of the solid curve.

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