Ab Initio Investigation of the Relativistic Effect in the Physical Properties of Intermetallic Superconductor TlBi 2 ${\rm TlBi}_2$ with AlB 2 ${\rm AlB}_2$ -Type Hexagonal Layer Structure

In this work, the role of spin-orbit coupling (SOC) in the physical properties of ${\mathrm{TlBi}}_2$ is reported. These electronic calculations reveal that the heavy elements Tl and Bi promote strong spin-orbit coupling, lifting some of the degeneracies at the high-symmetry points. The presence of SOC causes significant decreases in both elastic constants and elastic moduli, which in turn improve the compatibility of the calculated Debye temperature (${\Theta }_D$ = 80 K) with the recent experimental data of 83 K. Furthermore, our quasi-static harmonic approximation calculation, like this elastic constant calculation, confirms that taking SOC into account improves agreement with the experiment by decreasing value of Debye temperature from 95 to 85 K. Activating the SOC causes significant modifications in the phonon spectrum and the density of phonon states, as well as in the Eliashberg spectral function. As a result of these modifications, the electron–phonon coupling parameter ($\lambda$) increases from 1.178 to 1.259, by $\approx 7\%$. The calculated values of $\lambda$ both with and without SOC imply that ${\mathrm{TlBi}}_2$ can be treated as phonon-mediated superconductor with strong coupling. The SOC-induced increase in $\lambda$ brings the superconducting transition temperature of ${\mathrm{TlBi}}_2$ from 6.076 to 6.211 K, which is almost equal to the recent experimental value of 6.2 K.