Abstract
Resolving the microscopic pairing mechanism and its experimental identification in unconventional superconductors is among the most vexing problems of contemporary condensed matter physics. We show that Raman spectroscopy provides an avenue towards this aim by probing the structure of the pairing interaction at play in an unconventional superconductor. As we study the spectra of the prototypical Febased superconductor Ba_{1−x}K_{x}Fe_{2}As_{2} for 0.22 ≤ x ≤ 0.70 in all symmetry channels, Raman spectroscopy allows us to distill the leading swave state. In addition, the spectra collected in the B_{1g} symmetry channel reveal the existence of two collective modes which are indicative of the presence of two competing, yet subdominant, pairing tendencies of \(d_{x^2  y^2}\) symmetry type. A comprehensive functional Renormalization Group and randomphase approximation study on this compound confirms the presence of the two subleading channels, and consistently matches the experimental doping dependence of the related modes. The consistency between the experimental observations and the theoretical modeling suggests that spin fluctuations play a significant role in superconducting pairing.
Introduction
In superconductors such as the cuprates, ferropnictides, ruthenates, or heavyfermion systems, the pairing mechanism is believed to be unconventional and related to direct electronic interactions rather than conventional electron–phonon mediated couplings. Yet, the precise microscopic mechanism, the “glue” that binds electrons into Cooper pairs, remains elusive. Measurements of the superconducting ground state alone are insufficient to unambiguously determine whether a superconductor has a conventional or unconventional pairing mechanism. Raman spectroscopy provides the avenue for gathering the missing information in both dominant and subdominant pairing channels.
In comparison to other techniques, Raman spectroscopy (which involves inelastic scattering of light) is rather unique as it provides access to both the energy gaps of a superconductor and to bound states inside the gaps^{1} that serve as signposts marking the strength of a given pairing interaction.
These bound states were predicted a long time ago by Bardasis and Schrieffer (BS)^{2} and are collective excitations that correspond to the phase oscillations of the ground state order parameter triggered by the subdominant (dwave) interactions. The BS modes or particleparticle excitons couple to the Raman probe, but there is no consensus yet about their observation in conventional superconductors.^{3,4} Febased superconductors (FeSCs), however, presented a more favorable scenario to search for this physics as many of them are believed to exhibit s_{±} pairing (with an order parameter that may change sign between Fermi surface pockets^{5,6,7,8,9}) and also a subleading dwave pairing interaction that can be strongly competitive. Theoretical calculations based on spin fluctuations have even argued that dwave could become the ground state for sufficiently strong holedoping.^{10,11}
For these reasons, Scalapino and Devereaux^{12} performed a “barebones” calculation for a typical FeSC electronic structure with s_{±} symmetry of the ground state and anisotropic gaps, showing that the mode frequency should depend on 1/λ_{d} − 1/λ_{s}, where λ_{d} and λ_{s} are the respective coupling strengths of the electrons to the glue that binds the Cooper pair in the dwave and the swave channel. Recent measurements on Ba_{1−x}K_{x}Fe_{2}As_{2},^{4,13,14} NaFe_{1−x}Co_{x}As,^{15} Ba(Fe_{1−x}Co_{x})_{2}As_{2}^{16,17} found peaks in the B_{1g} spectrum which were consistent with a collective mode, but its direct association with a BS mode was unclear.
In this work, we confirm the presence of two subdominant pairing interactions, as predicted theoretically, by providing an identification of multiple BS modes in the B_{1g} spectrum of the prototypical ferropnictide Ba_{1−x}K_{x}Fe_{2}As_{2} (BKFA). Each subdominant pairing interaction results in a BS mode.^{18} This perspective underlies our identification of the two new peaks in the Raman spectrum with B_{1g} BS modes. The analysis of our experimental peak energies also supports this scenario and even allows us to empirically extract the relative coupling strengths, λ_{d(1)}/λ_{s} and λ_{d(2)}/λ_{s}, of the two distinct B_{1g} \(\left( {d_{x^2  y^2}} \right)\) pairing channels competing with the s_{±} ground state. We could reproduce the presence of all three pairing channels by performing a functional Renormalization Group (fRG) as well as a Random Phase Approximation (RPA) study. Since the fRG calculation includes the leading fluctuations (magnetic, superconducting, charge density wave etc.) whereas the RPA is distinctly based on magnetically driven (i.e., spinfluctuationinduced) pairing, the agreement of both approaches with each other and the experiment strongly points to a spinfluctuation scenario in BKFA. Since a direct observation of spin fluctuations below T_{c} is not achievable by Raman scattering (the relevant scattering states are gapped out) we study the BS modes which remain as the fingerprints of the microscopic pairing interactions.
Results
Experiments
To this end we measured eight samples of BKFA in the wide doping range 0.22 ≤ x ≤ 0.70 as indicated in Fig. 1a and described in detail in Sec. II of the Supplementary Information. BKFA forms high quality single crystals^{19,20,21} and fairly clean and isotropic gaps.^{22,23} In the samples with x = 0.22 and x = 0.25 superconductivity and the spin density wave (SDW) state coexist. The samples with x = 0.62 and x = 0.70 are above the doping level of x = 0.6, where E_{F} reaches the bottom of the inner electron band and the topology of the Fermi surface changes qualitatively.^{24} To present the case for the physics of subdominant pairing interactions, we wish to stay away from special effects arising from magnetism or disappearance of pockets and focus on the samples with x = 0.35, 0.40, 0.43, 0.48. In this range, the Raman spectra in the B_{1g} symmetry channel (1 Fe unit cell) change continuously as shown in Fig. 2a–d. Spectra of the other symmetries and outside the range 0.35 ≤ x ≤ 0.48 are compiled in Sec. IV of the Supplementary Information.
The spectra above the superconducting transition temperature T_{c} are dominated by the electronhole continua. Below T_{c} additional (symmetrydependent) structures appear in the energy range up to ~300 cm^{−1}, and the spectral weight is redistributed from below twice the superconducting gap 2Δ to energies above. New features arise from pair breaking, excitations across the gap, and excitonlike bound states.^{1,4,18} With increasing doping and a concomitant reduction of T_{c}, the peaks move to lower energies.
To illustrate why BKFA is a model superconductor for investigating BS modes we highlight the changes in the electronic spectra below T_{c}. For this purpose we subtract the normal state response from the superconducting spectra. This procedure elimantes temperatureindependent components of the spectra like phonons in A_{1g} and B_{2g} symmetry (see Sec. IV of the Supplementary Information). By plotting the difference \({\mathrm{\Delta }}R\chi ^{\prime\prime} ({\tilde{\mathrm \Omega }})\) ≡ \(R\chi ^{\prime\prime} ({\tilde{\mathrm \Omega }},T \leq 10{\kern 1pt} {\mathrm{K}})\) − \(R\chi ^{\prime\prime} ({\tilde{\mathrm \Omega }},T \geq T_c)\) in Fig. 2e with \({\tilde{\mathrm \Omega }}\) = ħΩ/k_{B}T_{c} we extract superconductivityinduced features of pure B_{1g} symmetry. Due to the full gap, the difference spectra become negative at low energies and three pronounced peaks are observed. The differences between normal and superconducting spectra disappear (ΔRχ″ → 0) close to \({\tilde{\mathrm \Omega }} = 8\). The highest peak (purple arrows in Fig. 2e) at ~6.2, which we identify with the maximal gap, depends weakly on doping. The range of 2Δ/k_{B}T_{c} ≃ 6.2 is in qualitative agreement with the results from other methods.^{22,23,25} There are two additional narrow lines in the ranges 1.5–3 (green arrows) and 4–5.5 (orange arrows) displaying a strong monotonic downshift with increasing K content. At optimal doping (x = 0.40), evidence was furnished that the narrow line at \({\tilde{\mathrm \Omega }} = 5.3\) (140 cm^{−1} in Fig. 2b) results from a bound state of two electrons of a broken Cooper pair.^{4}
Along with the line at \({\tilde{\mathrm \Omega }} = 5.3\), we find another narrow line in B_{1g} symmetry at \({\tilde{\mathrm \Omega }} = 2.8\) (75 cm^{−1} in Fig. 2b), which is difficult to properly assign on the basis of just one doping level. In ref.^{4} it was suggested that this peak originates in pairbreaking. However, upon studying several doping levels and all symmetries (Secs. IV and V of the Supplementary Information) we find the following systematics in favor of two BS modes: (i) The two ingap modes appear only in B_{1g} symmetry. (ii) As opposed to the pairbreaking maxima at ~6k_{B}T_{c} there are no other gap energies observed the two sharp modes could correspond to. (iii) The spectral weights of both modes depend on their binding energies as predicted by theory (see Sec. VI of the Supplementary Information). (iv) Upon doping K for Ba the ingap modes increasingly split off of the pairbreaking maximum. The nearly identical doping dependences of the two modes and the absence of pairbreaking features in other symmetries suggest that both modes are linked to the maximal gap. The unique appearance of narrow BS modes in B_{1g} symmetry for 0.35 ≤ x ≤ 0.48 indicates that there are subdominant interactions with dwave symmetry. We label the corresponding subleading B_{1g} channels as d(1) and d(2) for the lower and the higherenergy line, respectively.
In Fig. 3a we compile experimental peak energies derived from Fig. 2. The difference between 2Δ (purple) and the BS modes in the range 1.5–5.5k_{B}T_{c} (green and orange) corresponds to the binding energies E_{b(i)} = 2Δ − Ω_{BS(i)} with i = 1, 2 of the bound states. The ratios of the relative coupling strengths λ_{d(i)}/λ_{s} are estimated from E_{b(i)}/2Δ using the results of refs.^{3,4,12} and λ_{s} = 0.7 from refs.^{26,27}. Note that we used a dopingindependent value of 0.7 for this estimate as the ratios λ_{d(i)}/λ_{s} are weakly sensitive to small changes of λ_{s} (see Sec. VI of the Supplementary Information). This analysis enables us to check the validity of the RPA and fRG approaches in a system with intermediate coupling strength.
Theory
According to ref.^{18}, the presence of two BS modes in the same symmetry channel must imply the presence of two pairing interactions with different form factors competing with the ground state. Thus in addition to the ratios λ_{d(i)}/λ_{s} derived from experiment, we show in Fig. 3b, c the results of two microscopic studies using fRG and RPA schemes that precisely identify these pairing channels and also provide an estimate for λ_{d(i)}/λ_{s}.
In order to determine the hierarchy of pairing interactions from the effective pairing vertex V from either fRG or RPA, we decompose this pairing channel into eigenmodes, which is tantamount to solving an eigenvalue problem of the form
where k comprises momentum, band, and spin degrees of freedom, and α is the index consecutively numbering the different eigenvalues. We assume α to be ordered according to the magnitude of eigenvalues λ_{α}. g_{α}(k) is the pairing eigenvector along the Fermi surfaces specifying the symmetry of the pairing. More details can be found in Sec. I of the Supplementary Information.
From both fRG and RPA, we find λ_{s}, g_{s}(k) (α = 1) to be the dominant superconducting pairing of A_{1g} (s_{±}) type and λ_{d(1,2)}, g_{d(1,2)}(k) (α = 2, 3) the subleading B_{1g} type couplings. Schematic eigenvectors g_{α}(k) for α = 1, 2, 3 are shown as insets in Fig. 3a. These results apply to both \(V \equiv V_{{\mathrm{fRG}}}^{\mathrm{\Lambda }}\) and V ≡ V_{RPA} when used in Eq. (1), where Λ is the lowenergy cutoff in the fRG flow that serves as an upper bound for the transition temperature^{28,29} (see also Sec. I of the Supplementary Information). The leading eigenvalue λ_{s} ≡ λ_{1} in Eq. (1), which is a function of Λ in the case of fRG, then determines the leading Fermi surface instability. The ratios of the eigenvalues λ_{d(1,2)}/λ_{s} ≡ λ_{2,3}/λ_{1} determine the peak positions of the BS modes and are shown along with the experiments in Fig. 3b, c. Note that λ_{2} ≡ λ_{d1} fits the extended dwave harmonic form predicted in ref.^{10}.
Discussion
Arguably the most critical and presumably controversial part of this research is the identification of the ingap modes observed in B_{1g} symmetry. There are essentially four proposals for the explanation of narrow modes close to or below the gap edge 2Δ, where we assume that the gap on a given band is nearly isotropic in BKFA in accordance with experiment:^{25} (i) Josephsonlike numberphase oscillations of Cooper pairs between the electronic bands are expected for a multiband system (Leggett mode).^{30} In the ferropnictides they appear in A_{1g} symmetry close to 2Δ for the dominating interband pairing and are strongly damped.^{31} Experimentally they cannot be distinguished from the pairbreaking peak since the relative intensity of the two effects is not obvious. (ii) For an s_{±} gap an excitonlike narrow mode is predicted to appear in A_{1g} symmetry below the pairbreaking peak.^{32} Since the materials are very clean with the elastic scattering rate ħ/τ much smaller than Δ it should not be overdamped and be as clearly visible as the B_{1g} collective modes. We did not find indications thereof even upon using various laser lines (see Sec. III of the Supplementary Information). (iii) In the presence of nematic fluctuations the intensity close to the gap edge is predicted to be enhanced in the related B_{1g} channel at a putative quantum critical point.^{17} In Ba(Fe_{1−x}Co_{x})_{2}As_{2} the intensity of the B_{1g} response is indeed enhanced close to optimal doping. However, the A_{1g} intensity follows the B_{1g} intensity^{33} in contrast to the expectation. In NaFe_{1−x}Co_{x}As a very strong mode close to the gap edge was observed below T_{c}. The mode appears only along with the response of nematic fluctuations above T_{c}.^{15} Yet, the variation with doping of both intensity and energy of this mode is distinctly different from that in BKFA. In addition, the response from fluctuations in BKFA is already very weak for x = 0.22^{33} and can safely be excluded to exist for 0.35 ≤ x ≤ 0.7. Therefore, the modes in NaFe_{1−x}Co_{x}As have an origin different from that in BKFA. (iv) Phase oscillations of the order parameter first described by Bardasis and Schrieffer^{2} entail δlike ingap modes in the case of a clean gap appearing in symmetry channels orthogonal to that of the ground state. For the Fermi surface structure of the ferropnictides they are expected in B_{1g} symmetry as observed experimentally here. We provide additional arguments in favor of this interpretation now thus extending the detailed quantitative discussion of ref.^{4} to all doping levels relevant here.
Bound states are generally expected in the presence of competing interactions.^{2,12} They complete the excitation spectrum of a superconductor and are similar to excitons in a semiconductor. The identification of BS modes and their differentiation from other collective excitations is possible through various characteristic properties. These include the BCSlike temperature dependence of a resolutionlimited line in materials having a clean gap. In contrast, the pairbreaking maximum is broad and does not normally follow the BCS prediction^{4} since the peak energy depends on the gap, the concentration of impurities,^{34} and on interactions.^{35} In addition, the BS mode drains spectral weight from the pairbreaking maximum in agreement with theoretical predictions^{3,12} (see Fig. S10a1–d3 of the Supplementary Information). The transfer of spectral weight and the fitting of the two BS modes is only qualitatively captured by the phenomenology proposed earlier (see Fig. S10e of the Supplementary Information) and may eventually be improved by future 3D calculations. Finally, the spectral weight of BS modes does not increase monotonically with increasing coupling strength of the bound state but, rather, has a maximum for intermediate coupling (see Fig. S9 of the Supplementary Information). Obviously all criteria could be observed experimentally and we feel on safe ground for comparing the doping dependences of the observed modes with model calculations based on fRG and RPA schemes.
The comparison of the two independent theoretical approaches allows us to pin down the origin of the leading pairing channel since the fRG includes all interactions^{28,29}, whereas the RPA focuses on the spin sector as spelled out in detail in Sec. I of the Supplementary Information. Another difference becomes apparent in the procedure used to determine the effective interaction potential. The fRG analysis is designed to start its unbiased Renormalization Group flow already at energies above the bandwidth while the effective model scale entering the RPA resummation has to be chosen at comparably lower energies (see Sec. I of the Supplementary Information). As it turns out, however, in spite of these differing initializations, transcending further down to energies at which superconductivity occurs yields similar findings for both methods.
From the plethora of theories intended to describe the ironbased superconductors, the comparison with the experiment now enables us, as a first step, to verify the validity of fRG and RPA for the intermediately coupled electronic system of BKFA. We find in accordance with our experiments that both approaches predict an swave ground state and the two strongest subleading channels to be of dwave symmetry. Furthermore, the theoretical predictions for the relative coupling parameters as shown in Fig. 3 are in good agreement with the experiment. The fRG results are in quantitative agreement, the RPA values systematically underestimate the relative coupling strength but are still close to the experiment. Hence we conclude that fRG and RPA are suitable to describe the experiment around optimal doping, 0.35 ≤ x ≤ 0.48, where the two collective BS modes can be identified. Besides the agreement with the experiment the fRG interaction eigenvectors g_{α}(k) match very well with those obtained from the spinfluctuationbased RPA analysis in all three channels (α = 1, 2, 3). These agreements indicate that spin fluctuations are an important if not the leading interaction in the system under consideration.
The results presented here put narrow constraints on the description of the Raman data and render differing interpretations^{15,17} rather unlikely to be applicable to BKFA. Hence, the observation of two collective modes inside the gap of a superconductor establishes a novelty in terms of experimental analysis which promises to have an impact on the general understanding of unconventional superconductivity. Along with the magnitude of the gap, the modes reveal the hierarchy of pairing states in a prototypical material, in full agreement with microscopic predictions. As a result, our experiment demonstrates the unique possibilities of using light scattering as a probe for observing unconventional pairing fingerprints.
Methods
In this joint experimental and theoretical study we compare results of electronic Raman spectroscopy with predictions of two independent simulations, a fRG analysis and spinfluctuation theory in the RPA.
Light scattering
The experiments were performed with calibrated light scattering equipment.^{1} For excitation a solid state laser (Coherent, Genesis MX SLM) was used emitting at 575 nm. A few experiments at optimal doping (x = 0.40) were performed with additional laser lines at 532 (Coherent, Sapphire 532 SF), 514 and 458 nm (Coherent, Innova 304C) in order to scrutinize the resonance behavior as described in Sec. III of the Supplementary Information. The samples were mounted on the cold finger of a Heflow cryostat in a cryogenically pumped vacuum. The laserinduced heating was determined experimentally to be close to 1 K per mW absorbed laser power (see ref.^{36}). Spectra were measured in the four polarization configurations xy, x′y′, RR, and RL where x and y are along the FeFe bonds, \(x^\prime = 1{\mathrm{/}}\sqrt 2 \left( {x + y} \right)\), \(y^\prime = 1{\mathrm{/}}\sqrt 2 \left( {y  x} \right)\), and \(R{\mathrm{/}}L = 1{\mathrm{/}}\sqrt 2 \left( {x \pm iy} \right)\). All symmetry components (A_{1g}, A_{2g}, B_{1g}, and B_{2g} for tetragonal Ba_{1−x}K_{x}Fe_{2}As_{2}) can be extracted using linear combinations of the experimental spectra. For the symmetry assignment we use the 1 Fe per unit cell (cf. Fig. 1b for the corresponding BZ).^{16,37} The spectra we show within this work represent the response Rχ″(Ω, T) which is obtained by dividing the cross section by the Bose–Einstein factor {1 + n(T, Ω)} = [1 − exp(−ħΩ/k_{B}T)]^{−1} in which R is an experimental constant. In some cases we isolate superconductivityinduced contributions by subtracting the response measured at T ≳ T_{c} from the spectra taken at \(T \ll T_c\) and label the difference spectra ΔRχ″(Ω, T).
Theory
For analyzing the Cooper pairing in the ferropnictides we studied two microscopic models which allow us to disentangle the various contributions to the interaction potential \(V_{{\bf{k}},{\bf{k}}^\prime }\). This disentanglement becomes possible since the scheme of the fRG analysis^{28,29} includes all possible interactions a priori in an unbiased fashion whereas the RPA scheme focusses on spin fluctuations. We are aware that both models are valid only in the weak coupling limit but we believe that the essential physics is captured correctly. Either approach leads to an eigenvalue equation (see Eq. (1)) which yields a hierarchy of eigenvalues and the related eigenvectors. (For technical details see Sec. I of the Supplementary Information.) Upon comparing the results the relative influence of the various pairing tendencies can be estimated.
Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
We acknowledge useful discussions with L. Benfatto, A. Eberlein, D. Einzel, S. A. Kivelson, C. Meingast, and I. Tüttő. W.H. gratefully acknowledges the hospitality of the Institute for Theoretical Physics at the University of California Santa Barbara. Financial support for the work came from the Deutsche Forschungsgemeinschaft (DFG) via the Priority Program SPP 1458 (T.B., A.B., R.H., C.P. and W.H., project nos. HA 2071/72 and HA 1537/242), the Collaborative Research Centers SFB 1170 (W.H., C.P., and R.T.), and TRR 80 (F.K. and R.H.), the Bavaria California Technology Center BaCaTeC (T.B. and R.H., project no. A5 [20122]), the European Research Council (ERC) through ERCStGThomaleTOPOLECTRICS (R.T.), and from the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Contract Nos. DEAC0276SF00515 (B.M. and T.P.D.) and DEFG0205ER46236 (P.J.H. and S.M.). The RPA calculations were conducted at the Center for Nanophase Materials Sciences, which is a DOE Office of Science User Facility. The work in China (H.H.W.) was supported by the National Key Research and Development Program of China (2016YFA0300401), and the National Natural Science Foundation of China (NSFC) via projects A0402/11534005 and A0402/11374144.
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Contributions
T.B. and R.H. conceived the experiments. R.T., T.A.M., W.H., T.P.D., D.J.S., S.M. and P.J.H. developed the theoretical concept. P.A., T.W. and H.H.W. prepared the samples. T.B., F.K., A.B., M.R., D.J. and R.H.A. performed the experiments. T.B. developed the phenomeology and fitted the data. C.P., T.A.M., B.M. and S.M. performed the numerical work. T.B., R.T., W.H., B.M., T.P.D., D.J.S., S.M., P.J.H., and R.H. analyzed the results and wrote the manuscript.
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Correspondence to Rudi Hackl.
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Further reading

Scaling of the Fano Effect of the InPlane FeAs Phonon and the Superconducting Critical Temperature in Ba1−xKxFe2As2
Physical Review Letters (2019)