Fig. 1: Schematic diagram of spinodal decomposition in HfN-Hf₂ON₂ system.

a Ternary phase diagram of the Hf-O-N system. The circles filled with solid yellow denote the stoichiometric oxides and oxynitrides of Hf. They are HfO₂, Hf₇O₈N₄, Hf₂ON₂ and HfN, respectively. The non-stoichiometric oxynitrides, denoted as Hf₂O_1-xN₂ (0 < x < 1), vary in composition between HfN and Hf₂ON₂. The spinodal decomposition process initiates from such Hf₂O_1-xN₂ intermediate phases and eventually leads to the formation of the HfN/Hf₂ON₂ heterostructure. b Schematic Gibbs free energy graph and the corresponding phase diagram for the HfN-Hf₂ON₂ system. The Hf₂O_1-xN₂ intermediate phase falls into the region of negative curvature \(\it \it \it \it \it \it \it \left(\frac{{\partial }^{2}{\mbox{G}}}{{\partial {\mbox{c}}}^{2}} < 0\right)\), lying within the inflection points \(\it \it \left(\frac{{\partial }^{2}{\mbox{G}}}{{\partial {\mbox{c}}}^{2}}=0\right)\) of the curve.