Interpretation of the Hume-Rothery electron concentration rule in theT2Zn11(T=Ni, Pd, Co, and Fe)γbrasses based on first-principles FLAPW calculations

Abstract

The first-principles full-potential augmented plane wave (FLAPW) band calculations were performed for a series of ${T}_{2}{\mathrm{Zn}}_{11}$ ($T=\mathrm{Ni}$, Pd, Co, and Fe) $\ensuremath{\gamma}$ brasses to elucidate the Hume-Rothery electron concentration rule. The pseudogap is found immediately below the Fermi level ${E}_{F}$ in the ${\mathrm{Ni}}_{2}{\mathrm{Zn}}_{11}$ and ${\mathrm{Pd}}_{2}{\mathrm{Zn}}_{11}\phantom{\rule{0.3em}{0ex}}\ensuremath{\gamma}$ brasses. A resulting gain in the electronic energy is attributed to their stabilization in the same way as in ${\mathrm{Cu}}_{5}{\mathrm{Zn}}_{8}$ and ${\mathrm{Cu}}_{9}{\mathrm{Al}}_{4}$ previously studied. However, the pseudogap is essentially shifted above ${E}_{F}$ in both ${\mathrm{Co}}_{2}{\mathrm{Zn}}_{11}$ and ${\mathrm{Fe}}_{2}{\mathrm{Zn}}_{11}$. The Fourier analysis of the FLAPW wave function was made at the symmetry point $N$ of the reduced Brillouin zone in the energy range involving the pseudogap. It is found that the plane wave giving rise to the largest Fourier component always resonates with the {330} and {411} zone planes to produce the pseudogap near ${E}_{F}$. Moreover, a single-branch energy dispersion relation was constructed in the extended zone scheme by averaging the wave vector $2(\mathbf{k}+\mathbf{G})$ having the largest Fourier component of the FLAPW wave function over selected electronic states in the Brillouin zone. The $e∕a$ value thus deduced is found to be close to $21∕13=1.615$ for ${\mathrm{Cu}}_{5}{\mathrm{Zn}}_{8}$, ${\mathrm{Cu}}_{9}{\mathrm{Al}}_{4}$, ${\mathrm{Ni}}_{2}{\mathrm{Zn}}_{11}$, and ${\mathrm{Pd}}_{2}{\mathrm{Zn}}_{11}\phantom{\rule{0.3em}{0ex}}\ensuremath{\gamma}$ brasses but to be only 1.4 and 1.3 for ${\mathrm{Co}}_{2}{\mathrm{Zn}}_{11}$ and ${\mathrm{Fe}}_{2}{\mathrm{Zn}}_{11}$, respectively.