Effects of spin-orbit coupling on the electron-phonon superconductivity in the cubic Laves-phase compounds ${\mathrm{CaIr}}_2$ and ${\mathrm{CaRh}}_2$

We report our ab initio pseudopotential results for the structural, electronic, vibrational, and electron-phonon interaction properties of the cubic Laves-phase compounds ${\mathrm{CaIr}}_2$ and ${\mathrm{CaRh}}_2$. While the spin-orbit coupling (SOC) does not result in any appreciable changes in structural parameters, it lifts the degeneracies of some bands near the Fermi level, albeit with a much smaller amount for ${\mathrm{CaRh}}_2$. The effect of SOC on the vibrational properties of both materials is considerable. The SOC results in a slight decrease in the electronic density of states near the Fermi level $N\left(E_F\right)$ and makes low-frequency phonon branches harder, and the electron-phonon coupling parameter $\lambda$ is lowered from 1.43 to 1.05 for ${\mathrm{CaIr}}_2$ and from 1.17 to 0.96 for ${\mathrm{CaRh}}_2$. On the other hand, the logarithmically averaged phonon frequency ${\omega }_{ln}$ is enhanced from 79.60 to 100.97 K for ${\mathrm{CaIr}}_2$ and from 120.20 to 140.80 K for ${\mathrm{CaRh}}_2$ with the inclusion of SOC. Using the calculated values of $\lambda$ and ${\omega }_{ln}$, the superconducting critical temperature is determined to be 5.94 K (7.34 K without SOC) for ${\mathrm{CaIr}}_2$ and 6.97 K (9.08 K without SOC) for ${\mathrm{CaRh}}_2$. The superconducting critical-temperature values with SOC compare very well with corresponding experimental values of 5.80 and 6.40 K, indicating the importance of SOC for the physical properties of both materials.

Figure 1

The cubic Laves crystal structure of ${\mathrm{CaIr}}_2$ and ${\mathrm{CaRh}}_2$.

Figure 2

The calculated electronic band structures and electronic density of states for ${\mathrm{CaIr}}_2$ and ${\mathrm{CaRh}}_2$ with and without spin-orbit coupling (SOC). The inclusion of SOC has split several bands.

Figure 3

The calculated total and partial electronic densities of states of ${\mathrm{CaIr}}_2$ and ${\mathrm{CaRh}}_2$ with SOC.

Figure 4

The calculated phonon dispersion curves of ${\mathrm{CaIr}}_2$ and ${\mathrm{CaRh}}_2$ with and without SOC.

Figure 5

The calculated total and atom-projected phonon densities of states of ${\mathrm{CaIr}}_2$ with and without SOC.

Figure 6

The calculated total and atom-projected phonon densities of states of ${\mathrm{CaRh}}_2$ with and without SOC.

Figure 7

Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$ (solid line) and integrated electron-phonon coupling parameter $\lambda$ (dashed line) for ${\mathrm{CaIr}}_2$ and ${\mathrm{CaRh}}_2$.