Direct evidence for a pressure-induced nodal superconducting gap in the Ba0.65Rb0.35Fe2As2 superconductor

The superconducting gap structure in iron-based high-temperature superconductors (Fe-HTSs) is non-universal. In contrast to other unconventional superconductors, in the Fe-HTSs both d-wave and extended s-wave pairing symmetries are close in energy. Probing the proximity between these very different superconducting states and identifying experimental parameters that can tune them is of central interest. Here we report high-pressure muon spin rotation experiments on the temperature-dependent magnetic penetration depth in the optimally doped nodeless s-wave Fe-HTS Ba0.65Rb0.35Fe2As2. Upon pressure, a strong decrease of the penetration depth in the zero-temperature limit is observed, while the superconducting transition temperature remains nearly constant. More importantly, the low-temperature behaviour of the inverse-squared magnetic penetration depth, which is a direct measure of the superfluid density, changes qualitatively from an exponential saturation at zero pressure to a linear-in-temperature behaviour at higher pressures, indicating that hydrostatic pressure promotes the appearance of nodes in the superconducting gap.

The low temperature data clearly shows that the gap is nodal, but the data near T c seems to be better described by a nodeless state. (b) Fitting for the zero pressure case with the Fermi velocity ratio v h /v e being a free parameter. The fitting improves with respect to Fig. 6a, but the values of v h /v e and ρ h /ρ e seem to be too large or too small.

I. SUPPLEMENTARY NOTE 1: SAMPLE CHARACTERIZATION
The temperature dependence of the zero field-cooled (ZFC) and field-cooled (FC) diamagnetic susceptibility of Ba 0.65 Rb 0.35 Fe 2 As 2 measured in a magnetic field of µ 0 H = 1 mT is shown in Supplementary Figure 1a. From the diamagnetic response the SC transition temperature T c is determined from the intercept of the linearly extrapolated zero-field cooled (ZFC) susceptibility curve with χ m = 0 line, and it is found to be T c = 36.8(5) K. The temperature-dependent heat capacity data for this sample plotted as C p /T vs T is shown in Supplementary Figure 1b. The jump associated with the SC transitions is clearly seen.
Here the anomaly at the transition has been isolated from the phonon dominated background by subtracting a second order polynomial C p,n fitted above T c and extrapolated to  indicating that there is no long-range magnetic order. On the other hand, we observed a significant drop of the asymmetry, taking place within 0.2 µs. This is caused by the presence of diluted Fe moments as discussed in previous µSR studies [6]. In order to quantify the magnetic fraction, the ZF-µSR data were analyzed by the following function: (1) the first and the second terms describe the magnetic and nonmagnetic part of the signals, respectively. A 0 is the initial asymmetry, Ω is the magnetic volume fraction, and λ T (λ L ) is the transverse (longitudinal) depolarization rate of the µSR signal, arising from the magnetic part of the sample. The second term describing the paramagnetic part of the sample is the combination of a Lorentzian and a Gaussian Kubo-Toyabe depolarization functions [7,8].
σ and Λ are the depolarization rates due to the nuclear dipole moments and randomly oriented diluted local electronic moments, respectively. The temperature dependence of the magnetic fraction obtained for Ba 0.65 Rb 0.35 Fe 2 As 2 is plotted in Supplementary Figure 3b.
The magnetic fraction at the base temperature was found to be only 8 %. Bearing in mind that the signal from the magnetically ordered parts vanishes within the first 0.2 µs in the whole temperature region, the analysis of transverse field data was restricted to times t > 0.2 µs.
Supplementary Figure 4 shows the ZF-µSR time spectra for Ba 0.65 Rb 0.35 Fe 2 As 2 at various applied pressures. The ZF relaxation rate stays nearly unchanged between p = 0 GPa and 2.25 GPa, implying that there is no sign of pressure induced magnetism in this system. Model for s +− pairing: As a minimal model that accounts for the different superconducting states of the iron pnictides (nodeless s +− , nodal s +− , and d-wave), we consider a two-dimensional system with three isotropic Fermi pockets [9]: one hole pocket h centered around Γ = (0, 0) and two electron pockets e 1 and e 2 centered around M 1 = (π, 0) and M 2 = (0, π) (see Supplementary Figure 5). To describe the s +− state, the pairing interaction between the hole pocket h and the electron pocket e 1 is assumed to be angular dependent with the form: where φ is the polar angle measured relative to the center of the electron pocket, V 0 is the interaction energy scale, and r is the relative amplitude of the angular-independent and the angular-dependent pairing interactions. Due to the tetragonal symmetry of the system, the pairing interaction between h and e 2 is: Furthermore, to minimize the number of free parameters, we assume that the three where E e 1 (k), E e 2 (k), and E h (k) are the quasi-particle energy dispersions: To determine T c , we linearize the gap equations, yielding: To perform the fitting, we set T c to be fixed, and set the energy cutoff Λ c = 86meV (the results do not depend significantly on the choice of the cutoff). This provides a constraint on ρ e V 0 , η, and r. When T < T c , the gaps are calculated based on the BCS Eqs. (4) and (5).
The expression for the penetration depth of a single-band system is: where f is the Fermi distribution function, is the energy of the non-interacting system, and E k is the quasi-particle energy dispersion. Applying this formula to our three pocket model, we obtain In the fittings, we will focus on the normalized penetration depth λ −2 (T ) /λ −2 (0).

Model for d-wave pairing:
To describe the d-wave superconducting state within our three band model, we consider the following form of the pairing interaction: where θ is the angle around the hole pocket. The gap functions can then be written as: resulting in the BCS-like gap equations: Here, η = ρ h /ρ e , E e = 2 e + ∆ 2 e , and E h = 2 + ∆ 2 h cos 2 2θ. Repeating the same steps as for the s +− case, we obtain the penetration depth: Comparing the expressions for the d-wave case to the expressions we derived for the s +− case, Eqs. (4) and (6), we note that they can be mapped onto each other if r = 0. In this extreme case, changing η d → 4/η s , V 0,d → ηV 0,s /2, and ∆ h ↔ ∆ e leads to the same gap equations and penetration depth expression. With these replacements, both s and d pairing give the same λ −2 (T )/λ −2 (0). Therefore, we conclude that the penetration depth cannot distinguish between nodal-s +− and d-wave if the nodal-s +− is the extreme case with r = 0.
Fitting Results: We now fit the experimental data λ −2 (T ) /λ −2 (0) of optimally-doped Ba 1−x Rb x Fe 2 As 2 to find the values of ρ e V 0 , η, and r for different pressures. Note that the value of T c imposes another constraint on these three parameters, as explained above. Supplementary Figures   6a, b and c show the fitting for the s +− model for P = 0, P = 1.57 GPa, and P = 2.25 GPa, respectively. For the P = 0 case, we find equal gap amplitudes and no nodes, as seen by ARPES experiments in the related compound Ba 1−x K x Fe 2 As 2 . We see that the fitting is not as good in the region immediately below T c . We will discuss this issue in more details below.
For the pressurized samples, the fitting is overall better and indicates a nodal state (r < 1).
The value of the density of states ratio ρ h /ρ e is little affected by pressure (as expected, since no charge carriers are introduced), and is consistent with the value of a nearly compensated metal.
Surprisingly, the best fittings for both the P = 1.57 GPa and P = 2.25 GPa cases give r = 0, where the nodes on the electron pockets are fixed at θ = ±π/4. This is a very special case of the accidentally nodal s +− state, since by symmetry there is no reason for r to vanish. To make this point more transparent, in Supplementary Figure 7a we plot the non-zero pressure data and the theoretical urves for the penetration depth for various values of r -keeping all the other parameters constant. Clearly, 0 < r < 1 gives worst fittings than r = 0. What we also found is that r = 10 -i.e. a nodeless superconducting statedescribes the data better near T c , on the expense of a very bad fitting at low temperatures -where the nodal behavior is evident.
As we discussed in the previous section, a nodal-s +− state with r = 0 is indistinguishable -for fitting purposes -from a d-wave state. Since there is no symmetry reason to have r = 0 in our simple model, or even r 1 over a wide pressure range, we interpret this result as an indirect indication that a d-wave state is more likely to be the state of the pressurized samples.
Finally, we comment on the difficulty of the fittings to capture the behavior near T cparticularly for the sample at ambient pressure (see Supplementary Figure 6a). One reason could be the presence of inhomogeneities, which would require a distribution of gaps to be taken into account, instead of a single gap value. Another reason could be related to our choice of fixing the Fermi velocities to be the same for both the electron and hole pockets. To investigate this possibility, we lift this restriction and allow v h /v e to also be a fitting parameter. The result is shown in Supplementary Figure 7b. Clearly, we obtain a better fitting, but not only ρ e V 0 is relatively large, but the ratios ρ h /ρ e and v h /v e are very large or very small, which is difficult to reconcile with the Fermi surface of these materials.
Most likely, additional pockets are necessary to capture the full temperature dependence of the penetration depth. Nevertheless, our microscopic model provides results that agree with those obtained from the α-model fitting, particularly in the low-temperature regime, suggesting that a d-wave state is more likely to be realized than a nodal s +− state.