Local suppression of the superfluid density of PuCoGa${}_5$ by strong onsite disorder

We present superfluid density calculations for the unconventional superconductor PuCoGa${}_5$ by solving the real-space Bogoliubov–de Gennes equations on a square lattice within the Swiss-cheese model in the presence of strong onsite disorder. We find that, despite strong electronic inhomogeneity, one can establish a one-to-one correspondence between the local maps of the density of states, superconducting order parameter, and superfluid density. In this model, strong onsite impurity scattering punches localized holes into the fabric of $d$-wave superconductivity similar to a Swiss cheese. Already, a two-dimensional impurity concentration of $n_{\mathrm{imp}}=4\%$ gives rise to a pronounced short-range suppression of the order parameter and a suppression of the superconducting transition temperature $T_c$ by roughly 20% compared to its pure limit value $T_{c0}$, whereas the superfluid density ${\rho }_s$ is reduced drastically by about 70%. This result is consistent with available experimental data for aged (400-day-old) and fresh (25-day-old) PuCoGa${}_5$ superconducting samples. In addition, we show that the $T^2$ dependence of the low-$T$ superfluid density, a signature of dirty $d$-wave superconductivity, originates from a combined effect in the density of states of “gap filling” and “gap closing.” Finally, we demonstrate that the Uemuera plot of $T_c$ versus ${\rho }_s$ deviates sharply from the conventional Abrikosov-Gor’kov theory for radiation-induced defects in PuCoGa${}_5$, but follows the same trend of short-coherence-length high-$T_c$ cuprate superconductors.

Figure 1

(a1)–(c1) Visualization of the local behavior of superconducting properties for a single impurity at the center of the cell. The three-dimensional plots of $N_0\left(\mathbf{r}\right)$ in (a1), $\Delta \left(\mathbf{r}\right)$ in (b1), and ${\rho }_{\mathrm{s}}\left(\mathbf{r}\right)$ in (c1) show the real-space modulation of these properties in response to a single impurity at the center. The corresponding color-coded contour maps highlight the patterns of the modulations. All spectra are calculated on a $35\times 35$ lattice and then interpolated for visualization. The inset to (a1) gives the schematic view of various directions and locations of sites with respect to an impurity (red dot) at which the following plots are drawn. (a2)–(c2), Each plot in the middle row corresponds to a one-dimensional cut through the spectrum shown in the corresponding top panel (the color of each representative curve is the same as for the arrow drawn in the inset). The shading at the lattice boundaries delineates the region where finite-size effects are expected to affect the results. The black dashed line in (a2) depicts the $1/r^2$ behavior (with respect to the impurity site) of the LDOS along the nodal direction. (a3)–(c3) Similarly, the curves in the bottom row are drawn at four representative sites with respect to the location of the impurity and are compared with their average value. Site A is the nearest neighbor to the impurity along the antinodal direction. The next-nearest-neighbor site B is along the nodal direction. Sites C and D are farther away from the impurity location along the antinodal and nodal directions, respectively.

Figure 2

Visualization of the local behavior of superconducting properties for a randomly distributed impurity concentration of 2% (same notation as in Fig. 1). Brown cross symbols on the two-dimensional color maps in the top row depict the locations of impurities on the lattice. The selection of test sites used in the bottom row of panels is the same for all spectra and marked in panel (a1). Unlike for the single-impurity case in Fig. 1, the SC gap and superfluid density now show much stronger spatial fluctuations. All averaged calculations are done for 10 samplings over random configurations of impurities. The visualized results are for one particular disorder configuration.

Figure 3

The energy dependence of the DOS is related to the temperature dependence of ${\rho }_{\mathrm{s}}\left(T\right)$ (averaged over $35\times 35$ lattice) for different impurity concentrations. (a) Computed DOS $N\left(\omega \right)$ is plotted for increasing impurity concentration along the vertical axis (curves are not shifted vertically). The gray shaded area highlights the nature of the gap closing, while the colored filling in each spectrum illustrates the trend of gap filling with increasing disorder. Consequences of finite-size effects can be seen in the value of the zero-energy DOS for the pure sample (red line), which is expected to be zero in the absence of a numerical broadening term $\Gamma$. (b) Calculated results of ${\rho }_{\mathrm{s}}\left(T\right)$ (normalized to their corresponding zero-temperature value). The inset expands the low-$T$ region of ${\rho }_{\mathrm{s}}\left(T\right)$ vs $T^2$ to emphasize the quadratic temperature dependence. (c) Computed ${\rho }_{\mathrm{s}}\left(T\right)$ for 1% and 4% impurity concentrations are compared with the measured data for a fresh (25-day-old) and old (400-day-old) PuCoGa${}_5$ sample, respectively (Ref.27). The cyan diamond and red circle symbols are for field-cooled measurements in $H_0=60$ mT, while the cyan star symbols are for $H_0=300$ mT. Since no data are available at zero temperature and near $T_c$, we extrapolated each experimental curve into these regions for proper normalization. After extrapolating the data, we estimated the values of ${\rho }_s\left(0\right)$ and $T_c$, which are slightly higher than those predicted using a linear extrapolation method (Refs.25, 26, 27; see Table ).

Figure 4

Disorder-induced correlations between $N\left(0\right)$, $\Delta \left(0\right)$, $T_c$, and ${\rho }_{\mathrm{s}}\left(0\right)$. (a) Variation of different properties (see legend) as a function of impurity concentration $n_{\text{imp}}$. (b), (c) The same quantities are drawn as a function of total number of quasiparticles on the Fermi surface in (b) and as a function of order-parameter suppression.

Figure 5

Uemura plot of superfluid density in a disordered system. The two red open squares are the experimental data for PuCoGa${}_5$ (Pu-115) from Ref.27. The rest of the open symbols are taken from YBCO data under different conditions of disorder environments (Ref.12). Data for YBCO films are obtained from Ref.59, for YBCO crystals from Ref.60, for ceramic samples from Ref.61, and for He-irradiated YBCO films from Ref.62. The filled red dots are the present BdG theory. The blue solid and black dashed lines are the corresponding results from our “dirty” $d$-wave AG theory calculation for impurity scattering in the Born and unitarity limits.