Coherency effects on the mixing thermodynamics of cubic ${\mathrm{Ti}}_{1-x}{\mathrm{Al}}_x\mathrm{N}/\mathrm{TiN}$(001) multilayers

In this work, we discuss the mixing thermodynamics of cubic (B1) ${\mathrm{Ti}}_{1-x}{\mathrm{Al}}_x\mathrm{N}/\mathrm{TiN}\left(001\right)$ multilayers. We show that interfacial effects suppress the mixing enthalpy compared to bulk ${\mathrm{Ti}}_{1-x}{\mathrm{Al}}_x\mathrm{N}$. The strongest stabilization occurs for compositions in which the mixing enthalpy of bulk ${\mathrm{Ti}}_{1-x}{\mathrm{Al}}_x\mathrm{N}$ has its maximum. The effect is split into a strain and an interfacial (or chemical) contribution, and we show that both contributions are significant. An analysis of the local atomic structure reveals that the Ti atoms located in the interfacial layers relax significantly different from those in the other atomic layers of the multilayer. Considering the electronic structure of the studied system, we demonstrate that the lower Ti-site projected density of states at ${\epsilon }_{\mathrm{F}}$ in the ${\mathrm{Ti}}_{1-x}{\mathrm{Al}}_x\mathrm{N}/\mathrm{TiN}$ multilayers compared to the corresponding monolithic bulk explains a decreased tendency toward decomposition.

Figure 1

The mixing enthalpy of bulk ${\mathrm{Ti}}_{1-x}{\mathrm{Al}}_x\mathrm{N}$ (ab initio), the mixing enthalpy with the coherency strain effect, and the mixing enthalpy (ab initio) of ${\mathrm{Ti}}_{1-x}{\mathrm{Al}}_x\mathrm{N}/\mathrm{TiN}$ multilayer (ML). (a) 6/6 ML and (b) 12/6 ML. The dashed lines show the maximum of the mixing enthalpy values for the bulk and the different ML case.

Figure 2

In-plane lattice parameters of multilayers 6/6 and 12/6: $a_{{\mathrm{Ti}}_{1-\mathrm{x}}{\mathrm{Al}}_{\mathrm{x}}\mathrm{N}/\mathrm{TiN}}$ and bulk: $a_{{\mathrm{Ti}}_{1-\mathrm{x}}{\mathrm{Al}}_{\mathrm{x}}\mathrm{N}}$.

Figure 3

The calculated averaged interlayer distances $d$ using the fully relaxed structures of both MLs. (a) 6/6 ML and (b) 12/6 ML. The solid lines show the average of the interlayer distance for ${\mathrm{Ti}}_{1-x}{\mathrm{Al}}_x\mathrm{N}$ and TiN in the MLs, while the dashed lines show the bulk cases.

Figure 4

Comparison of the two chemical effects (normalized binding energy), the one from ${\mathrm{Ti}}_{1-x}{\mathrm{Al}}_x\mathrm{N}/\mathrm{TiN}$ and the other form AlN/TiN in the mixing enthalpy of the ${\mathrm{Ti}}_{1-x}{\mathrm{Al}}_x\mathrm{N}/\mathrm{TiN}$ multilayer. (a) shows the 6/6 ML while (b) displays the 12/6 ML.

Figure 5

The atomic relaxations with respect to the ideal positions in each layer for $x=0.55$ of bulk ${\mathrm{Ti}}_{1-x}{\mathrm{Al}}_x\mathrm{N}$ and 6/6 ML (a) in the $x$ direction and (b) in the $z$ direction. (c) The relaxations of the N atoms with different types of neighbors (Al-N-Al, Ti-N-Al, and Ti-N-Ti) along the $z$ direction. (d) The relaxations of the Ti and Al atoms along the $z$ direction. See the text for more details.

Figure 6

Bulk and the 6/6 multilayer $d$-orbital partial density of states (PDOS) at the Fermi energy (${\epsilon }_{\mathrm{F}}$) of the Ti atoms from only the ${\mathrm{Ti}}_{1-x}{\mathrm{Al}}_x\mathrm{N}$ slab.