Revisiting the epitaxial Si$_3$N$_4$ crystalline cap on AlGaN/GaN via evolutionary structure search

In our recent experimental work (Appl. Phys. Lett. 125, 122109 (2024)), we observed that crystalline Si$_3$N$_4$ cap layers a few nanometers thick can form in situ on GaN surfaces. Compared with amorphous SiO$_2$ and Al$_2$O$_3$ caps, these crystalline caps yield cleaner GaN/Si$_3$N$_4$ interfaces with fewer defects and improved device performance. These observations raise two questions: why does Si$_3$N$_4$ away from the interface become amorphous as the cap thickens, and what is the actual crystal structure of the interfacial Si$_3$N$_4$? Previous work proposed a defect-wurtzite (DW) model constructed heuristically from $\beta$-Si$_3$N$_4$ and the AlGaN lattice constants, but this model is significantly higher in energy than $\beta$-Si$_3$N$_4$ and disagrees with experiment in both interlayer spacings and electronic gap. Using a systematic structure-search approach under in-plane lattice constraints commensurate with AlGaN, we identify a lower-energy configuration, denoted Lam-Si$_3$N$_4$, with quasi-two-dimensional (laminar) stacking normal to the interface. Under AlGaN-matched metrics, Lam-Si$_3$N$_4$ is about 60 meV/atom more stable than DW-Si$_3$N$_4$ and reproduces the experimentally observed interlayer spacings more closely. The substantial lattice mismatch explains amorphization when the crystalline cap grows far from the interface. Upon full relaxation, both DW- and Lam-Si$_3$N$_4$ exhibit wide $\sim$4 eV band gaps. Under AlGaN constraints, the DW gap collapses to $\sim$1.88 eV whereas Lam-Si$_3$N$_4$ maintains a larger $\sim$2.70 eV gap (for reference, PBE gaps: GaN 1.73 eV, AlN 4.05 eV). The wider gap and improved structural match of Lam-Si$_3$N$_4$ rationalize the superior capping performance and provide guidance for optimizing AlGaN/GaN device encapsulation.