  In this paper we introduce Peierls-Nabarro type models for edge dislocations
at semi-coherent interfaces between two heterogeneous crystals, and prove the
optimality of uniformly distributed edge dislocations. Specifically, we show
that the elastic energy $\Gamma$-converges to a limit functional comprised of
two contributions: one is given by a constant $c_\infty>0$ gauging the minimal
energy induced by dislocations at the interface, and corresponding to a uniform
distribution of edge dislocations; the other one accounts for the far field
elastic energy induced by the presence of further, possibly not uniformly
distributed, dislocations. After assuming periodic boundary conditions and
formally considering the limit from semi-coherent to coherent interfaces, we
show that $c_\infty$ is reached when dislocations are evenly-spaced on the one
dimensional circle.
