Effect of the density of states at the Fermi level on defect free energies and superconductivity: A case study of ${\mathrm{Nb}}_3\mathrm{Sn}$

Although often ignored in first-principles studies of material behavior, electronic free energy can have a profound effect in systems with a high-temperature threshold for kinetics and a high Fermi-level density of states (DOS). ${\mathrm{Nb}}_3\mathrm{Sn}$ and many other members of the technologically important A15 class of superconductors meet these criteria. This is no coincidence: both electronic free energy and superconducting transition temperature $T_{\text{c}}$ are closely linked to the electronic density of states at the Fermi level. Antisite defects are known to have an adverse effect on $T_{\text{c}}$ in these materials because they disrupt the high Fermi-level density of states. We observe that this also locally reduces electronic free energy, giving rise to large temperature-dependent terms in antisite defect formation and interaction free energies. This work explores the effect of electronic free energy on antisite defect behavior in the case of ${\mathrm{Nb}}_3\mathrm{Sn}$. Using ab initio techniques, we perform a comprehensive study of antisite defects in ${\mathrm{Nb}}_3\mathrm{Sn}$, and find that their effect on the Fermi-level DOS plays a key role determining their thermodynamic behavior, their interactions, and their effect on superconductivity. Based on our findings, we calculate the A15 region of the Nb-Sn phase diagram and show that the phase boundaries depend critically the electronic free energy of antisite defects. In particular, we show that extended defects such as grain boundaries alter the local phase diagram by suppressing electronic free-energy effects, explaining experimental measurements of grain boundary antisite defect segregation. Finally, we quantify the effect of antisite defects on superconductivity with the first ab initio study of $T_{\text{c}}$ in ${\mathrm{Nb}}_3\mathrm{Sn}$ as a function of composition, focusing on tin-rich compositions observed in segregation regions around grain boundaries. As tin-rich compositions are not observed in bulk, their properties cannot be directly measured experimentally; our calculations therefore enable quantitative Ginzburg-Landau simulations of grain boundary superconductivity in ${\mathrm{Nb}}_3\mathrm{Sn}$. We discuss the implications of these results for developing new growth processes to improve the properties of ${\mathrm{Nb}}_3\mathrm{Sn}$ thin films.

Figure 1

Left, atomic composition of a sample cross section showing regions of niobium segregation (1 and 3) in contrast to 25% tin regions (2), reproduced from [36] (Lee et al.). Right, atomic composition measured by atom-probe tomography near a grain boundary showing tin segregation; the 3-nm-wide segregation region is significantly wider than the region of structural disorder at the boundary core. Reproduced from [37] (Lee et al.).

Figure 2

Density of states versus energy difference from Fermi level in pure ${\mathrm{Nb}}_3\mathrm{Sn}$. The high peak at the Fermi level plays a crucial role in determining defect and superconducting behaviors.

Figure 3

Fractional antisite defect concentration as a function of composition and temperature, calculated without (left) and with (right) electronic free-energy effects. Contours are at intervals of 0.005, with concentrations beyond 0.03 in dark pink.

Figure 4

Change in local density of states (LDOS) at the Fermi level versus position in a $1\times 1\times 12$ supercell with a ${\mathrm{Nb}}_{\mathrm{Sn}}$ defect at $\sim 1.8$ nm and a ${\mathrm{Sn}}_{\mathrm{Nb}}$ defect at $\sim 4.2$ nm. Both antisite defects affect the Fermi-level LDOS in a radius of just under 1 nm.

Figure 5

Convex hull construction of free energy versus tin content at $T=1100\mathrm{K}$: bcc phase (blue), A15 phase without (green) and with (red) ${\mathrm{Nb}}_{\mathrm{Sn}}\text{−}{\mathrm{Nb}}_{\mathrm{Sn}}$ interactions, linear tangents (black), compositional limits of phases ($\times$'s). The noninteracting approximation increasingly underestimates the A15 free energy at increasingly tin-poor compositions. The strong curvature of the A15 hull indicates that this phase favors compositional homogeneity in equilibrium.

Figure 6

Nb-Sn phase diagram: this work (shaded regions reflect estimated DFT uncertainties in phase boundaries), and experiment [49] (solid curves). Hash marks indicate the region where the ${\mathrm{Nb}}_6{\mathrm{Sn}}_5$ phase considered in this work is unstable experimentally (see text).

Figure 7

Density of states versus energy difference from Fermi level in pure ${\mathrm{Nb}}_3\mathrm{Sn}$ (black) and in a grain boundary calculation (red) [59].

Figure 8

Predicted tin-rich phase limit of A15 Nb-Sn versus temperature: expected bulk dependence (black), dependence in regions where the electronic free-energy contribution to the antisite defect formation free energy is reduced by 25% (dark red) and 50% (light red).

Figure 9

Phonon spectral function ${\alpha }^{2}F$ versus energy in A15 Nb-Sn: 25% tin content (black), 26.6% (dark red), 29.2% (red), 31.3% (light red). ${\alpha }^{2}F$ decreases with increasing tin content above 25% across all energies.

Figure 10

Superconducting transition temperature $T_{\text{c}}$ versus tin content of A15 Nb-Sn: experiment [54, 61, 68] (gray squares) and this work (black circles). We predict $T_{\text{c}}$ to reach a minimum of about 5 K near 31% tin content in the tin-rich regime. Multiple data points at the same composition represent calculations for supercells with different antisite defect configurations.

Figure 11

Predicted $T_{\text{c}}$ versus Fermi-level density of states in A15 Nb-Sn of varying composition: tin-poor compositions (red circles), tin-rich compositions (red triangles), ${\mathrm{Nb}}_3\mathrm{Sn}$ stoichiometry (black circle). The data illustrate a nearly linear relationship for off-stoichiometery calculations.

Figure 12

Map of the square of the superconducting order parameter ${\left|\psi \right|}^2$ in a ${\mathrm{Nb}}_3\mathrm{Sn}$ layer cross section with a tin-rich grain boundary (center), adapted from [69]. Distances are measured in units of the RF penetration depth $\lambda \approx$ 100 nm.

Figure 13

The Cornell coating recipe [27]. The annotated steps are described in the text.

Figure 14

Excess surface tin versus time for the Cornell coating recipe modeled using the temperature-dependent evaporation rates of tin and tin chloride [76] and the thickness-dependent growth rate [31] of the ${\mathrm{Nb}}_3\mathrm{Sn}$ layer. Tin-poor growth is expected when there is no excess surface tin ($\sim 12$ h and $\sim 15$ h in this case).