Anisotropic Energy Gaps of Iron-Based Superconductivity from Intraband Quasiparticle Interference in LiFeAs

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Science  04 May 2012:
Vol. 336, Issue 6081, pp. 563-567
DOI: 10.1126/science.1218726


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 Δi(k) should appear on the different electronic bands i. Here, we introduce intraband Bogoliubov quasiparticle scattering interference (QPI) techniques for determination of Δik.  in such materials, focusing on lithium iron arsenide (LiFeAs). We identify the three hole-like bands assigned previously as γ, α2, and α1, and we determine the anisotropy, magnitude, and relative orientations of their Δik.  These measurements will advance quantitative theoretical analysis of the mechanism of Cooper pairing in iron-based superconductivity.

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 (k-space) Brillouin zone (BZ) then contains five electronic bands; the hole-like α1, α2, and γ bands surround the Γ point, and the electron-like β1 and β2 bands surround the M̃ point (Fig. 1B). The superconductivity derives (1) from a commensurate antiferromagnetic and orthorhombic “parent” state [see supplementary materials (2)]. The highest superconducting critical temperatures (Tc) occur when the magnetic and structural transitions are suppressed toward zero temperature. Theories describing the FeAs superconductivity can be quite complex (1, 39), 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 Δi(k) on different bands i are anisotropic in k-space, with each exhibiting distinct 90° rotational (C4) 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.

Fig. 1

(A) Top view of crystal structure in the FeAs plane. Dashed lines represent the one-Fe unit cell that would exist if all As were coplanar, and the actual unit cell of dimension a0 ≈ 0.38 nm is shown using solid lines. (B) Schematic Fermi surface of an iron-based superconductor like LiFeAs in the tetragonal nonmagnetic Brillouin zone (solid line). The “one-Fe zone” is shown as a dashed line. The blue, yellow, and red curves show the hole-like pockets surrounding the Γ point, and the green curves show electron pockets surrounding the Embedded Image point. The gray crosses mark the Embedded Image points. (C) Model system exhibiting two distinct anisotropic energy gaps Embedded Image on two hole-like bands. (D) CCE of the Bogoliubov quasiparticle excitation spectrum for the two bands in (C), each in the same color as its Embedded Image. Contours enclose diminishing areas surrounding the gap minimum in each case. Black arrows indicate the octet scattering vectors Embedded Imagebetween the CCE banana tips (red dots). The red and yellow vectors indicate the high-symmetry directions along which g(|Embedded Image|,E) should be measured to determine Embedded Image for the respective bands separately; see (H). (E and F) Theoretically simulated g[Embedded Image,E = max+Δmin)/2)] for a single hole-like band with band- and gap-anisotropy parameters shown in (D), using the JDOS (E) and T-matrix (F) approaches. In Embedded Image-space, the gray crosses occur at the Embedded Image points. The arrows show the scattering vectors Embedded Image from the octet model in (D). The red dots indicate the vectors that lie along Embedded Image, the direction studied in (G) and (H). (G) JDOS simulation of dependence of scattering intensity in g(|Embedded Image|,E), with Embedded Image parallel to Embedded Image for a single hole-like band with gap anisotropy parameters shown in (C) and (D). Note the curved shapes of scattering intensity maxima, which we then extract in (H). (H) Red curves: Expected trajectory of maxima in scattering intensity g(|Embedded Image|,E) with Embedded Image parallel to Embedded Image for h3. Yellow curves: Expected maxima in scattering intensity g(|Embedded Image|,E) with Embedded Image parallel to Embedded Image for h2.

Although there is evidence for s± OP symmetry (10, 11), the structure of any anisotropic gaps Δi(k) on different bands and their relative k-space orientation is an open question for virtually all iron-based superconductors (1). However, it is the structure of these Δi(k) that is crucial for understanding the pairing interactions. Thermodynamic and transport studies [which cannot reveal Δi(k)] provide good evidence for electronic anisotropy (1, 1214). By contrast, almost all k-space angle-resolved photoemission spectroscopy (ARPES) studies of these materials, including LiFeAs (15), have reported the Δi(k) to be isotropic in the kx/ky 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 α2 band (surrounding Γ) were reported in LiFeAs (16, 17). For the electron-like pocket at M̃, an anisotropic 3meV→4.5meV gap is also described, but the reported k-space positions of gap maxima/minima appear mutually inconsistent (16, 17). High-resolution determination of Δi(k) should help to more accurately quantify these fundamental characteristics of the superconductivity.

Bogoliubov quasiparticle scattering interference (QPI) imaging is a suitable technique for high-resolution determination of Δi(k) (1824). The scattering interference patterns can be visualized in real space (r-space) using spectroscopic imaging scanning tunneling microscopy (SI-STM) in which the tip-sample differential tunneling conductance dI/dV(r,E) g(r, E) is measured as a function of location r and electron energy E. However, using QPI to determine the Δi(k) of iron-pnictide superconductors may be problematic because (i) the large field-of-view g(r,E) imaging [and equivalent high q-space resolution in g(q,E)] necessary to obtain Δi(k) is technically difficult; (ii) complex overlapping QPI patterns are expected from multiple bands; and (iii) most iron-pnictide compounds exhibit poor cleave-surface morphology. This latter point can be mitigated by using a material with a charge-neutral cleave plane (e.g., 11). An iron-pnictide that satisfies this requirement is LiFeAs (2528), 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 (2124). To explore the expected QPI signatures of the Δi(k), we consider the model two-gap structure in Fig. 1, C and D. Within one band, each energy Δ1minEΔ1max at which Δ1k=̇E picks out eight specific k-space locations kj(E). In a generalization of the “octet” model of QPI in copper-based superconductors (1820), scattering between these kj(E) should produce interference patterns with the seven characteristic QPI wave vectors q1….q7 shown in Fig. 1D. An equivalent set of QPI wave vectors, but now with different lengths and orientations, would be generated by a different Δ2(k) on the second band (Fig. 1, C and D, yellow). Thus, conventional octet analysis could be challenging for a multiband anisotropic superconductor, as many intraband QPI wave vectors coincide near the center of q-space.

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 Δ1minEΔ1max within a given band (e.g., red band in Fig. 1, C and D), the tips of the “bananas” still strongly influence QPI because of the enhanced JDOS for scattering between these locations and effects inherent to the structure of the coherence factors. Figure 1E shows g[q,E = max+Δmin)/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(q,E) relates to regions of high density of states in k-space, the T-matrix formalism is needed for a rigorous description (1824). In Fig. 1F, we show a T-matrix simulation of g[q,E = max+Δmin)/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 Δ1minEΔ1max). The key QPI signatures of an anisotropic but nodeless Δi(k) are therefore expected to be arcs of strong scattering centered along the direction of the gap minima, with regions of minimal scattering intensity located toward the gap maxima (green arrows, Fig. 1, E and F).

The complexity of the simulated g(q,E) (Fig. 1, E and F) also reveals the considerable practical challenges if measured g(q,E) are to be inverted to yield the underlying Δi(k). Therefore, we developed a restricted analysis scheme that still allows the pertinent Δi(k) information to be easily extracted. Instead of the usual constant energy g(q,E) images, this approach is based on measuring the maximum scattering intensity in a |q|-E plane along a specific high-symmetry direction—for example, q || ΓM̃ for the red band in Fig. 1D. Figure 1G shows a simulation of such a |q|-E intensity plot (q || ΓM̃) revealing two curved trajectories of maximal scattering intensity. They are extracted and plotted as red curves in Fig. 1H [(2), section III]. Such a plot of intensity maxima in a g(|q|,E) plane along a high symmetry direction actually contains all the information on Δk for that C4-symmetric band. Another band with a different q-space radius and whose gap maxima are rotated by 45° to the first (e.g., yellow in Fig. 1D) can be analyzed similarly; a plot of maxima in g(|q|,E) for q || ΓX̃ (yellow arrow, Fig. 1D) would then yield distinct interference maxima from the Δ(k) of that band (yellow curves, Fig. 1H). The magnitude and relative orientation of Δi(k) on multiple C4-symmetric bands of different |q| radius can be determined simultaneously by using this multiband QPI approach. Importantly, the Δi(k) are then determined experimentally, not from comparison to simulation but directly from a combination of (i) the measured normal-state band dispersions, (ii) the BZ symmetry, and (iii) the measured geometrical characteristics of the scattering intensity g(|q|,E) in a specific |q|-E plane [(2), section III].

We implement this approach using LiFeAs crystals with Tc ≈ 15 K. Their cleaved surfaces are atomically flat (Fig. 2A) and exhibit the a0 = 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(r,E) with atomic resolution and with a thermal energy resolution δE ≤ 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 k-space gap anisotropy.

Fig. 2

(A) An ~35 nm square topographic image of LiFeAs surface, taken at voltage VB = –20 mV junction resistance RJ = 2 GΩ, temperature T = 1.2 K. (Inset) Average spectrum over an area without impurities [see (2), section V]. The maximum gap superconducting coherence peaks and the zero conductance near EF are clear. (B) Typical differential conductance image g(Embedded Image,E = 7.7meV) at T = 1.2 K. (Inset) Typical QPI oscillations in g(Embedded Image,E) surrounding an impurity atom. (C) The g(Embedded Image,E = 7.7meV) that is the power-spectral-density Fourier transform of g(Embedded Image,E = 7.7meV), from (B). The high-intensity closed contour is a consequence of scattering interference within a large hole-like band h3. (D) The g(Embedded Image,E = –6.6meV) showing the high-intensity closed contours resulting from scattering interference from a smaller hole-like band h2, and h1 (inset). These data are measured at T = 1.2 K. (E) The measured energy dependence of the Embedded Image(E) = 2Embedded Image (E) for all three bands in h1 (blue line), h2 (yellow), and h3 (red) along the marked directions. Black dots are measured in the superconducting phase at T = 1.2 K; gray dots are measured in the normal state at T = 16 K.

Next, we image g(r,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 q-space resolution. Figure 2, B and C, shows a typical g(r,E) and its Fourier transform g(q,E) with E = 7.7 meV, and the inset shows a typical r-space example of the interference patterns. Figure 2, C and D, shows representative Fourier transforms g(q,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 q-space; we refer to them as h3, h2, and h1 throughout. The measured |q(E)| of these bands in Fig. 2E shows them all to be hole-like. A quantitative comparison of QPI to both ARPES (1517) and quantum oscillation measurements (31) [(2), section VII] identifies h1, h2, and h3 with the three hole-like bands assigned α1, α2, and γ. The QPI signatures of electron-like bands are weak and complex, perhaps because they share the same q-space regions as the signature of h2, or because of the stronger kz dispersion of these bands, or because of weak overlap between high-|k| states and the localized tip electron wave function; we do not consider them further here.

Figure 3, A and B, shows the g(q,E) measured within the energy gaps in Fig. 2E. Scattering interference is virtually nonexistent in the q-space direction parallel to ΓX̃ for the h3 band (Fig. 3A, green arrows) and is similarly faint in a direction parallel to ΓM̃ for the h2 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 k-space locations on these bands (see insets in Fig. 3, A and B), as predicted from both the T-matrix and JDOS simulations in Fig. 1, E and F. Figure 3C shows g(θ,E), the strongly anisotropic scattering intensity versus angle in q-space g(q,E) on the h3 Fermi surface. The g(q,E) data in Fig. 3, A and B, together with the g(θ,E) data in Fig. 3C, reveal unambiguous QPI signatures for anisotropic gaps of different orientations on different bands in LiFeAs.

Fig. 3

(A) g(Embedded Image,E = 2meV) measured at T = 1.2 K. The scattering on the h3 band has become highly anisotropic within its energy gap range, with vanishing intensity in the Embedded Image direction (green arrows). The inset shows schematically in Embedded Image-space how this is indicative of a lower energy gap along Embedded Image and a higher energy gap along Embedded Image. (B) g(Embedded Image,E = –5 meV) measured at T = 1.2 K. The scattering on the h2 band has become highly anisotropic within its energy gap range (green arrows), indicative of a lower energy gap along Embedded Imageand a higher energy gap along Embedded Image. (C) A projection of the QPI intensity g(Embedded Image,E) (a three-dimensional data set) onto the measured normal-state band dispersion Embedded Image . In the direction of minimal gap (Embedded Image), the intensity is gapped up to around 2meV ≈ Embedded Image whereas in the direction of maximal gap (Embedded Image), it is gapped up to around 3meV ≈ Embedded Image. (D) Extracted maximum scattering intensity trajectory from g(|Embedded Image|,E) for Embedded Image || Embedded Image[(2), section VIII] containing the information on Embedded Image for h3. (E) Extracted maximum scattering intensity trajectory from g(|Embedded Image|,E) for Embedded Image || Embedded Image[(2), section VIII] containing the information on Embedded Image for h2. Blue: Gap opening on the h1 band.

To quantify the Δi(k), we examine the scattering intensity in g(q,E) within a |q|-E plane defined first by q parallel to ΓM̃ and then by q parallel to ΓX̃. The locus of maximum intensity in the g(|q|,E) plane with |q| parallel to the ΓM̃ direction is extracted [(2), section VIII] and plotted in Fig. 3D. The evolution of the normal-state h3 band (red in Fig. 2E) is indicated here by red arrows. The superconducting energy gap minimum of h3 is seen, along with the curved trajectories of scattering intensity maxima expected from an anisotropic Δ(k) (Fig. 1, G and H) [(2), section VIII]. Similarly, by measuring g(|q|,E) in the plane with |q| parallel to ΓX̃, the locus of maximum intensity of the h2 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 Δ(k) (see Fig. 1H); the data for the upper branch cannot be obtained because of interference from other signals (possibly the electron-like band). The QPI signature of the third band h1 becomes unidentifiable within –6 ± 1.5 meV below EF 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 Δi(k) on bands h3, h2, and h1 are then determined from Fig. 3, D and E, using the previously described procedure [(2), section III]. The resulting anisotropic superconducting gaps on bands h3, h2, and h1 of LiFeAs are displayed in Fig. 4, A and B. Although our g(|q|,E) agree well with pioneering QPI studies of LiFeAs where common data exist, no studies of Δi(k) were reported therein (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 Δi reported recently in ARPES studies of LiFeAs (16, 17) appear in agreement with our observations for the h3 (Fig. 1, γ) and h2 (Fig. 1, α2) 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 Δi(k) for LiFeAs will advance the quantitative theoretical study of the mechanism of its Cooper pairing. Moreover, the multiband anisotropic-gap QPI techniques introduced here will allow equivalent Δi(k) observations in other iron-pnictide superconductors.

Fig. 4

(A) Anisotropic energy gap structure Embedded Image measured using QPI at T = 1.2 K on the three hole-like bands h3, h2, and h1 (Fig. 2E). These bands have been labeled γ, α1, and α2 before. Here, the 0.35-meV error bars stem from the thermal resolution of SI-STM at 1.2 K. (B) A three-dimensional rendering of the measured (solid dots) anisotropic energy gap structure Δ on the three hole-like bands at T = 1.2 K.

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S10

References (3236)

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
  2. 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.
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