Electrons and phonons in the ternary alloy ${\mathrm{CaAl}}_{2-x}{\mathrm{Si}}_x$ as a function of composition

We report a detailed first-principles study of the structural, electronic, and vibrational properties of the superconducting ${\mathrm{C}}_{32}$ phase of the ternary alloy ${\mathrm{CaAl}}_{2-x}{\mathrm{Si}}_x$, both in the experimental range $0.6⩽x⩽1.2$, for which the alloy has been synthesized, and in the theoretical limits of high aluminum and high silicon concentration. Our results indicate that, in the experimental range, the dependence of the electronic bands on composition is well described by a rigid-band model, which breaks down outside this range. Such a breakdown, in the (theoretical) limit of high aluminum concentration, is connected to the appearance of vibrational instabilities and results in important differences between ${\mathrm{CaAl}}_2$ and ${\mathrm{MgB}}_2$. Unlike ${\mathrm{MgB}}_2$, the interlayer band and the out-of-plane phonons play a major role on the stability and superconductivity of CaAlSi and related ${\mathrm{C}}_{32}$ intermetallic compounds.

Figure 1

(Color online) Upper panel: energy bands (black), DOS (blue-filled area) and integrated DOS (IDOS, red solid line) for CaAlSi. Middle panel: same as above for the disordered Ca(AlSi); the three arrows indicate the energies at which some relevant bands would begin to be emptied or filled in a rigid-band framework (see text). Lower panel: same as the upper panel, for the fictitious ${◻}^{2+}$ AlSi.

Figure 2

(Color online) Fermi surface and Brillouin zone of CaAlSi in the in-plane ordered (upper panels) and disordered (lower panels) structure. The outer sheet originates from both the ${\pi }^{*}$ (around $A$) and the interlayer band (around $\Gamma$). The inner sheet (a lenslike structure) exclusively originates from the interlayer band at $\Gamma$ and is much larger, with more evident hexagonal symmetry, in the ordered structure (upper panel). We chose the yellow (light gray) color for the outer surface of the gearlike structure and the blue (dark gray) color for its inner surface. The surface of the inner, lenslike structure, is also blue. With this choice, the top view (right panels) happens to show the ${\pi }^{*}$ part of the Fermi surface in yellow and the interlayer part in blue.

Figure 3

(Color online) Upper panel: Phonon dispersion, phonon DOS (blue-filled area), and Eliashberg function (red solid line) ${\alpha }^{2}F\left(\omega \right)$ for CaAlSi. Note that the Eliashberg function has a large peak around $50{\mathrm{cm}}^{-1}$, corresponding to the soft ${\mathrm{B}}_{1g}$ branch. Lower panel: same as above but in the fictitious ${◻}^{2+}$ AlSi; the ${\mathrm{B}}_{1g}$ branch has an equally large e-ph coupling, but now its frequency is imaginary (negative in the plot) in the entire BZ, revealing a dynamical instability (see text).

Figure 4

(Color online) Upper panel: optimized lattice parameters for the ${\mathrm{C}}_{32}$ crystalline structure of ${\mathrm{CaAl}}_{2-x}{\mathrm{Si}}_x$. Lower panel: black circles and solid lines indicate the self-consistent density of states at the Fermi level $g\left({ϵ}_F\right)$ of ${\mathrm{CaAl}}_{2-x}{\mathrm{Si}}_x$ in the fully relaxed structure, while red triangles and dashed lines show $g\left({ϵ}_F\right)$ obtained with a rigid-band filling of Ca(AlSi) appropriate to the electron number $N_{\mathrm{e}}\left(x\right)=8+x$ (see middle panel of Fig. 1).

Figure 5

(Color online) Selected phonon frequencies of ${\mathrm{CaAl}}_{2-x}{\mathrm{Si}}_x$ as a function of the silicon content $x$. The branch that corresponds to the out-of-plane displacement of the $\mathrm{Al}∕\mathrm{Si}$ sublattice along $z$ (${\mathrm{B}}_{1g}$ at $\Gamma$, ${\mathrm{B}}_{2u}$ at $A$) becomes unstable around ${\mathrm{k}}_z=\pm \pi ∕c$ for $x<0.2$ (imaginary frequencies are shown as negative).

Figure 6

(Color online) Upper panel: energy bands and DOS for ${\mathrm{CaAl}}_2$ in the structurally relaxed ${\mathrm{C}}_{32}$ structure. Because of the very small $c∕a$, the $\pi$ bands acquire a large dispersion in the $k_z$ direction and are driven well above ${ϵ}_F$ in the $k_z=0$ plane. Note that the resulting Fermi surface (not shown) is still similar to that of Ca(AlSi). Lower panel: phonon dispersion and density of states corresponding to the same structure.

Figure 7

(Color online) Energy bands (black), electronic DOS (blue-filled area) and integrated DOS (solid red line) at $x=0.6$, i.e., the lowest Si concentration for which CaAlSi is observed in the ${\mathrm{C}}_{32}$ structure.

Figure 8

(Color online) Band structure of ${\mathrm{CaAl}}_2$ in the ${\mathrm{C}}_{32}$ structure with the same $c∕a$ ratio as ${\mathrm{MgB}}_2(c∕a=1.142)$. In this case there are $\sigma$ holes at the Fermi level both at $A$ and $\Gamma$ and all the phonon frequencies (not shown) are real.