Pressure effects on the electronic structure and superconductivity of ${\left(\mathrm{TaNb}\right)}_{0.67}{\left(\mathrm{HfZrTi}\right)}_{0.33}$ high entropy alloy

Effects of pressure on the electronic structure, electron-phonon interaction, and superconductivity of the high entropy alloy ${(\mathrm{TaNb})}_{0.67}{\left(\mathrm{HfZrTi}\right)}_{0.33}$ are studied in the pressure range 0–100 GPa. The electronic structure is calculated using the Korringa-Kohn-Rostoker method with the coherent potential approximation. Effects of pressure on the lattice dynamics are simulated using the Debye-Grüneisen model and the Grüneisen parameter at ambient conditions. In addition, the Debye temperature and Sommerfeld electronic heat capacity coefficient were experimentally determined. The electron-phonon coupling parameter $\lambda$ is calculated using the McMillan-Hopfield parameters and computed within the rigid muffin-tin approximation. We find that the system undergoes the Lifshitz transition, as one of the bands crosses the Fermi level at elevated pressures. The electron-phonon coupling parameter $\lambda$ decreases above 10 GPa. The calculated superconducting $T_c$ increases up to 40–50 GPa and, later, is stabilized at the larger value than for the ambient conditions, in agreement with the experimental findings. Our results show that the experimentally observed evolution of $T_c$ with pressure in ${\left(\mathrm{TaNb}\right)}_{0.67}{\left(\mathrm{HfZrTi}\right)}_{0.33}$ can be well explained by the classical electron-phonon mechanism.

Figure 1

Pressure dependence of the unit-cell volume. Points correspond to the experimental data [6] and line is determined from the fitted Birch-Murnaghan equation of state.

Figure 2

Pressure dependence of ${\theta }_D$ of niobium from the direct phonon calculations and from the Grüneisen model for several values of $n$.

Figure 3

Pressure dependence of ${\theta }_D$ of tantalum from the direct phonon calculations, from the Grüneisen model for several values of $n$, and from the quasiharmonic calculations of Liu et al. [39].

Figure 4

Temperature dependence of a relative change of the unit-cell volume. The lattice parameter was obtained by the LeBail method. A cubic $Im-3m$ (space group No. 229) structure was used as a starting model.

Figure 5

(a) Temperature dependence of the electronic heat capacity $C_{\mathrm{el}}/T$ in zero (open circles) and 8 T (close circles) magnetic field. (b) Low-temperature experimental data $C_p/T$ vs $T^2$. The solid red line is a fit by the expression $C_p/T=\gamma +\beta T^2$.

Figure 6

Simulated pressure dependence of ${\theta }_D$ of TNHZT.

Figure 7

Total and atomic densities of states of TNHZT alloy, calculated under various pressures in the range of 0 to 100 GPa. Solid black line represents the total DOS. Atomic densities of states are plotted with colors and weighted over its concentrations.

Figure 8

Variation of the density of states at the Fermi level under hydrostatic pressure from 0 to 100 GPa. Dashed lines are the linear trend lines, described in the text.

Figure 9

Electronic dispersion relations for $P=0,50$, and 100 GPa. Shading describes the band smearing and corresponds to the imaginary part of the complex energy.

Figure 10

Electronic dispersion relations near $E_F$ in $\mathrm{\Gamma }-N$ direction, showing the Lifshitz transition. The distance between the center of the band and the Fermi level is marked in green. Shading describes the band smearing and corresponds to the imaginary part of the complex band energy.

Figure 11

Cross sections of Fermi surface, calculated under pressures of 50 and 100 GPa. $k$ is given in $2\pi /a$.

Figure 12

Pressure evolution of the McMillan-Hopfield parameters: concentration-weighted sum (top) and ${\eta }_i$ per atom (bottom).

Figure 13

Evolution of the McMillan-Hopfield parameters of each of the atoms, decomposed over the $l\to l+1$ scattering channels.

Figure 14

Calculated evolution of the electron-phonon coupling parameter $\lambda$ with pressure.

Figure 15

Variation of the Coulomb pseudopotential parameter ${\mu }^{*}$ with pressure, calculated using Eq. (20) and $N\left(E_F\right)$ as in Fig. 8.

Figure 16

Calculated pressure dependence of $T_c$ of ${\left(\mathrm{TaNb}\right)}_{0.67}{\left(\mathrm{HfZrTi}\right)}_{0.33}$ for two constant values of ${\mu }^{*}=0.13$ and $0.215$, variable ${\mu }^{*}\left(P\right)$ (see Fig. 15), and compared to experiment [6].