Abstract
Cooper pairing in the iron-based high-Tc superconductors1,2,3 is often conjectured to involve bosonic fluctuations. Among the candidates are antiferromagnetic spin fluctuations1,4,5 and d-orbital fluctuations amplified by phonons6,7. Any such electron–boson interaction should alter the electron’s ‘self-energy’, and then become detectable through consequent modifications in the energy dependence of the electron’s momentum and lifetime8,9,10. Here we introduce a novel theoretical/experimental approach aimed at uniquely identifying the relevant fluctuations of iron-based superconductors by measuring effects of their self-energy. We use innovative quasiparticle interference (QPI) imaging11 techniques in LiFeAs to reveal strongly momentum-space anisotropic self-energy 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 self-energy due to antiferromagnetic spin fluctuations, thereby distinguishing them as the predominant electron–boson interaction.
Similar content being viewed by others
Main
The microscopic mechanism for Cooper pairing in iron-based high-temperature superconductors has not been identified definitively1,2,3. Among the complicating features in these superconductors is the multiband electronic structure (Fig. 1a). However, it is believed widely that the proximity to spin order1,2,3,4,5 and/or orbital order6,7 plays a key role in the Cooper pairing. In particular, two leading proposals for fluctuation-exchange-pairing mechanisms focus on two distinct bosonic modes associated with specific broken-symmetry states: antiferromagnetic spin fluctuations carrying momentum Q = (π, π)/a0, and d-orbital fluctuations amplified by Eg-phonon lattice vibrations of the Fe ions. No conclusive evidence that either fluctuation couples strongly to electrons and is thus relevant to Fe-based superconductivity has been achieved within the plethora of proposals about theexisting data12,13,14,15,16,17,18.
Each type of electron–boson interaction should produce a characteristic electronic ‘self-energy’ representing its effect on every non-interacting electronic state |k〉 with momentum ℏ k and energy ℏω. Thus, the interacting Green’s function is given by
where represents non-interacting 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 self-energy is to use angle-resolved 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 momentum-resolved 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 self-energy effects19. Our QPI data are obtained by first visualizing scattering interference patterns in real-space (r-space) images of the tip–sample differential tunnelling conductance dI/dV (r, ω = eV) ≡ g(r, ω) using spectroscopic-imaging 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 principle19, using such data.
In a conventional single-band s-wave 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 momentum-independent self-energy 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 hole-like 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 momentum-dependent (anisotropic) self-energy. This is because, given a fermionic dispersion (k, ωkn) for different bands n and a spectrum of spin fluctuations whose intensity is strongly concentrated at (Q, Ω), the renormalization due to the self-energy at a point (k, ωkn) will be most intense when that point can be connected to another point (k − Q, ωk−Qm) 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)/a0 and energy Ω necessary for the resonant antiferromagnetic fluctuation to couple these bands can be analysed by shifting the electron-pocket-dispersion 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 γ-band dispersion (blue), is where the kinematic constraint of equation (1) can be satisfied and thus where the strongest self-energy 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 iron-based superconductor LiFeAs as a concrete example for which it should be possible to make a clear theoretical distinction between the self-energy effects driven by different types of bosonic fluctuations. We assume that BCS theory adequately describes the superconductor deep in the superconducting phase. Hence, the non-interacting Green’s function is given by
where and are the identity and the Pauli matrices in Nambu space, respectively. The superconducting gap structure Δk and the band structure Hk0 are taken from experiments11,12,17 and ab-initio calculations21 (Supplementary Section I). We then study the lowest order self-energy 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 in-plane lattice vibrations of the Fe ions with Eg symmetry (Fe-Eg phonon). It is the coupling of this Fe-Eg phonon to electrons that is proposed to enhance the d-orbital fluctuations which mediate Cooper pairing in the orbital fluctuation mechanism6,7. We take a perturbative approach of computing the self-energy to the lowest order9 (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)/a0 with the characteristic energy of Ω ≈ 6 meV to model the resonant AFSF of LiFeAs (refs 22, 23), whereas it is nearly momentum-independent for the optical Eg phonon20. We focus on the self-energy effects on the γ band (Fig. 1a, e) in the rest of this paper as its nearly uniform orbital character (dxy) greatly simplifies the theoretical study (Supplementary Section III) while at the same time being readily accessible to QPI studies11. 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, ωkγ on the γ band (blue surface in Fig. 1e) to a point with momentum k − Q on one of the two electron-like bands (yellow surfaces in Fig. 1e). Thus, the distinct anisotropic dispersions of each band mean that resonant AFSF should result in self-energy effects with a strong directional dependence (black curve on the γ band in Fig. 1e). Similarly, for the Fe-Eg phonons with a weak momentum dependence7, the self-energy effect for the γ band (which consists almost entirely of dxy orbitals24) is predicted to be angle-independent (Supplementary Section IV).
In Fig. 2a–d we present the predictions from equation (2) for g(q, ω) in LiFeAs, in the presence of self-energy 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 high-intensity region around q ≈ 2kFγ shows an anisotropy dictated by the gap anisotropy11,17,25, 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 self-energy effects for the AFSF self-energy (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 structure11 to the new effects of the AFSF-driven 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 Tc ≈ 15 K and with the superconducting energy gap maximum |Δmax| = 6.5 ± 0.1 meV. Clean and flat Li-termination surfaces (Li–Li unit cell a0 = 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 self-energy phenomena intervened one would expect this closed contour (Fig. 2f) to evolve continuously to smaller and smaller q-radius with increasing ω until the top of this hole-like band is reached. Instead, Fig. 2g shows the beginning of a very different evolution. Above ω ∼ 12 meV, the q-space 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 self-energy 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 Fe-Eg phonon (Fig. 3a–c) and AFSF (Fig. 3d–f) can provide a distinguishing ‘fingerprint’ of AFSF-driven 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–Eg-phonon interaction is predicted to yield isotropic self-energy 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 k-space directions are shown in q-space); these data are most clearly demonstrated in Supplementary Movie 1. (Data above Tc 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 q-space 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 self-energy 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 self-energy effects due to electron–boson-coupling phenomena are occurring in iron-based materials abounds18,26,27,28,29,30,31,32, 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 Eg-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) signatures19, we demonstrate that scattering interference at ω > Δmax on the γ band of LiFeAs is highly consistent with expected effects due to AFSF-driven 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 concentrated23,33. This is in excellent qualitative agreement with our predictions based on measured band/gap structures of LiFeAs for resonant-AFSF-driven 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 Fe-Eg phonon (Fig. 3a–c and Supplementary Section IV). Thus, our combined theoretical/experimental approach to ‘fingerprinting’ the electronic self-energy Σ(k, ω) discriminates directly between different types of bosonic fluctuations proposed to mediate pairing. In analogy to phonon-based superconductors, this novel approach may lead to a definite identification of the Cooper pairing mechanism of iron-based superconductivity—with the present result pointing strongly to antiferromagnetic spin fluctuations.
References
Wang, F. & Lee, D-H. The electron-pairing mechanism of iron-based superconductors. Science 332, 200–204 (2011).
Hirschfeld, P. J., Korshunov, M. M. & Mazin, I. I. Gap symmetry and structure of Fe-based superconductors. Rep. Prog. Phys. 74, 124508 (2011).
Scalapino, D. J. A common thread: The pairing interaction for unconventional superconductors. Rev. Mod. Phys. 84, 1383–1417 (2012).
Mazin, I. I., Singh, D. J., Johannes, M. D. & Du, M. H. Unconventional superconductivity with a sign reversal in the order parameter of LaFeAsO1−xFx . Phys. Rev. Lett. 101, 057003 (2008).
Kuroki, K. et al. Unconventional pairing originating from the disconnected Fermi surfaces of superconducting LaFeAsO1−xFx . Phys. Rev. Lett. 101, 087004 (2008).
Stanescu, T. D., Galitski, V. & Das Sarma, S. Orbital fluctuation mechanism for superconductivity in iron-based compounds. Phys. Rev. B 78, 195114 (2008).
Kontani, H. & Onari, S. Orbital-fluctuation-mediated superconductivity in iron pnictides: Analysis of the five-orbital Hubbard–Holstein model. Phys. Rev. Lett. 104, 157001 (2010).
Zhang, J., Sknepnek, R. & Schmalian, J. Spectral analysis for the iron-based superconductors: Anisotropic spin-fluctuations and fully gapped s±-wave superconductivity. Phys. Rev. B 82, 134527 (2010).
Heimes, A., Grein, R. & Eschrig, M. Electronic dispersion anomalies in iron pnictide superconductors. Phys. Rev. Lett. 106, 047003 (2011).
Heimes, A., Grein, R. & Eschrig, M. Effect of spin fluctuations on the electronic structure in iron-based superconductors. Phys. Rev. B 86, 064528 (2012).
Allan, M. P. et al. Anisotropic energy gaps of iron-based superconductivity from intraband quasiparticle interference in LiFeAs. Science 336, 563–567 (2012).
Knolle, J. et al. Incommensurate magnetic fluctuations and Fermi surface topology in LiFeAs. Phys. Rev. B 86, 174519 (2012).
Wray, L. et al. Momentum dependence of superconducting gap, strong-coupling dispersion kink, and tightly bound Cooper pairs in the high-Tc (Sr, Ba)1−x(K, Na)xFe2As2 superconductors. Phys. Rev. B 78, 184508 (2008).
Koitzsch, A. et al. Temperature and doping-dependent renormalization effects of the low energy electronic structure of Ba1−xKxFe2As2 single crystals. Phys. Rev. Lett. 102, 167001 (2009).
Richard, P. et al. Angle-resolved photoemission spectroscopy of the Fe-based Ba0.6K0.4Fe2As2 high-temperature superconductor: Evidence for an orbital selective electron-mode coupling. Phys. Rev. Lett. 102, 047003 (2009).
Kordyuk, A. A. et al. Angle-resolved photoemission spectroscopy of superconducting LiFeAs: Evidence for strong electron–phonon coupling. Phys. Rev. B 83, 134513 (2011).
Borisenko, S. V. et al. One-sign order parameter in iron based superconductor. Symmetry 4, 251–264 (2012).
Chi, S. et al. Scanning tunneling spectroscopy of superconducting LiFeAs single crystals: Evidence for two nodeless energy gaps and coupling to a bosonic mode. Phys. Rev. Lett. 109, 087002 (2012).
Dahm, T. & Scalapino, D. J. Quasi-particle interference probe of the self-energy. New J. Phys. 16, 023003 (2014).
Marsiglio, F. & Carbotte, J. P. in Superconductivity Vol. 2 (eds Bennemann, K. H. & Ketterson, J. B.) Ch. 3, (Springer, 2008).
Eschrig, H. & Koepernik, K. Tight-binding models for the iron-based superconductors. Phys. Rev. B 80, 104503 (2009).
Taylor, A. E. et al. Antiferromagnetic spin fluctuations in LiFeAs observed by neutron scattering. Phys. Rev. B 83, 220514 (2011).
Qureshi, N. et al. Inelastic neutron-scattering measurements of incommensurate magnetic excitations on superconducting LiFeAs single crystals. Phys. Rev. Lett. 108, 117001 (2012).
Hajiri, T. et al. Three-dimensional electronic structure and interband nesting in the stoichiometric superconductor LiFeAs. Phys. Rev. B 85, 094509 (2012).
Umezawa, K. et al. Unconventional anisotropic s-wave superconducting gaps of the LiFeAs iron-pnictide superconductor. Phys. Rev. Lett. 108, 037002 (2012).
Yang, J. et al. Optical spectroscopy of superconducting Ba0.55K0.45Fe2As2: Evidence for strong coupling to low-energy bosons. Phys. Rev. Lett. 102, 187003 (2009).
Fasano, Y. et al. Local quasiparticle density of states of superconducting SmFeAsO1−xFx single crystals: Evidence for spin-mediated pairing. Phys. Rev. Lett. 105, 167005 (2010).
Wu, D. et al. Eliashberg analysis of optical spectra reveals a strong coupling of charge carriers to spin fluctuations in doped iron-pnictide BaFe2As2 superconductors. Phys. Rev. B 82, 144519 (2010).
Choi, K-Y. et al. Self-energy effects and electron–phonon coupling in Fe–As superconductors. J. Phys. Condens. Matter 22, 115802 (2010).
Shamoto, S. et al. Inelastic neutron scattering study of the resonance mode in the optimally doped pnictide superconductor LaFeAsO0.92F0.08 . Phys. Rev. B 82, 172508 (2010).
Shan, L. et al. Evidence of a spin resonance mode in the iron-based superconductor Ba0.6K0.4Fe2As2 from scanning tunneling spectroscopy. Phys. Rev. Lett. 108, 227002 (2012).
Cai, P. et al. Visualizing the microscopic coexistence of spin density wave and superconductivity in underdoped NaFe1−xCoxAs. Nature Commun. 4, 1596 (2013).
Wang, M. et al. Antiferromagnetic spin excitations in single crystals of nonsuperconducting Li1−xFeAs. Phys. Rev. B 83, 220515 (2011).
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 DE-SC0010313 (K.L. and E-A.K.); NSF DMR-1120296 to the Cornell Center for Materials Research and NSF CAREER grant DMR-0955822 (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 DE-2009-BNL-PM015; by the UK EPSRC; by a Grant-in-Aid for Scientific Research C (No. 22540380) from the Japan Society for the Promotion of Science. T-M.C. acknowledges support by NSC101-2112-M-001-029-MY3.
Author information
Authors and Affiliations
Contributions
M.P.A., A.W.R., F.M. and T-M.C. performed the experiments and analysed the data; K.K., A.I., C-H.L. and H.E. synthesized the samples; K.L. and M.H.F. performed the theoretical calculations of the self-energy 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 self-energy effects; J.C.D. and E-A.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.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Information (PDF 1810 kb)
Supplementary Movie
Supplementary Movie 1 (MOV 2750 kb)
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nphys3187
This article is cited by
-
Ordered and tunable Majorana-zero-mode lattice in naturally strained LiFeAs
Nature (2022)
-
Andreev Reflection Spectroscopy on SnAs Single Crystals
Journal of Superconductivity and Novel Magnetism (2022)
-
Strong enhancement of magnetic order from bulk to stretched monolayer FeSe as Hund’s metals
npj Computational Materials (2020)
-
Causal analysis of competing atomistic mechanisms in ferroelectric materials from high-resolution scanning transmission electron microscopy data
npj Computational Materials (2020)
-
Direct comparison of ARPES, STM, and quantum oscillation data for band structure determination in Sr2RhO4
npj Quantum Materials (2020)