Investigating the normal state and superconducting state properties of orthorhombic and hexagonal ZrRuP: A first-principles study

We have executed ab initio pseudopotential calculations on the structural, electronic, elastic, mechanical, vibrational, and electron-phonon interaction properties of hexagonal ZrRuP (h-ZrRuP) and orthorhombic ZrRuP (o-ZrRuP). The electronic states of both phases near the Fermi energy are found to be dominated by the $d$ electrons of transition metal atoms, suggesting that they play a more active role in the generation of superconducting state for both phases of ZrRuP. A critical assessment of their elastic and mechanical properties reveals that the lattice of h-ZrRuP is softer than that of o-ZrRuP. A comparison of phonon dispersion curves for both phases indicates that the lower transverse acoustic branch of h-ZrRuP is much softer than that of o-ZrRuP. The soft character of this phonon branch gives rise to strong electron-phonon interaction in h-ZrRuP. Therefore the electron-phonon coupling parameter for h-ZrRuP equals to 1.25 which is considerably larger than the corresponding value of 0.57 for o-ZrRuP. As a consequence, phonon and electron-phonon interaction properties are crucial in making superconducting transition temperature much higher for h-ZrRuP than o-ZrRuP. At the end, the value of this temperature is found to be 12.49 K for h-ZrRuP and 3.89 K for o-ZrRuP which coincide with their experimental values of 12.93 and 3.82 K.

Figure 1

(a) The layer crystal structure of h-ZrRuP with a hexagonal ZrNiAl-type form. Either Zr and P atoms or Ru and P atoms occupy each layer in the hexagonal lattice. (b) The layer crystal structure of o-ZrRuP with an orthorhombic TiNiSi-type form. On the contrary, the orthorhombic phase contains layers which are filled with Zr, Ru, and P atoms and thus all layers are equivalent.

Figure 2

(a) The calculated electronic band structure of h-ZrRuP along the high-symmetry lines in the first Brillouin zone of hexagonal lattice. The bands crossing the Fermi level are shown with red, blue, and green solid lines. (b) The calculated total and partial electronic density of states for h-ZrRuP.

Figure 3

(a) The calculated electronic band structure of o-ZrRuP along the high-symmetry lines in the first Brillouin zone of simple orthorhombic lattice. The bands crossing the Fermi level are shown with red, blue, green and magenta solid lines. (b) The calculated total and partial electronic density of states for o-ZrRuP.

Figure 4

The calculated Fermi surfaces for (a) h-ZrRuP and (b) o-ZrRuP.

Figure 5

(a) The calculated phonon spectrum of o-ZrRuP along the high-symmetry lines in the first Brillouin zone of simple orthorhombic lattice. (b) The calculated total and partial phonon density of states for o-ZrRuP.

Figure 6

(a) The calculated phonon spectrum of h-ZrRuP along the high-symmetry lines in the first Brillouin zone of hexagonal lattice. (b) The dispersion of electron-phonon coupling parameter for the lower transverse acoustic branch (${\mathrm{TA}}_1$) of h-ZrRuP along the $\mathrm{\Gamma }-K-M-\mathrm{\Gamma }$ directions. (c) The calculated total and partial phonon density of states for h-ZrRuP.

Figure 7

The frequency variations of the Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$ (red lines) and the average electron-phonon coupling parameter $\lambda$ (blue lines) for ZrRuP. Solid lines and dashed lines present our results for the orthorhombic and the hexagonal phases, respectively.