Electronic and lattice properties of noncentrosymmetric superconductors $\mathrm{Th}T\mathrm{Si}$ $(T=\mathrm{Co}, \mathrm{Ir}, \mathrm{Ni}, \mathrm{and} \mathrm{Pt})$

Theoretical studies on the electronic and lattice properties of the series of noncentrosymmetric superconductors $\text{Th}T\text{Si}$, where T = Co, Ni, Ir, and Pt, are presented. The electronic band-structure and crystal parameters were optimized within the density functional theory. Spin-orbit coupling leads to the splitting of the electronic bands and Fermi surfaces, with the stronger effect observed for the compounds with the heavier atoms Ir and Pt. The possible mixing of the spin-singlet and spin-triplet pairing in the superconducting state is discussed. The phonon dispersion relations and phonon density of states were obtained using the direct method. The dispersion curves in ThCoSi and ThIrSi exhibit low-energy modes along the $\mathrm{S}\text{−}\mathrm{N}\text{−}{\mathrm{S}}_0$ line with the tendency for softening and dynamic instability. Additionally, we calculate and analyze the contributions of phonon modes to lattice heat capacity.

Figure 1

Relation between the conventional standard cell and the primitive unit of the $\text{Th}T\text{Si}$ compounds (marked by the bold dashed line and the solid line with the transparent filling, respectively). The image was rendered using vesta software [25].

Figure 2

The Brillouin zone of the $I{4}_{1}md$ structure with the high-symmetry points [26]. The red line demonstrates the path used for the electronic band-structure presentation.

Figure 3

The band structure of studied systems (as labeled) obtained in the absence and in the presence of the spin-orbit coupling (dashed red and solid green lines, respectively). The Fermi level is located at zero energy.

Figure 4

The Fermi surfaces of studied systems (as labeled) obtained in the absence of the spin-orbit coupling.

Figure 5

Cross sections of the Fermi surfaces of studied systems (as labeled) in two chosen planes shown in the top panel (cf. with Fig. 2). Results obtained in the absence and in the presence of the spin-orbit coupling (red and green lines, respectively).

Figure 6

DOS with projections on the orbitals (as indicated in labels) for the studied systems (as labeled). The Fermi level is located at zero energy.

Figure 7

Phonon dispersion curves and density of states of studied systems (as labeled).

Figure 8

An example of atomic displacements in the wagging mode with the lowest energy at the N point in ThIrSi. The solid lines represent schematically the three-dimensional Si–T framework based on the perpendicular T–Si–T zigzag chains. The image was rendered using vesta software [25].

Figure 9

The theoretical phonon heat capacity $C_V$ (black dashed lines), averaged over all degrees of freedom (d.o.f.), compared with the measured total heat capacity $C_p$ (red open squares) of studied systems. Additionally, the calculated partial $C_V$ per degree of freedom is presented, averaged for Th (turquoise dotted line) and Si (magenta dash-dotted line) atoms as well as all components for T atoms (green, violet, and orange solid lines correspond to $xy$ wagging type, other $xy$, and $z$ contributions, respectively).