Positron annihilation spectroscopy has been combined with Auger electron depth profiling, elastic recoil detection, Fourier-transform infrared spectroscopy, and capacitance-voltage measurements to study silicon nitride films grown by plasma-enhanced chemical vapor deposition. For silicon-rich films the positron lineshape parameter is very close to that of undefected silicon, consistent with a strong hydrogen passivation effect. With increasing nitrogen content in the films, there is an increase in the number of unpassivated vacancy complexes available to trap positrons prior to annihilation. Detailed analysis gives a measure of the electric field induced in the silicon substrate by the presence of charge trapped in the silicon nitride near the interface. These results agree qualitatively with electrical measurements. Both the charge and the electric field are found to decrease with increasing nitrogen content. Incorporation of a small amount of oxygen in the films leads to a suppression of the feature in the positron spectrum associated with vacancy complexes in favor of a spectrum typical of oxygen related defects.