Longitudinal piezoelectric, elastic, and dielectric properties of rare-earth aluminum nitride alloys determined by density-functional perturbation theory

The superior piezoelectric (PE) performance of devices based on thin-films of wurtzite (w) w-${\mathrm{Sc}}_{x}{\mathrm{Al}}_{1\ensuremath{-}x}\mathrm{N}$ alloys has motivated research efforts to determine the material properties of other alloys of aluminum nitride and rare-earth nitride (w-${\mathrm{RE}}_{x}{\mathrm{Al}}_{1\ensuremath{-}x}\mathrm{N}$). The longitudinal PE, elastic and dielectric properties of substitutional w-${\mathrm{RE}}_{x}{\mathrm{Al}}_{1\ensuremath{-}x}\mathrm{N}$ alloys have been calculated with density-functional perturbation theory. The rare-earth elements in the simulations stand for scandium, yttrium and seven lanthanides: lanthanum, neodymium, gadolinium, dysprosium, erbium, ytterbium, and lutetium. The structure of the investigated alloys is modeled with special quasirandom $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ supercells at rare-earth nitride fractions $x$ of 0.0625 and 0.25. The $4f$ electrons of the lanthanides are incorporated in the pseudopotential core, an approach supported by the standard model of lanthanide chemistry. An increase of the PE stress coefficient ${e}_{33}$ and a softening of the stiffness coefficient ${c}_{33}^{E}$ are found for all investigated alloys for $x$ = 0.25, when compared to w-AlN. This leads to significant improvements of the PE strain effective coefficient ${d}_{33,f}$ of films of these alloys. For w-${\mathrm{Lu}}_{0.25}{\mathrm{Al}}_{0.75}\mathrm{N}$, the ${d}_{33,f}$ increases by $\ensuremath{\sim}100%$ compared to pure w-AlN. Increased displacement-response internal-strains and Born effective charges are demonstrated for all alloys at $x$ = 0.25, in a manner consistent with the behavior of w-${\mathrm{Sc}}_{x}{\mathrm{Al}}_{1\ensuremath{-}x}\mathrm{N}$ alloys, suggesting shared intrinsic properties for the entire w-${\mathrm{RE}}_{x}{\mathrm{Al}}_{1\ensuremath{-}x}\mathrm{N}$ alloy family. Distortions of the alloy structure, correlated with the ionic radius of the rare earths, negatively affect their PE properties. At a small fraction $x$ = 0.0625, the alloys incorporating Sc, Y, Er, Yb, and Lu possess PE coupling coefficients ${k}_{33,f}^{2}$ greater than for w-AlN.