Abstract
Cooper pairing in the ironbased highT_{c} superconductors^{1,2,3} is often conjectured to involve bosonic fluctuations. Among the candidates are antiferromagnetic spin fluctuations^{1,4,5} and dorbital fluctuations amplified by phonons^{6,7}. Any such electron–boson interaction should alter the electron’s ‘selfenergy’, and then become detectable through consequent modifications in the energy dependence of the electron’s momentum and lifetime^{8,9,10}. Here we introduce a novel theoretical/experimental approach aimed at uniquely identifying the relevant fluctuations of ironbased superconductors by measuring effects of their selfenergy. We use innovative quasiparticle interference (QPI) imaging^{11} techniques in LiFeAs to reveal strongly momentumspace anisotropic selfenergy signatures that are focused along the Fe–Fe (interband scattering) direction, where the spin fluctuations of LiFeAs are concentrated. These effects coincide in energy with perturbations to the density of states N(ω) usually associated with the Cooper pairing interaction. We show that all the measured phenomena comprise the predicted QPI ‘fingerprint’ of a selfenergy due to antiferromagnetic spin fluctuations, thereby distinguishing them as the predominant electron–boson interaction.
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The microscopic mechanism for Cooper pairing in ironbased hightemperature superconductors has not been identified definitively^{1,2,3}. Among the complicating features in these superconductors is the multiband electronic structure (Fig. 1a). However, it is believed widely that the proximity to spin order^{1,2,3,4,5} and/or orbital order^{6,7} plays a key role in the Cooper pairing. In particular, two leading proposals for fluctuationexchangepairing mechanisms focus on two distinct bosonic modes associated with specific brokensymmetry states: antiferromagnetic spin fluctuations carrying momentum Q = (π, π)/a_{0}, and dorbital fluctuations amplified by E_{g}phonon lattice vibrations of the Fe ions. No conclusive evidence that either fluctuation couples strongly to electrons and is thus relevant to Febased superconductivity has been achieved within the plethora of proposals about theexisting data^{12,13,14,15,16,17,18}.
Each type of electron–boson interaction should produce a characteristic electronic ‘selfenergy’ representing its effect on every noninteracting electronic state k〉 with momentum ℏ k and energy ℏω. Thus, the interacting Green’s function is given by
where represents noninteracting electrons and the detailed structure of encapsulates the Cooper pairing process. Here, a hat denotes a matrix in particle–hole space (Nambu space) for Bogoliubov quasiparticles in the superconducting state. The real part then describes changes in the electron’s dispersion k(ω) and the imaginary part describes changes in its inverse lifetime τ^{−1}(k, ω). The simplest diagrammatic representation of this electron–boson interaction is shown in Fig. 1b. One way to detect the experimental signature of such a selfenergy is to use angleresolved photoemission spectroscopy (ARPES) to measure the spectral function A(k, ω) ∝ ImG(k, ω) of the states with ω < 0. However, it has recently been realized that quasiparticle interference imaging, which can access momentumresolved information of both filled and empty states with excellent energy resolution (δω < 0.35 meV at T = 1.2 K), might prove especially advantageous for detecting selfenergy effects^{19}. Our QPI data are obtained by first visualizing scattering interference patterns in realspace (rspace) images of the tip–sample differential tunnelling conductance dI/dV (r, ω = eV) ≡ g(r, ω) using spectroscopicimaging scanning tunnelling microscopy, and then Fourier transforming g(r, ω) to obtain the power spectral density g(q, ω) (ref. 11). The g(q, ω) can then be used to reveal the electron dispersion k(ω) because elastic scattering of electrons from −k(ω) to +k(ω) results in high intensity at q(ω) = 2k(ω) in g(q, ω). Sudden changes in the energy evolution k(ω) due to Σ(k, ω) can then be determined, in principle^{19}, using such data.
In a conventional singleband swave superconductor with isotropic energy gap magnitude Δ, it has been well established that coupling to an optical phonon with frequency Ω can lead to a renormalization of the electronic spectra at energy Δ + Ω(ℏ = 1) due to a singularity in the momentumindependent selfenergy Σ(k, ω) = Σ(ω) at ω = Δ + Ω (ref. 20). This classic case is illustrated in Fig. 1c, d through a model spectral function A(k, ω) ∝ ImG(k, ω) and the associated density of states N(ω) = ∫ dk A(k, ω). In Fig. 1c, the ‘free’ dispersion of a holelike band is represented by the red dashed line, while the renormalized dispersion k(ω) due to Σ(ω) is highlighted by the locus of maxima in A(k, ω). These effects can be understood from the conservation of energy and momentum during scattering processes (Fig. 1b), where the flat dispersion of an optical phonon presents constraints only on energy, without any momentum dependence.
In developing our new approach to ‘fingerprinting’ different electron–boson interactions using QPI, we use the realization that the kinematic constraints for a multiband electronic system coupled to resonant AFSF with a sharp momentum structure should result in a strongly momentumdependent (anisotropic) selfenergy. This is because, given a fermionic dispersion (k, ω_{k}^{n}) for different bands n and a spectrum of spin fluctuations whose intensity is strongly concentrated at (Q, Ω), the renormalization due to the selfenergy at a point (k, ω_{k}^{n}) will be most intense when that point can be connected to another point (k − Q, ω_{k−Q}^{m}) on a different band m, such that
This is the constraint from conservation of both energy and momentum in the electron–AFSF interaction and its consequence is shown schematically in Fig. 1e. Here the blue (yellow) surfaces represent the hole (electron) bands. The transfer of momentum Q = (π, π)/a_{0} and energy Ω necessary for the resonant antiferromagnetic fluctuation to couple these bands can be analysed by shifting the electronpocketdispersion surface (horizontally) by Q and (vertically) by Ω in the k–ω space, to obtain the transparent red surface. The black curve, showing the intersection of this red surface with the central γband dispersion (blue), is where the kinematic constraint of equation (1) can be satisfied and thus where the strongest selfenergy effect due to coupling to AFSF is predicted. The resulting strongly anisotropic renormalization due to electron–AFSF coupling is in strong contrast to what is expected as a consequence of the electron–phonon coupling case discussed in the previous paragraph.
Here we study the representative ironbased superconductor LiFeAs as a concrete example for which it should be possible to make a clear theoretical distinction between the selfenergy effects driven by different types of bosonic fluctuations. We assume that BCS theory adequately describes the superconductor deep in the superconducting phase. Hence, the noninteracting Green’s function is given by
where and are the identity and the Pauli matrices in Nambu space, respectively. The superconducting gap structure Δ_{k} and the band structure H_{k}^{0} are taken from experiments^{11,12,17} and abinitio calculations^{21} (Supplementary Section I). We then study the lowest order selfenergy due to the coupling between Bogoliubov quasiparticles and two bosonic modes: a resonant AFSF (refs 22, 23) and an optical phonon of the type driving orbital fluctuations due to inplane lattice vibrations of the Fe ions with E_{g} symmetry (FeE_{g} phonon). It is the coupling of this FeE_{g} phonon to electrons that is proposed to enhance the dorbital fluctuations which mediate Cooper pairing in the orbital fluctuation mechanism^{6,7}. We take a perturbative approach of computing the selfenergy to the lowest order^{9} (Supplementary Section II):
where the repeated indices are summed over. Given independent quantitative knowledge of the gap structure, such a perturbative treatment can accurately capture the salient features of renormalization due to electron–boson coupling (Supplementary Section III). In equation (2), the bosonic Green’s function D(q, ν) is sharply peaked around Q = (π, π)/a_{0} with the characteristic energy of Ω ≈ 6 meV to model the resonant AFSF of LiFeAs (refs 22, 23), whereas it is nearly momentumindependent for the optical E_{g} phonon^{20}. We focus on the selfenergy effects on the γ band (Fig. 1a, e) in the rest of this paper as its nearly uniform orbital character (d_{xy}) greatly simplifies the theoretical study (Supplementary Section III) while at the same time being readily accessible to QPI studies^{11}. Given the geometry of the Fermi surfaces, the kinematic constraint for coupling to resonant AFSF with momentum Q and energy Ω (red arrows in Fig. 1e) connects a given k, ω_{k}^{γ} on the γ band (blue surface in Fig. 1e) to a point with momentum k − Q on one of the two electronlike bands (yellow surfaces in Fig. 1e). Thus, the distinct anisotropic dispersions of each band mean that resonant AFSF should result in selfenergy effects with a strong directional dependence (black curve on the γ band in Fig. 1e). Similarly, for the FeE_{g} phonons with a weak momentum dependence^{7}, the selfenergy effect for the γ band (which consists almost entirely of d_{xy} orbitals^{24}) is predicted to be angleindependent (Supplementary Section IV).
In Fig. 2a–d we present the predictions from equation (2) for g(q, ω) in LiFeAs, in the presence of selfenergy effects due to coupling to AFSF (Suplementary Sections IV and V). Just below the maximum gap value on the γ band of 3 meV (Fig. 2a), the highintensity region around q ≈ 2k_{F}^{γ} shows an anisotropy dictated by the gap anisotropy^{11,17,25}, with the QPI intensity suppressed along the gap maximum (Fe–As) direction. At energies exceeding the maximum gap values, the predicted g(q, ω) at first becomes isotropic (Fig. 2b) as one might expect from the fact that the Bogoliubov energy is dominated by the kinetic energy over the gap at high energies. However, at energies ω ≥ 12 meV the predicted selfenergy effects for the AFSF selfenergy (Fig. 2c, d) are seen and, in fact, strongly suppress the g(q, ω) intensity in the Fe–Fe direction relative to the Fe–As direction. The complete predicted evolution of g(q, ω), from being dominated by the anisotropic gap structure^{11} to the new effects of the AFSFdriven Σ(k, ω) introduced here, is shown in the left panels of the Supplementary Movie 1.
The experimental search for such signatures of Σ(k, ω) in QPI data consists of imaging g(r, ω) at T = 1.2 K with 0.35 meV energy resolution on LiFeAs samples exhibiting T_{c} ≈ 15 K and with the superconducting energy gap maximum Δ_{max} = 6.5 ± 0.1 meV. Clean and flat Litermination surfaces (Li–Li unit cell a_{0} = 0.38 nm) allowed our atomic resolution/register g(r, ω) measurements to be carried out over the energy range ω < 30 meV (Supplementary Section VI). We then derive the g(q, ω) in Fig. 2e–h from the measured g(r, ω) at each energy, as shown in Fig. 2i–l. In Fig. 2e we see the expected QPI signature of the anisotropic energy gaps on multiple bands (compare Fig. 2a). Figure 2f shows the characteristic signature of the complete Fermi surface of the γ band of LiFeAs at ω just outside the superconducting gap edge on that band (compare Fig. 2b). If none of the electron–boson selfenergy phenomena intervened one would expect this closed contour (Fig. 2f) to evolve continuously to smaller and smaller qradius with increasing ω until the top of this holelike band is reached. Instead, Fig. 2g shows the beginning of a very different evolution. Above ω ∼ 12 meV, the qspace features become strongly anisotropic in a fashion highly unexpected for unrenormalized states. Indeed, the strongly suppressed g(q, ω) intensity in the Fe–Fe direction relative to the Fe–As direction is very similar to the predictions for Σ(k, ω) due to AFSF (Fe–Fe directionFig. 2d).
We compare these results to the predicted g(q, ω) signatures of a selfenergy Σ(k, ω) due to phonons whose strong coupling to electrons is a central premise for the orbital fluctuation scenario (Supplementary Section III). Clearly, comparison of predictions due to the two different boson couplings presented in Fig. 3 through the ω and q dependence of g(q, ω) for the FeE_{g} phonon (Fig. 3a–c) and AFSF (Fig. 3d–f) can provide a distinguishing ‘fingerprint’ of AFSFdriven effects. The AFSF cause maximum renormalization (peaks of the blue curve) in relatively narrow ‘beams’ in the Fe–Fe directions, precisely where the resonant spin fluctuations are concentrated owing to interband scattering (see Fig. 3g). By contrast the electron–E_{g}phonon interaction is predicted to yield isotropic selfenergy signatures (red curve) in QPI data.
In Fig. 4a we show a complete representation of our measured data using a combined q–ω presentation of g(q, ω) for 0 < ω < 30 meV (Γ–X and Γ–M kspace directions are shown in qspace); these data are most clearly demonstrated in Supplementary Movie 1. (Data above T_{c} and for 0 < ω < 30 meV are shown in Supplementary Section VII.) Most striking in the g(q, ω) are the anisotropic ‘kinks’ in q(ω) indicated by red arrows. Figure 4b shows the simultaneously measured normalized conductance (∼density of states N(ω)), with the characteristic features of pairing interactions indicated by red arrows; these occur within the energy range of the ‘kinks’ in q(ω). Figure 4c–e show plots of g(q, ω) data along different directions. Figure 4f shows the measured dispersion of the maxima of these g(q, ω) (Supplementary Section VIII). The inflection points of the g(q, ω) dispersion seen in Fig. 4a, f, which are directly related to the band renormalization from ReΣ(k, ω), are obviously strongly anisotropic in qspace and strongest in the Fe–Fe direction. Finally, Fig. 4g shows measured values of ΔE, the departure of the dispersion of the maxima in g(q, ω) from a model with no selfenergy effect, versus the angle θ around the γ band. This is to be compared with the theoretical prediction in Fig. 3g. The good correspondences between our theoretical prediction for ReΣ(k, ω) effects from coupling to AFSF (Fig. 3g) and the QPI measurements (Fig. 2e–h) are evident. If the optical phonon conjectured to exist in the same energy range is strongly coupling to electrons, a far more isotropic dependence would be expected.
Although evidence that selfenergy effects due to electron–bosoncoupling phenomena are occurring in ironbased materials abounds^{18,26,27,28,29,30,31,32}, a direct comparison between a theoretical prediction with realistic band/gap structure that distinguishes effects of coupling to AFSF from those due to coupling to E_{g}phonons generating the orbital fluctuations, has not been achieved. Here, by combining new theoretical insight into QPI discrimination between Σ(k, ω) from resonant AFSF and Σ(k, ω) due to alternative scenarios, together with novel QPI techniques designed to visualize the Σ(k, ω) signatures^{19}, we demonstrate that scattering interference at ω > Δ_{max} on the γ band of LiFeAs is highly consistent with expected effects due to AFSFdriven Σ(k, ω). Crucially the apparent changes in the dispersion (Figs 2 and 4) show a strong directional dependence, being focused along the Fe–Fe direction where the spin fluctuations of LiFeAs are concentrated^{23,33}. This is in excellent qualitative agreement with our predictions based on measured band/gap structures of LiFeAs for resonantAFSFdriven Σ(k, ω) effects (Figs 2a–d and 3 and Supplementary Section IV). Further, we demonstrate that such anisotropic Σ(k, ω) effects studied here cannot be caused by a FeE_{g} phonon (Fig. 3a–c and Supplementary Section IV). Thus, our combined theoretical/experimental approach to ‘fingerprinting’ the electronic selfenergy Σ(k, ω) discriminates directly between different types of bosonic fluctuations proposed to mediate pairing. In analogy to phononbased superconductors, this novel approach may lead to a definite identification of the Cooper pairing mechanism of ironbased superconductivity—with the present result pointing strongly to antiferromagnetic spin fluctuations.
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Acknowledgements
We are especially grateful to A. P. Mackenzie and D. J. Scalapino for key guidance with this project. We acknowledge and thank D. H. Lee, A. Chubukov, P. J. Hirschfeld, M. Norman and J. Schmalian for helpful discussions and communications. Theoretical studies are supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award DESC0010313 (K.L. and EA.K.); NSF DMR1120296 to the Cornell Center for Materials Research and NSF CAREER grant DMR0955822 (M.H.F.). Experimental studies are supported by the Center for Emergent Superconductivity, an Energy Frontier Research Center, headquartered at Brookhaven National Laboratory and funded by the US Department of Energy, under DE2009BNLPM015; by the UK EPSRC; by a GrantinAid for Scientific Research C (No. 22540380) from the Japan Society for the Promotion of Science. TM.C. acknowledges support by NSC1012112M001029MY3.
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M.P.A., A.W.R., F.M. and TM.C. performed the experiments and analysed the data; K.K., A.I., CH.L. and H.E. synthesized the samples; K.L. and M.H.F. performed the theoretical calculations of the selfenergy and simulation of quasiparticle interference. This project was initiated by the experimental discovery of the strongly anisotropic QPI features in the electron–boson energy range (A.W.R.) and by the resulting hypothesis that they are selfenergy effects; J.C.D. and EA.K. supervised the investigation and wrote the paper with contributions from M.P.A., A.W.R., F.M., K.L. and M.H.F. The manuscript reflects the contributions of all authors.
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Allan, M., Lee, K., Rost, A. et al. Identifying the 'fingerprint' of antiferromagnetic spin fluctuations in iron pnictide superconductors. Nature Phys 11, 177–182 (2015). https://doi.org/10.1038/nphys3187
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DOI: https://doi.org/10.1038/nphys3187
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