cavity mode. In order to maintain high-$Q_o$, tapering to highly elliptical holes is beneficial, as they better match the rectangular shape of the gap [@Hryciw:13]. When designing SBC’s, it is critical to understand the optical mode structure to know what gap width $g$ to replace the centre hole with. With SBC’s in Si, it was previously found that the dielectric mode of the gap had to be matched to the air mode of the centre hole due to a crossing of the waveguide band edges [@Hryciw:13]. As shown in Fig. \[figureThree\](b), due to the lower refractive index of the Si$_{3}$N$_{4}$, the dielectric and air mode bandedge frequencies of the gap unit cell as a function of nanobeam width in Si$_{3}$N$_{4}$ do not cross, even with the same $w/t$ ratio as in Ref. [@Hryciw:13]. Knowing this, we then substitute a gap with $g$ = 110 nm that has the same fill factor as the cavity centre hole that it replaces (given from Fig. \[figureTwo\](c)). After local optimization of $g$, we find that for $g$ = 112 nm the SBC supports an air-mode with $Q_o$ = $2.2~\times~10^5$. The bandstructure of a unit cell containing this gap is shown in Fig. \[figureTwo\] (blue 