## 3 Discussion
A maximum tensile stress of the Ag film, as originating from grain-boundary formation (\"zipping\") upon coalescence of neighboring grains, was found to occur at a thickness of about 10 nm [12] in the presence of Ge, for thermally evaporated Ag thin films. This corresponds to the Ag nanolayers thickness of the non-bias Ag[Ge]/AlN NMLs and could explain the observed tensile stress measured in Ag layers in the presence of Ge. The NMLs deposited with bias have a much higher Ag adatom mobility during growth (i.e., the effect of Ge on the surface adatom mobility is much less pronounced), which results in a compressive intrinsic stress state and a very high interface roughness. The compressive intrinsic stress in the as-deposited bias NMLs can originate from incomplete coalescence of neighboring Aggrains and/or a compressive stress build-up during the post-coalescence growth stage, which is typical for thicker PVD grown films [25]. The thermal expansion coefficient of Ag is more than a factor 2 times larger than that of AlN and the Al\({}_{2}\)O\({}_{3}\) substrate (note: the thermal expansion coefficients of AlN and Al\({}_{2}\)O\({}_{3}\) are similar) [26]. Consequently, the intrinsic stress in the Ag nanolayers of the as-deposited NMLs (either compressive or tensile; see Figure 7) will be superimposed by extrinsic compressive stress contributions during heating. The resulting compressive stresses during annealing of the bias samples are released by buckling of the NML with concurrent accumulation of Ag at the substrate/NML underneath the cusp of the buckles (Figure 5d). The buckling is associated with a partial delamination of the NML from its parent substrate. The Ag nanolayers of the non-bias samples exhibit a large tensile stress in the deposited state, which will also be superimposed by compressive stress contributions during heating. However, the eventual compressive stress level in the Ag nanolayers, as attained at the annealing temperature, will be significantly lower as compared to the bias samples. Moreover, the non-bias samples have continuous AlN barrier layers of uniform thickness, which will experience a higher tensile thermal strain during heating as compared to the irregular and distorted AlN barrier structure of the bias samples (compare Figure 2a,b respectively). Consequently, buckling during annealing is not observed for the non-bias samples, as their initial stress state is tensile and not compressive. The tensile stress generated or maintained in Ag layer after annealing at 400 \({}^{\circ}\)C can be due to crystalline grain growth upon thermal treatment. Thermal energy provided with the high temperature allows Ag layer to release the compressive stress (in bias samples). The larger grains formed at high temperature will experience a tensile stress when cooled down to room temperature as the thermal expansion coefficient of substrate and AlN is smaller than the one of Ag (see above). The thermal stress built up due to the Ag-AlN thermal coefficient mismatch upon cooling stays trapped into the Ag layer at room temperature as tensile stress. The stresses in the non-bias samples are partially released by surface cracking of the (tensile strained) AlN barriers (already initiated at \(T\geq 400\)\({}^{\circ}\)C) with concurrent outflow of Ag along the cracks to the NML surface (see Figures 5 and 6). The crack formation is typical for tensile stressed systems [27; 28]: tensile stress relaxation in the coating occurs through thickness microcracking. As stated in Ref. [8], the mobility of Ag is enhanced in the presence of oxygen due to the O-induced  
Figure 7: Ag in-plane stress results for the different annealing temperatures and atmospheres, measured ex-situ, at room temperature, for samples with and without bias. reduction of the activation energy for vacancy formation in Ag. Hence, the outflow of Ag along the surface cracks is accelerated in air with respect to Ar.