The physics behind is in principle the same so that you can read any book about EDS, i.e. Goldstein "SEM and Microanalysis" in order to learn the fundamentals. The difference in TEM is simply given by the size of the sample. It is no bulk so that the correction factors are different and the detector is optimized in order to capture much more signal because of the tiny probe emitting X-radiation. If you have a detector you should have as well a software which enables a interpretation. M. Jalilian already touched the problem significance of the signal. Of course, your interest is the locally resolved chemical information so that the typical discussion of a representative composition is in contrast to SEM investigations theoretically irrelevant. But you have a very specific problem if you need to prepare your sample. The sample prep can have an effect if it is not done carefully. On the other hand, the used quantification procedures consider an ideal sample, as for SEM where the sample is assumed to be absolutely flat and well aligned. in fact this is the same for TEM. If you want investigate fibers or single particles you cannot absolutely prevent an interaction with backscattered electrons. This effect is not that dramatic as with lower energetic electrons in SEM but cannot be completely neglected. Backscattered electrons also generate signal from the TEM, e.g. the pole piece or - if existing - the copper grid. All these minor effects may affect your signal and therefore also your results.
SEM equipped with EDS can be utilized to provide a simi-quantitative elemental analysis and it can be employed to determine the elemental composition of the nanomaterials.
@ S. Al-Jaroudi: you are only right if your nano-material is chemically homogeneous and appears as bulk. But then you can also investigate SEM-stable liquids or glass. This is not the point of this question. The challenge for nanoparticles in a chemically different matrix or single nanoparticles on a stub is simply impossible in SEM, already because of the non-sufficient size of the covered solid angle of the detector. There also the term: Size matters in not correct. You need to be able to cover a huge solid angle in order to collect as many as possible x-ray quants. Therefore, TEM-detectors have a different design although their detection principle is more or less identical. I big difference (and challenge as well) is the much higher acceleration voltage!
I don't think EDS is a quantitative method to analyze elemental composition because it only cover a very small area of the sample (which can be problematic for composites or other chemically heterogeneous materials).
EDS is of course a technique for quantification of elemental composition. If you have a heterogeneous material with grains in the scale of several µm...how do you want use ICP for this investigation? Here we are even talking about nanomaterials!
A property of a technically used material which is the sum of all phases and not the sum of all elements, You need to identify the phases in order to understand which is perhaps the weakest candidate. Failure analysis is doing nothing else. On the other hand, of course there are applications where you need to know the overall elemental composition, e.g. in case of the evaluation of an alloy or a certain steel. But this is a totally different applications. And of course there are more accurate techniques than EDS, e.g. WDS. However, you also don't measure the length of your car with a micrometer caliper since it is OK if you know the length in cm.
Again: EDS is a widely used analytical technique which works qualitatively in a precision and accuracy which is absolutely sufficient for many applications. And the question here is, does it work for (heterogeneous) nanomaterials.
EDS surely give us the existence of different elements but based on my experience I don't think the EDS is accurate technique for quantification of elemental composition. For example I have synthesized nano-porous silica (MCM-41) and the element ratio in SiO2 is Si:1,O:2 however when I used EDS for this sample the molar ratio was different from SiO2 formula and EDS analysis showed Si:O= 1:1.
I would be surprized if this doesn't work for Si and O if you consider the identification of Quartz or silicates. Did you checked bulk SiO2? Might you have charging? No technique is without compromises, artefacts, limitations. First I would recommend you to test the software with something like quartz before make such drastic conclusions. In Prinziple, EDS and WDS are very similar techniques and they certainly do not have problems. Only energy resolution is different and therefore precision.