## Abstract

If strong electron-electron interactions between neighboring Fe atoms mediate the Cooper pairing in iron-pnictide superconductors, then specific and distinct anisotropic superconducting energy gaps *i*. Here, we introduce intraband Bogoliubov quasiparticle scattering interference (QPI) techniques for determination of _{2}, and α_{1}, and we determine the anisotropy, magnitude, and relative orientations of their

In typical FeAs-based materials, every second As atom lies above or below the FeAs layer (Fig. 1A) so that the crystallographic unit cell, instead of being a square with an Fe atom at each corner (Fig. 1A, dashed box), is rotated by 45° and has an As at each corner (Fig. 1A, solid box). The corresponding momentum space (_{1}, α_{2}, and γ bands surround the Γ point, and the electron-like β_{1} and β_{2} bands surround the *1*) from a commensurate antiferromagnetic and orthorhombic “parent” state [see supplementary materials (*2*)]. The highest superconducting critical temperatures (*T*_{c}) occur when the magnetic and structural transitions are suppressed toward zero temperature. Theories describing the FeAs superconductivity can be quite complex (*1*, *3**–**9*), but they typically contain two essential ingredients: (i) the predominant superconducting order parameter (OP) symmetry is *s*±, that is, it has *s*-wave symmetry but changes sign between different bands; and (ii) the superconducting energy gap functions *i* are anisotropic in _{4}) symmetry and a specific relationship of gap minima/maxima relative to the BZ axes. Figure 1C shows a schematic of such a situation for just two electronic bands, with contours of constant energy (CCE) for their Bogoliubov quasiparticles shown in Fig. 1D.

Although there is evidence for *s*± OP symmetry (*10*, *11*), the structure of any anisotropic gaps *1*). However, it is the structure of these *1*, *12*–*14*). By contrast, almost all *15*), have reported the *k*_{x}/*k*_{y} plane (*1*). Subsequent to our submission, however, nodeless anisotropic gaps with values spanning the range 2meV→4meV on the γ band and 5meV→6meV on the α_{2} band (surrounding Γ) were reported in LiFeAs (*16*, *17*). For the electron-like pocket at *16*, *17*). High-resolution determination of

Bogoliubov quasiparticle scattering interference (QPI) imaging is a suitable technique for high-resolution determination of *18*–*24*). The scattering interference patterns can be visualized in real space (*I*/d*V*(*E*) *g*(*E*) is measured as a function of location *E*. However, using QPI to determine the *g*(*E*) imaging [and equivalent high *g*(*E*)] necessary to obtain *11*). An iron-pnictide that satisfies this requirement is LiFeAs (*25*–*28*), which has a glide plane between two Li layers [(*2*), section I].

Bogoliubov QPI can be influenced by a variety of effects in the iron-pnictides (*21*–*24*). To explore the expected QPI signatures of the _{j}(*E*). In a generalization of the “octet” model of QPI in copper-based superconductors (*18*–*20*), scattering between these _{j}(*E*) should produce interference patterns with the seven characteristic QPI wave vectors _{1}…._{7} shown in Fig. 1D. An equivalent set of QPI wave vectors, but now with different lengths and orientations, would be generated by a different

To achieve a more robust QPI prediction, we next consider the joint-density-of-states (JDOS) approximation for the whole Bogoliubov quasiparticle spectrum (Fig. 1D). In this case, for *g*[*E = *(Δ_{max}*+*Δ_{min})/2] simulated by calculating the JDOS for a single hole-like band (red band of Fig. 1C) [(*2*), section II]. Although the JDOS approximation gives an intuitive picture of how *g*(*,E*) relates to regions of high density of states in *18*–*24*). In Fig. 1F, we show a T-matrix simulation of *g*[*,E = *(Δ_{max}*+*Δ_{min})/2] for equivalent parameters as those in Fig. 1E [(*2*), section II]. By comparison of Fig. 1, E and F, we see that the two types of simulations are virtually indistinguishable and that the expected octet wave vectors (overlaid by black arrows) are in excellent agreement (as is true for all

The complexity of the simulated *g*(*,E*) (Fig. 1, E and F) also reveals the considerable practical challenges if measured *g*(*,E*) are to be inverted to yield the underlying *g*(*,E*) images, this approach is based on measuring the maximum scattering intensity in a |*E* plane along a specific high-symmetry direction—for example, *E* intensity plot (*2*), section III]. Such a plot of intensity maxima in a *g*(|*E*) plane along a high symmetry direction actually contains all the information on _{4}-symmetric band. Another band with a different *g*(|*E*) for _{4}-symmetric bands of different |*g*(|*E*) in a specific |*E* plane [(*2*), section III].

We implement this approach using LiFeAs crystals with *T*_{c} ≈ 15 K. Their cleaved surfaces are atomically flat (Fig. 2A) and exhibit the *a*_{0} = 0.38 nm periodicity of either the As or Li layer [(*2*), section I]. As the glide plane is between Li layers we are probably observing a Li-termination layer. We image *g*(*E*) with atomic resolution and with a thermal energy resolution δ*E* ≤ 350 μeV at T = 1.2 K. Figure 2A inset shows the spatially averaged differential conductance *g*(*E*) far from in-gap impurity states [(*2*), section V]. The density of electronic states *N*(*E*)* ∝ g*(*E*) (Fig. 2A, inset) is in agreement with the *N*(*E*) first reported for LiFeAs (*29*, *30*). It indicates (i) a fully gapped superconductor because *g*(*E*) ~ 0 for |E| < 1 meV, (ii) a maximum energy gap of ~6 meV, and (iii) a complex internal structure to *N*(*E*) consistent with strong

Next, we image *g*(*E*) in ~90 nm square fields of view (FOV) at temperatures between 1.2 K and 16 K and in the energy range associated with Cooper pairing [(*2*), section VI]. The large FOV is required to achieve sufficient *g*(*E*) and its Fourier transform *g*(*E*) with *E* = 7.7 meV, and the inset shows a typical *g*(*E*) from energies above the maximum superconducting gap magnitude (*E* = +7.7 meV, –6.6 meV). The scattering interference signatures for three distinct bands can be detected as closed contours in *h*_{3}, *h*_{2}, and *h*_{1} throughout. The measured |*E*)| of these bands in Fig. 2E shows them all to be hole-like. A quantitative comparison of QPI to both ARPES (*15*–*17*) and quantum oscillation measurements (*31*) [(*2*), section VII] identifies *h*_{1}, *h*_{2}, and *h*_{3} with the three hole-like bands assigned α_{1}, α_{2}, and γ. The QPI signatures of electron-like bands are weak and complex, perhaps because they share the same *h*_{2,} or because of the stronger *k*_{z} dispersion of these bands, or because of weak overlap between high-|

Figure 3, A and B, shows the *g*(*E*) measured within the energy gaps in Fig. 2E. Scattering interference is virtually nonexistent in the *h*_{3} band (Fig. 3A, green arrows) and is similarly faint in a direction parallel to *h*_{2} band (Fig. 3B, green arrows). The disappearance in the superconducting phase of QPI in these two directions is a result of an anisotropic gap opening at relevant *g*(*E*), the strongly anisotropic scattering intensity versus angle in *g*(*E*) on the *h*_{3} Fermi surface. The *g*(*E*) data in Fig. 3, A and B, together with the *g*(*E*) data in Fig. 3C, reveal unambiguous QPI signatures for anisotropic gaps of different orientations on different bands in LiFeAs.

To quantify the *g*(*E*) within a |*E* plane defined first by *g*(|*E*) plane with |*2*), section VIII] and plotted in Fig. 3D. The evolution of the normal-state *h*_{3} band (red in Fig. 2E) is indicated here by red arrows. The superconducting energy gap minimum of *h*_{3} is seen, along with the curved trajectories of scattering intensity maxima expected from an anisotropic *2*), section VIII]. Similarly, by measuring *g*(|*E*) in the plane with |*h*_{2} band is extracted [(*2*), section VIII] and plotted in Fig. 3E. The energy gap minimum is again obvious along with the curved scattering intensity maxima expected from a different anisotropic *h*_{1} becomes unidentifiable within –6 ± 1.5 meV below *E*_{F} in the superconducting phase, consistent with the opening of a gap of this magnitude (Figs. 2E and 3E), but we cannot yet resolve any gap modulations.

The magnitude, anisotropy, and relative position of *h*_{3}, *h*_{2}, and *h*_{1} are then determined from Fig. 3, D and E, using the previously described procedure [(*2*), section III]. The resulting anisotropic superconducting gaps on bands *h*_{3}, *h*_{2}, and *h*_{1} of LiFeAs are displayed in Fig. 4, A and B. Although our *g*(|*E*) agree well with pioneering QPI studies of LiFeAs where common data exist, no studies of *30*). Moreover, although field-dependent Bogoliubov QPI can reveal OP symmetry (*11*), these techniques were not applied here to LiFeAs, and no OP symmetry conclusions were drawn herein. The anisotropic *16*, *17*) appear in agreement with our observations for the *h*_{3} (Fig. 1, γ) and *h*_{2} (Fig. 1, α_{2}) bands. Lastly, our measurements are quite consistent with deductions on LiFeAs band structure from quantum oscillation studies (*31*). Overall, the growing confidence and concord in the structure of

## Supplementary Materials

www.sciencemag.org/cgi/content/full/336/6081/563/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S10

## References and Notes

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**Acknowledgments:**We are particularly grateful to D.-H. Lee for advice and discussions, and we acknowledge and thank F. Baumberger, A. Carrington, P. C. Canfield, A. V. Chubukov, A. I. Coldea, M. H. Fischer, T. Hanaguri, P. J. Hirschfeld, B. Keimer, E.-A. Kim, M. J. Lawler, C. Putzke, J. Schmalian, H. Takagi, Z. Tesanovic, R. Thomale, S. Uchida, and F. Wang for helpful discussions and communications. Studies were supported by the Center for Emergent Superconductivity, an Energy Frontier Research Center, funded by the U.S. Department of Energy under DE-2009-BNL-PM015; by the UK Engineering and Physical Sciences Research Council (EPSRC); and by a Grant-in-Aid for Scientific Research C (no. 22540380) from the Japan Society for the Promotion of Science. Y.X. acknowledges support by the Cornell Center for Materials Research (CCMR) under NSF/DMR-0520404. T.-M.C. acknowledges support by Academia Sinica Research Program on Nanoscience and Nanotechnology, and A.P.M. the receipt of a Royal Society-Wolfson Research Merit Award. The data described in the paper are archived by the Davis Research Group at Cornell University.