Through some kind of stoichiometric analysis (eg. EDX)

Use FTIR spectroscopy either in the transmission or the reflection mode.

Deposit your films on KBr pellets if possible. Try to deposit a thin film, and see if FTIR spectra reveals the interesting chemical information.

I would take it to u-Raman. I think Raman will give you the most information from your sample.

I dont know if you have your film hydrogenated, but according to enclosed article you should see any kind of defects in your film by u-Raman

http://scitation.aip.org/content/aip/journal/jap/85/11/10.1063/1.370604

What is the thickness of Si3N4 film, if its very thin than SIMS would be a good choice. But EDX measurements will give some idea about the stoichiometric of the film.

My suggestion is to carry out Raman spectroscipy and the peak broadening of phonon modes is related to the amorphous nature of  the material.

Electron diffraction by TEM is available.

You do not mention meither the thickness of your film nor other possible elements present in it. IF your film is thin enough you could try EELS (electron energy loss spectroscopy) in the electron microscope. It should be able to distinguish both, Si and N. 

If it is a TEM thinfilm, you can  aquire an HR image.. In Digital micrograph in "process" do a FFT. In the FFT apply a circular mask on the amorphous ring if prresent. Do an IFFT, and bingo your amorphours areas will light up

Please, I have a question:

Does the Energy-dispersive X-ray spectroscopy (EDX or EDS) provide the chemical composition of the material as metal or metal oxide? i.e the results obtained from the EDX are in percent for metal or for metal oxide?

I found in different published papers that the data obtained from EDX are in % of metals and other papers as % metal oxide. When we use it as metals and when as metal oxide?

Is the following sentence right or false:  "EDX determines the chemical composition of materials as metals and exposes them as oxides but it doesn't mean that they are oxides". Many thanks for your reply.

@Metwally

I don't have much knowledge about EDX.

But from my experiment I observed that EDX determines the chemical composition as metals but detected them from their standard oxides value.

I am attaching one of my EDX file.