## 2 Materials and Methods
### Results of Samples ZrN H, ZrN L, ZrN M and ZrN N
Often satellite positions and their intensities have reported a certain firmness, but in contrast to each other. The topic is controversial [26] and far from simple. As referred by [PERSON] et al. [27], the need for an accurate peak-fitting on the transition-metal region is welcome to point out humps or hidden satellite features, considering both branches of the first row of transition metals and not only the main branch, as reported countless times. The authors believe that, apart from the specific characteristics of the system under consideration, an accurate peak-fitting procedure on both branches of the transition metal doublet does not close the cycle. A theoretical model is needed for the specific complex system under investigation, [PERSON] et al. [28] established by quantitatively interpreting the shape and the intensities of shake-up structures originating from the photoexcitation process with their own model on silver and gold. In our case, the system consists of metallic stoichiometric c-ZrN and no tailored theorical model to apply was found. Referring to an experimental paper [17] where the authors refer that their report \"is the first on satellites in the Zr 3d spectrum for ZrN films\", the authors have been encouraged to go on with Zr 3d satellite deconvolution and interpretation without any theorical support. The starting points for the peak-fitting processing of each Zr 3d spectrum of the depth profiling for ZrN films analyzed (only six of those ZrN films were single binary compounds) are:  
1. the separation of shake-up satellites from the line-Zr 3d-doublet associated with zirconium nitride was fixed at + 2.5 eV [16], where the sign indicates the shift at higher binding energies with respect to the main peak (j = 5/2 as well as j = 3/2). The reference value was also supported via an internal laboratory database, the TiN system, about which the authors know well, and presents affinities with the ZrN system [21];
2. no constrains were imposed to the satellite's components differently from the main doublet to better observe where the peak position falls for both shake-up satellites;
3. for area quantifications after peak fit, the same RSF factor of the main doublet was used and the LA (a,b,c) was employed as built-in function of CasaXPS, but the a,b,c parameters were left free in the reiterations (4-6) applied to the deconvolution process. It was observed that their quantification features changed according to the sample structural order and the chemical composition of the pure ZrN or pure ZrNx with x \(<\) 1 and x \(>\) 1. The peak position of the shake-up associated with the photopeak Zr 3d stands at 182.4 \(\pm\) 0.02 for stoichiometric films M, N, but changes for sub stoichiometric samples (i.e., L) or where there is little oxygen in-depth detected. The line movement is represented in Figure 1b and a constant behavior for peaks position of the two satellites was revealed along the depth profiling for each film, suggesting a structural and chemical stability for each film. The total area percentage goes from 20-24% of the total Zr 3d signal in sample ZrN A (group 1) to 21-31% for ZrN G (group 2) and finally from 30-37.5% for sample ZrN N (group 3) as reported in the histogram of Figure 12. ZrN stoichiometric, pure and homogeneous films present a higher quantitative contribute of satellites respect to sub or over stochiometric samples. In the quantitative N/Zr ratios, the satellite's contribution was not included, as should happen more generally for all transition metals' composites. Further details about shake-ups satellites in the Zr 3d spectra of theZrN system will be considered in another publication.