RSAVS superconductors: Materials with a superconducting state that is robust against large volume shrinkage

The transition temperature ($T_{\mathrm{C}}$) between normal and superconducting states usually exhibits a dramatic increase or decrease with increasing applied pressure. Here we present, in contrast, a type of superconductor that exhibits the exotic feature that $T_{\mathrm{C}}$ is robust against large volume shrinkages (RSAVS, so naming them “RSAVS superconductors”) induced by applied pressure. Extraordinarily, our previous studies found that the $T_{\mathrm{C}}$ in the two materials stays almost constant over a large pressure range, e.g., over 136 GPa in the ${\left(\mathrm{TaNb}\right)}_{0.67}{\left(\mathrm{HfZrTi}\right)}_{0.33}$ high-entropy alloy and 141 GPa in the NbTi commercial alloy. Here, we show that the RSAVS behavior also exists in another high-entropy alloy, ${\left(\mathrm{ScZrNbTa}\right)}_{0.6}{\left(\mathrm{RhPd}\right)}_{0.4}$, and in superconducting elemental Ta and Nb, indicating that this behavior occurs universally in a certain kind of superconductor, composed of only transition metal elements, with a body-centered cubic lattice. Our electronic structure calculations indicate that in the RSAVS state the contribution of the degenerate $d_{x^2-y^2}$ and $d_{z^2}$ orbital electrons remains almost unchanged at the Fermi level, suggesting that these are the electrons that may play a crucial role in stabilizing the $T_{\mathrm{C}}$ in the RSAVS state. We preliminarily analyzed the reasonability and validity of this suggestion by the Homes law.

Figure 1

Superconductivity and crystal structure for the RSAVS superconductors. (a) The pressure-dependent change in the superconducting transition temperature $\left(T_{\mathrm{C}}\right)$ of the ${\left(\mathrm{TaNb}\right)}_{0.67}{\left(\mathrm{HfZrTi}\right)}_{0.33}$ and ${\left(\mathrm{ScZrNbTa}\right)}_{0.6}{\left(\mathrm{RhPd}\right)}_{0.4}$ high-entropy alloys, the NbTi alloy, and the elemental Ta and Nb. In order to facilitate the comparison of the different materials, we use the volume shrinkage $(–\mathrm{\Delta }V/V)$ as a variable. Arrows in the diagram indicate the critical pressure ($P_{\mathrm{C}}$) where the RSAVS state emerges. $P_{\mathrm{C}}$ is about 30 GPa [the corresponding volume $(–\mathrm{\Delta }V/V_0)$ change is about 15.5%] for the ${\left(\mathrm{ScZrNbTa}\right)}_{0.6}{\left(\mathrm{RhPd}\right)}_{0.4}$ superconductor, 60 GPa $(–\mathrm{\Delta }V/V_0=21.6\%)$ for the ${\left(\mathrm{TaNb}\right)}_{0.67}{\left(\mathrm{HfZrTi}\right)}_{0.33}$ superconductor, and 120 GPa $(–\mathrm{\Delta }V/V_0=34.7\%)$ for the NbTi superconductor, while $P_{\mathrm{C}}$ is 1 bar for the elemental Ta and Nb superconductors. $P_{\mathrm{E}}$ and $P^{*}$ represent the end pressure of the RSAVS state and the highest pressure measured for the RSAVS state, respectively. (b) Sketches for the lattice structure of the ${\left(\mathrm{TaNb}\right)}_{0.67}{\left(\mathrm{HfZrTi}\right)}_{0.33}$ and ${\left(\mathrm{ScZrNbTa}\right)}_{0.6}{\left(\mathrm{RhPd}\right)}_{0.4}$ high-entropy alloys, NbTi alloy, and elemental Ta and Nb, which all possess body-centered cubic structure.

Figure 2

The calculated pressure dependence of the density of states (DOS) at the Fermi level (FL) contributed by different orbitals in the RSAVS superconductors. (a), (b) Partial DOSs at the FL as a function of pressure for the elemental Ta and Nb. (c), (d) Pressure dependence of the partial DOSs at the FL for the Nb and Ti, respectively, in the NbTi alloy. (e)–(i) Plots of partial DOSs at the FL versus pressure for the Ta, Nb, Hf, Zr, and Ti in the ${\left(\mathrm{TaNb}\right)}_{0.67}{\left(\mathrm{HfZrTi}\right)}_{0.33}$ high-entropy alloy. Although a $5\times 5\times 5$ supercell, which involves 250 atoms and reaches the limit state of our computing, is adopted for the calculations on the high-entropy alloy, it seems still not enough for achieving a perfect degeneracy for the $d$ orbitals. The trend of the DOSs versus pressure, however, is analogous to those obtained in elemental Ta and Nb as well as NbTi alloy—the DOSs contributed by $d_{x^2-y^2}$ and $d_{z^2}$ orbital electrons remain almost unchanged in the pressure regimes of the RSAVS state.

Figure 3

The superfluid density (${\rho }_{\mathrm{s}}$) as a function of the product of the conductivity $\left(\sigma \right)$ and the superconducting transition temperature $\left(T_{\mathrm{C}}\right)$ for a variety of copper oxides and elemental metals. The data denoted by open squares are taken from the Ref. [31]. The data presented by solid balls are our results, in which the ${\rho }_{\mathrm{s}}$ values are derived from the partial DOSs contributed by the $d_{x^2-y^2}$ and $d_{z^2}$ orbitals. The data presented by stars are also our results, in which the ${\rho }_{\mathrm{s}}$ values are derived from the partial DOSs contributed by the $d_{xy}, d_{xz}$, and $d_{yz}$ orbitals. The $e_g$ represents $d_{x^2-y^2}$ and $d_{z^2}$ orbitals, and the $t_{2g}$ stands for $d_{xy}, d_{xz}$, and $d_{yz}$ orbitals.