First-principles study of the Hume-Rothery electron concentration rule in Al-Cu-(Fe,Ru)-Si 1/1-cubic approximants*

To elucidate the Hume-Rothery electron concentration rule, we determined the self-consistent electronic structures of the Al$_{108}$Ru$_{24}$Cu$_{6}$Si$_{6}$ and Al$_{108}$Fe$_{24}$Cu$_ {6}$Si$_{6}$ 1/1-1/1-1/1 approximants containing 144 atoms in each \textit {Pm}-3 cubic unit cell using the full-potential linearized augmented plane wave (FLAPW) method [1], now running on massively parallel computer platforms. A significant pseudogap was found around the Fermi level for both alloys in the calculated densities of states, which should contribute to stabilization of the system. The FLAPW wave functions provide a direct observation of the Brillouin zone resonance in the Fermi surface [2]: a Fourier analysis of the wave functions confirms the Hume-Rothery matching rule 2$k_{F}=K$ where the reciprocal lattice vectors $K$ consist of {\{}543{\}}, {\{}550{\}}, and {\{}710{\}} planes highly degenerate at the $N$ point. Consequently, an effective electron concentration per atom ($e/a)$ was evaluated to be 0.8 for both Ru and Fe in these structures making a sharp contrast with the previously assumed empirical value of -2.7 proposed by Raynor [3]. [1] Wimmer et al., Phys. Rev. B \textbf{24}, 864 (1981). [2] Asahi et al., Phys. Rev. B \textbf{72}, 125102 (2005). [3] Raynor, Prog. Metal Phys. \textbf{1}, 1 (1949).