Theoretical investigation of superconductivity in ${\mathrm{SrPd}}_2{\mathrm{Ge}}_2$, ${\mathrm{SrPd}}_2{\mathrm{As}}_2$, and ${\mathrm{CaPd}}_2{\mathrm{As}}_2$

Ab initio pseudopotential calculations have been performed to investigate the structural, electronic, and vibrational properties of ${\mathrm{SrPd}}_2{\mathrm{Ge}}_2, {\mathrm{SrPd}}_2{\mathrm{As}}_2$, and ${\mathrm{CaPd}}_2{\mathrm{As}}_2$ crystallizing in the ${\mathrm{ThCr}}_2{\mathrm{Si}}_2$-type body-centered tetragonal structure. Our electronic results show that the density of states at the Fermi level is mainly dominated by the strong hybridization of Pd $d$ states and Ge (or As) $p$ states. The linear response method and the Migdal-Eliashberg approach have been used to calculate the Eliashberg spectral function for all these compounds. By integrating the Eliashberg spectral function, the average electron-phonon coupling parameter $\left(\lambda \right)$ is found to be 0.74 for ${\mathrm{SrPd}}_2{\mathrm{Ge}}_2$, 0.66 for ${\mathrm{SrPd}}_2{\mathrm{As}}_2$, and 0.72 for ${\mathrm{CaPd}}_2{\mathrm{As}}_2$. Using the calculated values of $\lambda$ and the logarithmically averaged phonon frequency ${\omega }_{ln}$ the superconducting critical temperature $\left(T_c\right)$ values for ${\mathrm{SrPd}}_2{\mathrm{Ge}}_2, {\mathrm{SrPd}}_2{\mathrm{As}}_2$, and ${\mathrm{CaPd}}_2{\mathrm{As}}_2$ are found to be 3.20, 2.05, and 2.48 K, respectively, which are in acceptable agreement with the corresponding experimental values. The relative differences in the $T_c$ values between the Ge and As compounds have been explained in terms of some key physical parameters.

Figure 1

The crystal structure of $\mathrm{AP}{\mathrm{d}}_2{X}_2$ showing $X-\mathrm{P}{\mathrm{d}}_2-X-$type layers and $A$ ionic sheets alternatively stacked along the $c$ axis.

Figure 2

The electronic band structure and density of states for ${\mathrm{SrPd}}_2{\mathrm{Ge}}_2$. The Fermi level corresponds to 0 eV. The high$-$symmetry points in the Brillouin zone in Cartesian coordinates are: $G1=\frac{2\pi }{a}((\frac{1}{2}+\frac{a^2}{2c^2}),0.00,0.00), Z=\frac{2\pi }{a}(0.00,0.00,\frac{a}{c}), X=\frac{2\pi }{a}(0.50,0.50,0.00), P=\frac{2\pi }{a}(0.50,0.50,\frac{a}{2c})$, and $N=\frac{2\pi }{a}(0.0,0.50,\frac{a}{2c})$.

Figure 3

The electronic band structure and density of states for ${\mathrm{SrPd}}_2{\mathrm{As}}_2$. The Fermi level corresponds to 0 eV.

Figure 4

The electronic band structure and density of states for ${\mathrm{CaPd}}_2{\mathrm{As}}_2$. The Fermi level corresponds to 0 eV.

Figure 5

An schematic illustration of the eigendisplacements pattern of the lower $E_g$ phonon mode for ${\mathrm{SrPd}}_2{\mathrm{Ge}}_2$.

Figure 6

(a) The calculated phonon dispersion curves along high symmetry directions in the Brillouin zone for the body-centered tetragonal ${\mathrm{SrPd}}_2{\mathrm{Ge}}_2$. (b) Total and partial phonon density of states for ${\mathrm{SrPd}}_2{\mathrm{Ge}}_2$.

Figure 7

(a) The calculated phonon dispersion curves along high symmetry directions in the Brillouin zone for the body-centered tetragonal ${\mathrm{SrPd}}_2{\mathrm{As}}_2$. (b) Total and partial phonon density of states for ${\mathrm{SrPd}}_2{\mathrm{As}}_2$.

Figure 8

(a) The calculated phonon dispersion curves along high symmetry directions in the Brillouin zone for the body-centered tetragonal ${\mathrm{CaPd}}_2{\mathrm{As}}_2$. (b) Total and partial phonon density of states for ${\mathrm{CaPd}}_2{\mathrm{As}}_2$.

Figure 9

The calculated electron-phonon spectral function ${\alpha }^{2}F\left(\omega \right)$ (solid line) and the accumulative electron-phonon coupling parameter $\lambda$ (dashed line) for $A\mathrm{P}{\mathrm{d}}_2{X}_2$ superconductors.