Abstract
The superconducting gap structure in ironbased hightemperature superconductors (FeHTSs) is nonuniversal. In contrast to other unconventional superconductors, in the FeHTSs both dwave and extended swave 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 highpressure muon spin rotation experiments on the temperaturedependent magnetic penetration depth in the optimally doped nodeless swave FeHTS Ba_{0.65}Rb_{0.35}Fe_{2}As_{2}. Upon pressure, a strong decrease of the penetration depth in the zerotemperature limit is observed, while the superconducting transition temperature remains nearly constant. More importantly, the lowtemperature behaviour of the inversesquared magnetic penetration depth, which is a direct measure of the superfluid density, changes qualitatively from an exponential saturation at zero pressure to a linearintemperature behaviour at higher pressures, indicating that hydrostatic pressure promotes the appearance of nodes in the superconducting gap.
Introduction
After 6 years of intensive research on the Febased hightemperature superconductors (FeHTSs), no consensus on a universal gap structure has been reached. There is evidence that small differences in electronic or structural properties can lead to a strong diversity in the superconducting (SC) gap structure. On the one hand, nodeless isotropic gap functions were observed in optimally doped Ba_{1−x}K_{x}Fe_{2}As_{2}, Ba_{1−x}Rb_{x}Fe_{2}As_{2} and BaFe_{2−x}Ni_{x}As_{2} as well as in BaFe_{2−x}Co_{x}As_{2}, K_{x}Fe_{2−y}Se_{2} and FeTe_{1−x}Se_{x} (refs 1, 2, 3, 4, 5, 6, 7, 8). On the other hand, signatures of nodal SC gaps were reported in LaOFeP, LiFeP, KFe_{2}As_{2}, BaFe_{2}(As_{1−x}P_{x})_{2}, BaFe_{2−x}Ru_{x}As_{2}, FeSe as well as in overdoped Ba_{1−x}K_{x}Fe_{2}As_{2} and BaFe_{2−x}Ni_{x}As_{2} (refs 7, 9, 10, 11, 12, 13, 14, 15, 16, 17). Understanding what parameters of the systems control the different SC gap structures observed experimentally is paramount to elucidate the microscopic pairing mechanism in the FeHTSs and, more generally, to provide a deeper understanding of the phenomenon of hightemperature superconductivity. On the theoretical front, it has been proposed that both the s^{+−}wave and dwave states are close competitors for the SC ground state^{18,19,20,21,22,23,24,25}. Although the former generally wins, it has been pointed out that a dwave state may be realized on removing electron or hole pockets. On the experimental front, a subleading dwave collective mode was observed by Raman experiments inside the fully gapped SC state of optimally doped Ba_{1−x}K_{x}Fe_{2}As_{2} (refs 26, 27). In KFe_{2}As_{2}, a change of the SC pairing symmetry by hydrostatic pressure has been recently proposed, based on the Vshaped pressure dependence of T_{c} (ref. 28). However, no direct experimental evidence for a pressureinduced change of either the SC gap symmetry or the SC gap structure in the FeHTSs has been reported until now.
Measurements of the magnetic penetration depth λ, which is one of the fundamental parameters of a superconductor, since it is related to the superfluid density n_{s} via 1/λ^{2}=μ_{0}e^{2}n_{s}/m* (where m* is the effective mass), are a sensitive tool to study multiband superconductivity. Most importantly, the temperature dependence of λ is particularly sensitive to the presence of SC nodes: while in a fully gapped SC Δλ^{−2}(T)≡λ^{−2}(0)−λ^{−2}(T) vanishes exponentially at low T, in a nodal SC it vanishes as a power of T. The muon spin rotation (μSR) technique provides a powerful tool to measure λ in type II superconductors^{29}. A μSR experiment in the vortex state of a type II superconductor allows the determination of λ in the bulk of the sample, in contrast to many techniques that probe λ only near the surface.
For the compound Ba_{0.65}Rb_{0.35}Fe_{2}As_{2} investigated here, and for the closely related system Ba_{1−x}K_{x}Fe_{2}As_{2}, previous μSR measurements of λ(T) revealed a nodeless multigap SC state^{2,3}, in agreement with angleresolved photoemission spectroscopy (ARPES) measurements^{1,30,31}. In this article, we report on μSR studies of λ(0) and of the temperature dependence of Δλ^{−2} in optimally doped Ba_{0.65}Rb_{0.35}Fe_{2}As_{2} under hydrostatic pressures. This system exhibits the highest T_{c}≃37 K among the extensively studied 122 family of FeHTSs. We observe that while T_{c} stays nearly constant on application of pressure, λ(0) decreases substantially. In view of previous works in another 122 compound that reported a sharp peak of λ(0) at a quantum critical point^{32}, we interpret the observed suppression of λ(0) as evidence that pressure moves the system away from a putative quantum critical point in Ba_{0.65}Rb_{0.35}Fe_{2}As_{2}. More importantly, we find a qualitative change in the lowtemperature behaviour of Δλ^{−2}(T) as pressure is increased. While at p=0 an exponential suppression characteristic of a nodeless superconductivity is observed, for p=2.25 GPa a clear powerlaw behaviour is found. Because pressure does not affect the impurity concentration, which could promote powerlaw behaviour even for a nodeless system^{33}, our findings are suggestive of a nodeless to nodal SC transition. Our fittings to microscopic models reveal that this behaviour is more compatible with a dwave state rather than an s^{+−} state with accidental nodes, suggesting that pressure promotes a change in the pairing symmetry.
Results
Probing the vortex state as a function of pressure
Figure 1a,b exhibit the transversefield μSR time spectra for Ba_{0.65}Rb_{0.35}Fe_{2}As_{2}, measured at ambient p=0 GPa and maximum applied pressure p=2.25 GPa, respectively. The spectra above (45 K) and below (1.7 K) the SC transition temperature T_{c} are shown. Above T_{c} the oscillations show a small relaxation due to the random local fields from the nuclear magnetic moments. Below T_{c} the relaxation rate strongly increases with decreasing temperature due to the presence of a nonuniform local magnetic field distribution as a result of the formation of a fluxline lattice in the SC state. Figure 1c,d show the Fourier transforms of the μSR time spectra shown in Fig. 1a,b, respectively. At T=5 K the narrow signal around μ_{0}H_{ext}=50 mT (Fig. 1c,d) originates from the pressure cell, while the broad signal with a first moment μ_{0}H_{int}<μ_{0}H_{ext}, marked by the solid arrow in Fig. 1c, arises from the SC sample.
Below T_{c} a large diamagnetic shift of μ_{0}H_{int} experienced by the muons is observed at all applied pressures. This is evident in Fig. 2a, where we plot the temperature dependence of the diamagnetic shift ΔB_{dia}=μ_{0}[H_{int,SC}−H_{int,NS}] for Ba_{0.65}Rb_{0.35}Fe_{2}As_{2} at various pressures, where μ_{0}H_{int,SC} denotes the internal field measured in the SC state and μ_{0}H_{int,NS} the internal field measured in the normal state at 45 K. Note that μ_{0}H_{int,NS} is temperature independent. This diamagnetic shift indicates the bulk character of superconductivity and excludes the possibility of fieldinduced magnetism^{34} in Ba_{0.65}Rb_{0.35}Fe_{2}As_{2} at all applied pressures. The SC transition temperature T_{c} is determined from the intercept of the linearly extrapolated ΔB_{dia} curve to its zero line (we used the same criterium for determination of T_{c} from ΔB_{dia}(T) as from the susceptibility data χ_{m}(T), presented in Supplementary Fig. 1a). It is found to be T_{c}=36.9(7) K and 35.9(5) K for p=0 and 2.25 GPa, respectively. The ambient pressure value of T_{c} is in perfect agreement with T_{c}=36.8(5) K obtained from susceptibility and specific heat measurements (Supplementary Note 1 and Supplementary Figs 1a,b and 2). At the highest pressure of p=2.25 GPa applied, T_{c} decreases only by ≃1 K, indicating only a small pressure effect on T_{c} in Ba_{0.65}Rb_{0.35}Fe_{2}As_{2}. The temperature dependence of the muon spin depolarization rate σ_{sc} of Ba_{0.65}Rb_{0.35}Fe_{2}As_{2} in the SC state at selected pressures is shown in Fig. 2b; note that σ_{sc} is proportional to the second moment of the field distribution, which was extracted using the equations described in the Method section. At all applied pressures, below T_{c} the relaxation rate σ_{sc} starts to increase from zero with decreasing temperature due to the formation of the fluxline lattice (we note that no pressureinduced magnetism is observed in Ba_{0.65}Rb_{0.35}Fe_{2}As_{2}, as shown in Supplementary Note 2 and Supplementary Figs 3a,b and 4). It is interesting that the lowtemperature value σ_{sc}(5 K) increases substantially under pressure (Fig. 2b): σ_{sc}(5 K) increases about 30% from p=0 to 2.25 GPa. Interestingly, the form of the temperature dependence of σ_{sc}, which reflects the topology of the SC gap, changes as a function of pressure. The most striking change is in the lowtemperature behaviour of σ_{sc}(T). At ambient pressure σ_{sc}(T) shows a flat behaviour below T/T_{c}≃0.4, whereas the highpressure data exhibit a steeper (linear) temperature dependence of σ_{sc}(T) below T/T_{c}≃0.4. We show in the following how these behaviours indicate the appearance of nodes in the gap function.
Pressuredependent magnetic penetration depth
To investigate a possible change of the symmetry of the SC gap, we note that λ(T) is related to the relaxation rate σ_{sc}(T) by the equation^{35}:
where γ_{μ} is the gyromagnetic ratio of the muon and Φ_{0} is the magneticflux quantum. Thus, the flat T dependence of σ_{sc} observed at p=0 for low temperatures (Fig. 2b) is consistent with a nodeless superconductor, in which λ^{−2}(T) reaches its zerotemperature value exponentially. On the other hand, the linear T dependence of σ_{sc} observed at p=2.25 GPa (Fig. 2b) indicates that λ^{−2}(T) reaches λ^{−2}(0) linearly, which is characteristic of line nodes. This is the main result of this communication: pressure in an optimally doped FeHTS can tune a nodeless gap into a nodal gap. Although this qualitative analysis is robust, and independent of any fitting models for the gap function, it does not elucidate whether these nodes arise due to a nodal s^{+−} state or a dwave state.
To proceed with a quantitative analysis, we consider the local (London) approximation (λ≫ξ, where ξ is the coherence length) and first employ the empirical αmodel. The latter, widely used in previous investigations of the penetration depth of multiband superconductors^{3,36,37,38,39,40,41}, assumes that the gaps occurring in different bands, besides a common T_{c}, are independent of each other. Then, the superfluid density is calculated for each component separately^{3} and added together with a weighting factor. For our purposes a twoband model suffices yielding:
where λ(0) is the penetration depth at zero temperature, Δ_{0,i} is the value of the ith SC gap (i=1, 2) at T=0 K and ω_{i} is the weighting factor, which measures their relative contributions to λ^{−2} (that is, ω_{1}+ω_{2}=1).
The results of this analysis are presented in Fig. 3a–f, where the temperature dependence of λ^{−2} for Ba_{0.65}Rb_{0.35}Fe_{2}As_{2} is plotted at various pressures. We consider two different possibilities for the gap functions: either a constant gap, Δ_{0,i}=Δ_{i}, or an angledependent gap of the form Δ_{0,i}=Δ_{i}cos2ϕ, where ϕ is the polar angle around the Fermi surface. The resulting functions λ(T) are shown in the Methods section. The data at p=0 GPa are described remarkably well by two constant gaps, Δ_{1}=2.7(5) meV and Δ_{2}=8.4(3) meV. These values are in perfect agreement with our previous results^{3} and also with ARPES experiments^{30}, pointing out that most FeHTSs exhibit twogap behaviour, characterized by one large gap with 2Δ_{2}/k_{B}T_{c}=7(2) and one small gap with 2Δ_{1}/k_{B}T_{c}=2.5(1.5). In contrast to the case p=0 GPa, for all applied pressures λ^{−2}(T) is better described by one constant gap and one angledependent gap, consistent with the presence of gap nodes, as inferred from our qualitative analysis. Note that a fitting to two angledependent gaps is inconsistent with the data.
To understand the implications of the fitting to a constant and an angledependent gap for finite pressures, we analyse the two different scenarios in which nodes can emerge: a nodal s^{+−} state (with gap functions of different signs in the hole and in the electron pockets) and a dwave state. In the former, the position of the nodes are accidental, that is, not enforced by symmetry, while in the latter the nodes are enforced by symmetry to be on the Brillouin zone diagonals. Schematic representations of both scenarios are shown in Fig. 4, where a density plot of the gap functions is superimposed to the typical Fermi surface of the iron pnictides, consisting of one or more hole pockets at the centre of the Brillouin zone, and electron pockets at the border of the Brillouin zone. In this figure, we set the accidental nodes of the s^{+−} state to be on the electron pockets, as observed by ARPES in the related compound BaFe_{2}(As_{1−x}P_{x})_{2} (ref. 17). Note that in the dwave state, while nodes appear in the hole pockets, the electron pockets have nearly uniform gaps. Thus, the fact that the fitting to the αmodel gives a constant and an angledependent gap is consistent with a dwave state.
To contrast the scenarios of a nodal s^{+−} gap and a dwave gap, we consider a microscopic model (Supplementary Note 3) that goes beyond the simplifications of independent gap functions of the αmodel discussed above. In this microscopic model, the fully coupled nonlinear gap equations are solved for a hole pocket h and two electron pockets e_{1,2}, and the penetration depth is calculated at all temperatures (Supplementary Fig. 5). The free parameters are then the density of states of the pockets, the amplitude of the pairing interaction and the gap functions themselves (Supplementary Note 3). For simplicity, the anisotropies of the electron pockets are neglected, the Fermi velocities of the pockets are assumed to be nearly the same and the gaps are expanded in their leading harmonics. Thus, for the nodal s^{+−} state we have Δ_{h}=Δ_{h,0} and , whereas for the dwave state it follows that Δ_{h}=Δ_{h,0} cos2ϕ_{h} and . Note the difference in the position of the nodes in each case: while for the dwave case they are always at ϕ_{h}=±π/4, for the nodal s^{+−} the nodes exist only when r<1 at arbitrary positions . The results of the fittings for the pressures p=1.57 and 2.25 GPa imposing a nodal s^{+−} state are shown in Fig 3c,f (Supplementary Fig. 6a–c). Remarkably, we find in both cases that the best fit gives r→0. This extreme case is, within our model, indistinguishable from the fitting to the dwave state, since in both cases the nodes are at ϕ=±π/4 (albeit in different Fermi pockets). We note that from the fits one cannot completely rule out the possibility of small but nonvanishing values of r. Therefore, at least within our model, a nodal s^{+−} state is compatible with the data only if the accidental nodes are fine tuned to lie either at or very close to the diagonals of the electron pockets for a broad pressure range. Since the position of the accidental nodes is expected to be sensitive to the topology of the Fermi surface, and consequently to pressure, it seems more plausible that the gap state is dwave, since in that case the position of the gaps is enforced by symmetry to be along the diagonals of the hole pockets regardless of the value of the pressure (Supplementary Fig. 7a,b).
The pressure dependence of all the parameters extracted from the data analysis within the αmodel are plotted in Fig. 5a–c. From Fig. 5a a substantial decrease of λ(0) with pressure is evident. At the maximum applied pressure of p=2.25 GPa the reduction of λ(0) is ∼15% compared with the value at p=0 GPa. Both Δ_{1} and Δ_{2} show a small reduction on increasing the pressure from p=0 to 1.17 GPa, while above p=1.17 GPa the gaps values stay constant. On the other hand, the relative contribution ω_{2} of the small gap to the superfluid density increases by approximately factor of 2 for the maximum applied pressure of p=2.25 GPa (Fig. 5c), indicating a spectral weight shift to the smaller gap. The parameters extracted from the microscopic model are discussed in Supplementary Note 3.
Discussion
The main finding of our paper is the observation that pressure promotes a nodal SC gap in Ba_{0.65}Rb_{0.35}Fe_{2}As_{2}. This conclusion is model independent, as it relies on the qualitative change in the lowtemperature behaviour of Δλ^{−2} from exponential to linear in T on applied pressure. To our knowledge this is the first direct experimental demonstration of a plausible pressureinduced change in the SC gap structure in a FeHTS. Two possible gap structures could be realized at finite pressures: a nodal s^{+−} state and a dwave state. In the first case, the change from nodeless s^{+−} to nodal s^{+−} is a crossover rather than a phase transition^{42,43}, whereas in the latter it is an actual phase transition that could harbour exotic pairing states, such as s+id (refs 21, 23, 24) or s+d (ref. 44).
Additional results provide important clues of how pressure may induce either a nodal s^{+−} or a dwave state. In the closely related optimally doped compound Ba_{0.6}K_{0.4}Fe_{2}As_{2}, Raman spectroscopy^{27}, as well as theoretical calculations^{21,20}, reveal a subdominant dwave state close in energy to the dominant s^{+−} state. Pressure may affect this intricate balance, and tip the balance in favour of the dwave state. On the other hand, theoretical calculations have shown that the pnictogen height is an important factor in determining the structure of the s^{+−} SC order parameter^{18,45}. A systematic comparison of the quasiparticle excitations in the 1111, 122 and 111 families of FeHTSs showed that the nodal s^{+−} state is favoured when the pnictogen height decreases below a threshold value of ≃1.33 Å (ref. 46). Hydrostatic pressure may indeed shorten the pnictogen height and consequently modify the s^{+−} gap structure from nodeless to nodal. Although our fitting of the penetration depth data to both a microscopic model and an effective αmodel suggest that the dwave state is more likely to be realized than the nodal s^{+−} state, further quantitative calculations of the pressure effect are desirable to completely discard a nodal s^{+−} state.
Besides the appearance of nodes with pressure, another interesting observation is the reduction of λ(0) under pressure, despite the fact that T_{c} remains nearly unchanged. Interestingly, in the compound BaFe_{2}As_{2−x}P_{x}, a sharp enhancement of λ(0) is observed as optimal doping is approached from the overdoped side^{32}, which has been interpreted in terms of a putative quantum critical point (QCP) inside the SC dome^{47,48,49}. In Ba_{0.65}Rb_{0.35}Fe_{2}As_{2}, if such a putative QCP is also present, pressure is likely to move the system away from the putative QCP, which, according to the results of BaFe_{2}As_{2−x}P_{x}, would explain the observed suppression of the penetration depth at T=0. This scenario does not explain why T_{c} stays nearly constant under pressure, but this could be due to the intrinsic flatness of T_{c} around optimal doping in Ba_{1−x}Rb_{x}Fe_{2}As_{2}. Note that a similar behaviour for λ(0) and T_{c} with pressure has been recently observed in LaFeAsO_{1−x}F_{x} (ref. 50), but interpreted in terms of the interplay between impurity scattering and pressure. To distinguish between these two scenarios, pressuredependent studies of the quasiparticle mass in Ba_{0.65}Rb_{0.35}Fe_{2}As_{2} are desirable to probe whether a putative QCP is present or not in this compound.
In conclusion, the zerotemperature magnetic penetration depth λ(0) and the temperature dependence of λ^{−2} were studied in optimally doped Ba_{0.65}Rb_{0.35}Fe_{2}As_{2} by means of μSR experiments as a function of pressure up to . The SC transition temperature stays nearly constant under pressure, whereas a strong reduction of λ(0) is observed, possibly related to the presence of a putative quantum critical point. Our main result is the observation of a qualitative change in the lowtemperature behaviour of Δλ^{−2}(T) from exponential to linear in the investigated Febased superconductor as pressure is increased. This most likely indicates that a nodal SC gap is promoted by hydrostatic pressure. Model calculations favour a dwave over a nodal s^{+−}wave pairing as the origin for the nodal gap. The present results offer important benchmarks for the elucidation of the complex microscopic mechanism responsible for the observed nonuniversaltiy of the SC gap structure and of hightemperature superconductivity in the FeHTSs in general.
Methods
Sample preparation
Polycrystalline samples of Ba_{0.65}Rb_{0.35}Fe_{2}As_{2} were prepared in evacuated quartz ampoules by a solidstate reaction method. Fe_{2}As, BaAs and RbAs were obtained by reacting highpurity As (99.999 %), Fe (99.9%), Ba (99.9%) and Rb (99.95%) at 800, 650 and 500 °C, respectively. Using stoichiometric amounts of BaAs or RbAs and Fe_{2}As, the terminal compounds BaFe_{2}As_{2} and RbFe_{2}As_{2} were synthesized at 950 and 650 °C, respectively. Finally, samples of Ba_{1−x}Rb_{x}Fe_{2}As_{2} with x=0.35 were prepared from appropriate amounts of singlephase BaFe_{2}As_{2} and RbFe_{2}As_{2}. The components were mixed, pressed into pellets, placed into alumina crucibles and annealed for 100 h under vacuum at 650 °C with one intermittent grinding. Powder Xray diffraction analysis revealed that the synthesized samples are singlephase materials.
Method for creation and measurement of high pressures
Pressures up to 2.4 GPa were generated in a doublewall pistoncylinder type of cell made of MP35N material, especially designed to perform μSR experiments under pressure^{51,52}. As a pressuretransmitting medium Daphne oil was used. The pressure was measured by tracking the SC transition of a very small indium plate by a.c. susceptibility. The filling factor of the pressure cell was maximized. The fraction of the muons stopping in the sample was ∼40%.
μSR experiment
Zerofield and transversefield (TF) μSR experiments at ambient and under various applied pressures were performed at the μE1 beamline of the Paul Scherrer Institute, Switzerland, using the dedicated general purpose decay channel instrument (GPD) spectrometer, where an intense highenergy (p_{μ}=100 MeVc^{−1}) beam of muons is implanted in the sample through the pressure cell. A gasflow ^{4}He (base temperature ∼4 K) and a VARIOX cryostat (base temperature ∼1.3 K) were used. Highenergy muons (p_{μ}=100 MeVc^{−1}) were implanted in the sample. Forward and backward positron detectors with respect to the initial muon spin polarization were used for the measurements of the μSR asymmetry time spectrum A(t). The typical statistics for both forward and backward detectors were 6 millions. All zerofield and transversefield μSR experiments were performed by stabilizing the temperature in before recording the μSR time spectra. Note that a precise calibration of the GPD results was carried out at the πM3 beamline using the lowbackground GPS. The μSR time spectra were analysed using the free software package MUSRFIT^{36}.
In a μSR experiment nearly 100% spinpolarized muons μ^{+} are implanted into the sample one at a time. The positively charged μ^{+} thermalize at interstitial lattice sites, where they act as magnetic microprobes. In a magnetic material the muon spin precesses in the local field B_{μ} at the muon site with the Larmor frequency ν_{μ}=γ_{μ}/(2π)B_{μ} (muon gyromagnetic ratio γ_{μ}/(2π)=135.5 MHz T^{−1}). By means of μSR important length scale of superconductor can be measured, namely the magnetic penetration depth λ. When a type II superconductor is cooled below T_{c} in an applied magnetic field ranging between the lower (H_{c1}) and the upper (H_{c2}) critical field, a vortex lattice is formed, which in general is incommensurate with the crystal lattice, and the vortex cores will be separated by much larger distances than those of the unit cell. Because the implanted muons stop at given crystallographic sites, they will randomly probe the field distribution of the vortex lattice. Such measurements need to be performed in a field applied perpendicular to the initial muon spin polarization (so called transversefield configuration).
Analysis of transversefieldμSR data
Our zerofield μSR experiments (Supplementary Note 2) reveal a pressureindependent magnetic fraction of about 10% in the sample, caused by the presence of diluted Fe moments as discussed in previous μSR studies. The signal from the magnetically ordered parts vanishes within the first 0.2 μs. Thus, the fits of transversefield data were restricted to times t>0.2 μs for all temperatures.
The transversefield μSR data were analysed by using the following functional form^{36}:
A_{pc} denote the initial assymmetries of the sample and the pressure cell, respectively. MHz T^{−1} is the muon gyromagnetic ratio, ϕ is the initial phase of the muon spin ensemble and B_{int} represents the internal magnetic field at the muon site. The relaxation rates σ_{sc} and σ_{nm} characterize the damping due to the formation of the vortex lattice in the SC state and of the nuclear magnetic dipolar contribution, respectively. In the analysis σ_{nm} was assumed to be constant over the entire temperature range and was fixed to the value obtained above T_{c}, where only nuclear magnetic moments contribute to the muon relaxation rate σ. The Gaussian relaxation rate σ_{pc} reflects the depolarization due to the nuclear magnetism of the pressure cell. It can be seen from the Fourier transforms shown in Fig. 1c,d that the width of the pressure cell signal increases below T_{c}. As shown previously^{53}, this is due to the influence of the diamagnetic moment of the SC sample on the pressure cell, leading to a temperaturedependent σ_{pc} below T_{c}. To consider this influence, we assume a linear coupling between σ_{pc} and the field shift of the internal magnetic field in the SC state: σ_{pc}(T)=σ_{pc}(T>T_{c})+C(T)(μ_{0}H_{int,NS}−μ_{0}H_{int,SC}), where σ_{pc}(T>T_{c})=0.35 μs^{−1} is the temperatureindependent Gaussian relaxation rate. μ_{0}H_{int,NS} and μ_{0}H_{int,SC} are the internal magnetic fields measured in the normal and in the SC state, respectively. As indicated by the solid lines in Fig. 1a–d, the μSR data are well described by equation (1). The solid lines in Fig. 1c,d are the Fourier transforms of the fitted curves shown in Fig. 1a,b. The model used describes the data rather well.
Analysis of λ(T)
As pointed out in the manuscript, for polycrystalline samples the temperature dependence of the London magnetic penetration depth λ(T) is related to the muon spin depolarization rate σ_{sc}(T) by equation (1) (see the main text). Equation (1) is valid when the separation between the vortices is smaller than λ and the applied field small with respect to the second critical field B_{c2}. In this case according to the London model σ_{sc} is field independent^{35}. Fielddependent measurements of σ_{sc} at ambient pressure was reported previously^{3}. It was observed that first σ_{sc} strongly increases with increasing magnetic field until reaching a maximum at μ_{0}H≃0.03 T and then above 0.03 T stays nearly constant up to the highest field (0.64 T) investigated. Such a behaviour is expected within the London model and is typical for polycrystalline HTSs^{54}.
λ(T) was calculated within the local (London) approximation by the following expression^{36,37}:
where f=[1+exp(E/k_{B}T)]^{−1} is the Fermi function, ϕ is the angle along the Fermi surface and Δ_{i}(T, ϕ)=Δ_{0,i}Γ(T/T_{c})g(ϕ) (Δ_{0,i} is the maximum gap value at T=0). The temperature dependence of the gap is approximated by the expressions Γ(T/T_{c})=tanh{1.82[1.018(T_{c}/T−1)]^{0.51}} (ref. 38), while g(ϕ) describes the angular dependence of the gap and it is replaced by 1 for both an swave and an s+swave gap, and cos(2ϕ) for a dwave gap^{39}. The fitting of the T dependence of the penetration depth with αmodel was performed using the library BMW^{36}.
Additional information
How to cite this article: Guguchia, Z. et al. Direct evidence for a pressureinduced nodal superconducting gap in the Ba_{0.65}Rb_{0.35}Fe_{2}As_{2} superconductor. Nat. Commun. 6:8863 doi: 10.1038/ncomms9863 (2015).
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Acknowledgements
Experimental work was performed at the Swiss Muon Source (SμS) Paul Scherrer Institute, Villigen, Switzerland. Z.G. acknowledges the support by the Swiss National Science Foundation. R.M.F. and J.K. were supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under award number DESC0012336. A.S. acknowledges support from the SCOPES grant No. IZ73Z0_128242. G.P. is supported by the Humboldt Research Fellowship for Postdoctoral Researchers.
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Project planning: Z.G.; sample growth: Z.B.; μSR experiments: Z.G., R.K., A.A., H.L., P.K.B., E.M., A.S., G.P., H.K. and F.v.R.; magnetization experiment: Z.G. and F.v.R.; μSR data analysis: Z.G.; analysis of the penetration depth data with αmodel: Z.G.; analysis of the penetration depth data with the microscopic model: J.K. and R.M.F.; data interpretation: Z.G., R.M.F. and R.K.; draft writing: Z.G. with contributions and/or comments from all authors.
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Supplementary Figures 17, Supplementary Notes 13, and Supplementary References. (PDF 403 kb)
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Guguchia, Z., Amato, A., Kang, J. et al. Direct evidence for a pressureinduced nodal superconducting gap in the Ba_{0.65}Rb_{0.35}Fe_{2}As_{2} superconductor. Nat Commun 6, 8863 (2015). https://doi.org/10.1038/ncomms9863
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