  We present a first-principles investigation of the localization properties of
vibrational modes in amorphous silicon nitride ($a$-Si$_3$N$_4$). Our
investigation further confirms that the vibrational modes underlying the peak
at $\sim$471 cm$^{-1}$ in the infrared spectrum of silicon nitride mainly
consist of nitrogen motion in the direction normal to the plane defined by the
three Si nearest neighbors. In-plane stretching of N--Si bonds becomes largely
dominant above $\sim$700 cm$^{-1}$. In particular vibrational modes underlying
the infrared peak at 825 cm$^{-1}$ arise from the N--Si bond stretching
motions. If N--N homopolar bonds were present, we show that N--N bond
stretching occurs above $\sim$1100 cm$^{-1}$. Furthermore, we investigate the
localization properties of vibrational modes by calculating their inverse
participation ratio (IPR) and phase quotient. From this analysis we infer that
modes above 600 cm$^{-1}$ shows a progressive increase of the localization
degree and optic-like behavior, especially above 1000 cm$^{-1}$. At about 650
cm$^{-1}$, given the considerable IPR value, and on the basis of projectional
analysis on silicon-breathing-like motions of the NSi$_3$ units, we suggest
that vibrational modes may involve correlated motions of neighboring SiN$_4$
tetrahedra.
