Plastic Relaxation without Dislocations in $\beta$-Ga$_2$O$_3$ Heteroepitaxy: A Structural Peculiarity of Ga$_2$O$_3$ Polymorphs
$\beta$-Ga$_2$O$_3$ grows as three-dimensional islands on mismatched c-plane sapphire, often on a thin $\alpha$-Ga$_2$O$_3$ wetting layer, and eventually forms a continuous film by island coalescence. Experiments reveal the $\beta$ film to be relaxed. Surprisingly no dislocations are observed, suggesting an unusual plastic relaxation path. We reveal the key mechanisms underlying this process by combining density functional theory, continuum nucleation theory, and molecular dynamics simulations. Plastic relaxation occurs through rearrangement of the Ga atoms beneath the oxygen plane shared by both the wetting layer and the (-201) $\beta$-Ga$_2$O$_3$ structures. A local $\alpha$ to $\beta$-relaxed phase transition is thus realized within a single plane, allowing full relaxation of the $\beta$-Ga$_2$O$_3$ island.