Using Gap Symmetry and Structure to Reveal the Pairing Mechanism in Fe-based Superconductors
UUsing Gap Symmetry and Structure to Reveal the Pairing Mechanism in Fe-basedSuperconductors
P. J. Hirschfeld Department of Physics, University of Florida, Gainesville, FL 32611, U.S.A.
I review theoretical ideas and implications of experiments for the gap structure and symmetry ofthe Fe-based superconductors. Unlike any other class of unconventional superconductors, one hasin these systems the possibility to tune the interactions by small changes in pressure, doping ordisorder. Thus, measurements of order parameter evolution with these parameters should enable adeeper understanding of the underlying interactions. I briefly review the ”standard paradigm” for s -wave pairing in these systems, and then focus on developments in the past several years which havechallenged this picture. I discuss the reasons for the apparent close competition between pairing ins- and d-wave channels, particularly in those systems where one type of Fermi surface pocket – holeor electron – is missing. Observation of a transition between s - and d -wave symmetry, possibly viaa time reversal symmetry breaking “ s + id ” state, would provide an important confirmation of theseideas. Several proposals for detecting these novel phases are discussed, including the appearanceof order parameter collective modes in Raman and optical conductivities. Transitions between twodifferent types of s -wave states, involving various combinations of signs on Fermi surface pockets, canalso proceed through a T -breaking “ s + is ” state. I further discuss recent work that suggests pairingmay take place away from the Fermi level over a surprisingly large energy range, as well as the effectof glide plane symmetry of the Fe-based systems on the superconductivity, including various exotic,time and translational invariance breaking pair states that have been proposed. Finally, I addressdisorder issues, and the various ways systematic introduction of disorder can (and cannot) be usedto extract information on gap symmetry and structure.January 12, 2016 a r X i v : . [ c ond - m a t . s up r- c on ] J a n CONTENTS
I. Introduction 2II. Concepts of Gap Symmetry and Structure in Fe-based systems 3III. Experimental overview: SC state of iron pnictides 5A. Singlet vs. triplet 5B. s ± vs. s ++ η -pairing 24IX. Disorder 27A. Intra- vs. interband scattering 27B. T c suppression 27C. Impurity bound states 30D. Gap evolution vs. disorder 30E. Quasiparticle interference 31X. Conclusions 33XI. Acknowledgements 35References 35 I. INTRODUCTION
The primary interest in any newly discovered superconducting material is understanding the mechanism by whichsuperconductivity arises. Although not always sufficient in this regard, determining the symmetry and structure ofthe energy gap function in momentum space is often an important clue. Over the past thirty years, a number ofexperimental probes, together with simple theoretical modeling, have been devised to deduce information on the gapfunction in novel superconductors. In this short review, I consider the Fe-based superconductors (FeSC), discoveredin 2008 in the laboratory of Hideo Hosono , with the intention of collecting what is known about the order parameterin these systems, and discussing the implications for the pairing mechanism .Both the question and the answer in Fe-based systems are inevitably more complicated than in, say cupratesuperconductors, where it was possible within a few years of their discovery to make a fairly strong case that thepairing symmetry was d -wave, and furthermore rather close to the idealized form ∆ k ∝ cos k x − cos k y for k on theFermi surface for a wide range of materials and dopings . The FeSC exhibit electronic structures which involveseveral Fe d -bands near the Fermi level, and small electron- and hole-like pockets which are quite sensitive to externalperturbations such as pressure, doping, etc. Thus it is perhaps not surprising that across the range of materialsconsidered to fall in this category, the superconducting state appears also to be quite sensitive, exhibiting e.g. in somematerials a full gap, and in others clear evidence for low energy nodal quasiparticle excitations.The argument was made early on that these observations of “sensitivity” or ”nonuniversality” of low-energyproperties were prima facie evidence for s ± pairing, a superconducting state with the full crystal symmetry but withsign change between Fermi surface sheets. This form of order was proposed as a ground state for the Fe-pnictidesuperconductors by several authors as a natural way of obtaining unconventional pairing from repulsive Coulombinteractions, based on the structure of the Fermi surface . Nodes in an s ± state are perforce accidental , i.e.determined not by symmetry but by details of the pairing interaction, and can thus be created or destroyed withoutany phase transition at finite temperatures. Thus such a state is itself a natural candidate to explain the variabilityof low-energy excitations across the space of materials. However, it was also found within spin fluctuation pairingtheories that s ± pairing competes closely with d -wave pairing , and thus possibilities for new states which mix s and d channels in various ways, including time reversal symmetry breaking phases are also allowed. In recent years,theories which pair electrons in an orbital basis have emerged, and led to predictions of even more exotic states whichbreak parity or translational invariance. I will argue in this review that there is little microscopic reason to believethat such states can be stabilized.The general form of superconducting gap function ∆ ν ( k ), where ν is a band index and k is momentum on theFermi surface, will be difficult to pin down using a single experimental probe. As has been the case in all previousclasses of unconventional superconductors, to the extent a consensus on the gap structure has been established, ithas been because of information gleaned from a variety of experimental probes of thermodynamics, transport, and,only in some cases, phase-sensitive information from pair tunneling. Choosing among the various candidates in theFe-based system is certain to be a nontrivial task, and will require extensive, careful experiments on a wide range ofmaterials.The reader new to the field may by now have the impression that the situation in Fe-based systems is so complicatedthat nothing useful can be learned by studying gap symmetry and structure. However, the very nonuniversality ofthe superconducting state in the Fe-based systems, which makes them seem complicated, can be used to distinguishdifferent gap structures and provide tests of theories of pairing. Consider the variability of superconducting propertiestuned by doping or pressure. While any number of theories may predict an s ± state generically (e.g., at optimaldoping), as is indeed currently the case (see Sec. V below), it is likely that only the correct, materials-specific pairingtheory will be able to predict the correct doping and pressure dependence. Thus, with respect e.g. to the cuprates,there are new types of data available that will constrain theory and thus provide information on how electronspair. In the process, one will certainly take steps towards a quantitative theory of some classes of unconventionalsuperconductors that will aid in the search for new, higher- T c systems.This is not intended as a comprehensive review of theory or experiment on the FeSC, nor even on the superconductingstate of the FeSC. Instead, I would like it to be provocative, to collect some of the interesting concepts in unconventionalsuperconductivity that have been inspired by these fascinating materials, with a special emphasis on those newerdevelopments since the appearance of Ref. 5. It also represents a rather personal perspective on which ideas representproductive lines of inquiry. In this spirit, I have certainly neglected a number of novel proposals, particularly thoseoutside the framework of unconventional superconductivity due to spin fluctuations of itinerant electrons. With thesame motivation, I comment only on those experiments which seem best to illustrate the relevant concepts. I apologizeto the authors of the many important contributions neglected here. II. CONCEPTS OF GAP SYMMETRY AND STRUCTURE IN FE-BASED SYSTEMS
The classification of gap symmetries by group theory has been reviewed elsewhere in the general case of unconven-tional superconductors , as well as in the specific case of Fe-based systems, where the unusual glide plane symmetryof the Fe-pnictide or chalcogenide layer and the multiple orbital degrees of freedom introduce some new possibilities .Roughly speaking, the possible states correspond to the irreducible representations of the space group of the crys-tal. In the simplest cases, the representations of the point group play the essential role. Let us consider first avery simple model of the Fermi surface of a tetragonal crystal with 1 Fe/unit cell, as in Fig. 1a), with a single,Γ-centered hole pocket, and X - and Y -centered electron pockets, and postpone consideration of the subtleties ofthe glide plane symmetry to Section VIII. For the moment I assume spin singlet pairing (see however Sec. III A). FIG. 1. (Color online.) Schematic gaps ∆( k ) in FeSC. Color represents phase of ∆( k ). (a)-(d) Model Fermi surface withone hole and two electron pockets. (a) Conventional s -wave ( s ++ ) state; (b) s ± state with gap on hole pocket minus thaton electron pockets; (c) antiphase s -wave state possible when two or more hole pockets are present, showing gaps with threedifferent phases ∆ i ; (d) similar to (b), but with accidental nodes on electron pockets; (e) d -wave state; (f) d wave state insituation with no central hole pocket. I focus first on simple tetragonal point group symmetry. In a 2D tetragonal system, group theory allows only forfour one-dimensional irreducible representations: A g (“ s -wave”), B g (“ d -wave” [ x − y ]), B g (“ d -wave” [ xy ]), and A g (“ g -wave” [ xy ( x − y )]), depending on how the order parameter transforms under crystal rotations by 90 ◦ andother operations of the tetragonal group. In Figure 1 I have illustrated the two symmetry classes, namely s -wave and d x − y -wave, which are most relevant to the discussion of Fe-based systems. Within the s -wave class there are severalpossibilities shown in Fig. 1a)-d): s ++ , where the gap has the same sign on both electron and hole pockets, s ± wherethe sign changes between the pockets, a more complicated phase-changing s -wave state allowed if one has more thanone hole pocket, and finally a nodal s ± where the overall sign changes, but the gap varies around at least one set ofpockets such that it has (accidental) zeros.Note all such s wave cases have the same symmetry , i.e. none changes sign if the crystal axes are rotated by 90 ◦ ; thedifferences among the four examples are in gap structure . By contrast, the d -wave states shown in Fig. 1e),f) changesign under a 90 ◦ rotation. If central hole pockets are present, the gap must have nodes on these sheets, as in e).On the other hand, if such pockets are altogether absent at the Fermi level, as seems to occur for several interestingmaterials, it is possible to obey d -wave symmetry without creating nodes, as shown in f) .In materials with inversion symmetry the superconducting order parameter can be characterized by its parity, sincethe spatial part of its wave function (the order parameter) can either change sign or remain the same under theinversion operation. Since electrons obey the Pauli principle, the full wave function must be antisymmetric underexchange, which also involves spin exchange, requiring ∆ αβ ( k ) = − ∆ βα ( − k ), where α and β are the spin indicesof the two electrons . Thus in the absence of spin-orbit coupling the pair wave function ∆ αβ ( k ) ∝ f ( k ) χ αβ canhave total spin S = 1 (triplet), with the spatial part of the wave function f ( k ) odd, ∆ αβ ( k ) = − ∆ αβ ( − k ) or S = 0(singlet) with spatial part even (∆ αβ ( k ) = ∆ αβ ( − k )).Some interesting aspects of this argument in multiorbital systems have described by Fischer . In essence, if one pairsin orbital space one can fulfill the Pauli principle by having the single-particle orbital degrees of freedom help enforcethe antisymmetry under particle exchange. For example, if ∆ αa ; βb ( k ) describes a pairing between single-particlestates characterized by (momentum, spin, orbital) [ k , α, a ] and [ − k , β, b ], ∆ αa ; βb ( k ) may be taken ∝ f ( k ) χ α,β φ ( a, b )in the absence of spin-orbit coupling. This allows one to construct states consistent with the Pauli principle whichare, e.g. even parity, even under spin exchange, and odd under orbital exchange, or odd parity, odd under spinexchange, odd under orbital exchange. Such exotic states are generally expected to be energetically disfavored, sincethey generically involve substantial amplitudes to pair electrons at different single-particle energies. This is becauseafter performing the unitary transformation to bring the orbital pair ab into band space, one will inevitably generatelinear combinations involving contributions from electrons in different bands. e.g. one electron at the Fermi levelat k ν , and one at − k µ , where µ (cid:54) = ν are band indices. Note that it is possible to imagine pair potentials whichtake advantage of such pairings to lower the energy of the system, but for standard BCS-like potentials such energyimbalances in pairing typically leads to strong suppression of the condensation energy. It is for this reason that mosttheoretical researchers have made the ansatz of paired electrons in like bands at the Fermi level; however as I arguebelow in Sec. VI C, there are circumstances in multiband systems where even standard pairing interactions can leadto pairing in incipient bands, and there may be similar arguments favoring the stability of interband pairs (see, e.g.Ref. 19). These problems certainly deserve closer investigation. III. EXPERIMENTAL OVERVIEW: SC STATE OF IRON PNICTIDES
Here I briefly review the basic conclusions of Ref. 5 and many others regarding the basic symmetry and structureaspects of the “original” pnictide-based FeSC, i.e. the 1111, 122 and 111 arsenides and phosphides.
A. Singlet vs. triplet
An equal-spin triplet superconductor ( S z = ±
1) should polarize in an external magnetic field, just as free spinsin a normal metal. Thus one expects in such a system that the spin susceptibility (NMR Knight shift) should befeatureless at T c . Spin-orbit coupling to the lattice can suppress this for some directions, but not for others. In asinglet superconductor, on the other hand, or for the S z = 0 component of the triplet, the bound pair cannot bepolarized by the applied field, so that magnetic susceptibility vanishes as T →
0. Thus for singlet superconductivityone can expect the uniform spin susceptibility to diminish below T c , although the same occurs qualitatively for a stateincluding an S z = 0 triplet component, and vanishing susceptibility is often difficult to determine due to backgroundvan Vleck contributions.The conventional way to determine the spin susceptibility is by measuring the Knight shift. Early on, this ex-periment was performed on several FeSC including Ba(Fe − x Co x ) As , LaO − x F x FeAs , PrFeAsO . F . ,Ba − x K x Fe As , LiFeAs , and BaFe (As . P . ) . These works reported that the Knight shift decreasesfor all angles of the field with respect to the crystallographic axes. This effectively excluded triplet symmetries suchas p -wave or f -wave, and there is a general consensus that all FeSC have spin singlet gap symmetry. Some earlyreports regarding the intriguing material LiFeAs (Sec. V B) suggested signs of proximate ferromagnetism, which,when combined with measurements of weak or absent T -dependence of the Knight shift below T c led to the proposalof possible spin triplet pairing in this material . However these measurements have not been confirmed on othersamples to my knowledge and remain a mystery. B. s ± vs. s ++ Thus for many, and probably all of the FeSC, the spin structure of the gap is of the singlet type. In addition,there are strong theoretical reasons to believe that the momentum dependent part of the gap function is even parity,and probably s -wave (see Sec. V). Possible exceptions to this last conclusion include strongly overdoped systems,where there is some evidence for d -wave, discussed in Sec. VI A 1). Remember from Sec. II, the term “ s -wave” isused loosely in a crystalline system to refer to a gap with the full symmetry of the lattice. This does not tell us, forexample, whether it changes sign between Fermi surfaces, nor whether it has nodes on individual Fermi surfaces. Thisquestion acquires a particular urgency when one recognizes that the theoretical community, while largely endorsingthe spin-fluctuation driven s ± pairing notion, includes a significant component arguing that orbital fluctuations playan essential role, and that they lead to s ++ pairing, at least in some FeSC (Sec. V C 1). Thus this important questionof gap structure is also directly related to questions about the origins of superconductivity.Let us first ask what evidence exists that the gap changes sign at all. In this regard, there were a few isolatedexperiments of relevance when the forerunner of this review was written in 2011: neutron spin resonance , Joseph-son π -junctions , and quasiparticle interference (QPI) experiments . In the meantime, further examples of theseexperiments have turned up on other materials, but no really noteworthy experimental results nor serious new ideasfor “smoking gun” experiments have arisen, with the exception of some involving disorder. These include a) new STMmeasurements of impurity bound states (Sec. IX C); b) a class of systematic disorder experiments employing electronirradiation (Sec. IX D); and c) some possible new ways to analyze QPI data (Sec. IX E). It has also been noted thatthe observed coexistence of superconductivity and spin density wave order in some FeSC is strong evidence for s ± character of the pair state .Replicating the pioneering cuprate phase sensitive experiments that provided hard evidence for d -wave pairinghas proven extremely difficult, for technical reasons related to fabricating clean junctions, but more importantly forfundamental reasons. Because the s ++ and s ± states in question have the same angular symmetry, there is no wayto distinguish them by, e.g. tunneling into different faces of a crystal in the analog of the famous corner-junctionexperiment of Wollman et al. . Although other schemes have been proposed, they mostly depend on quantitativemodel calculations of the Josephson critical currents. An exception is an test involving a thin film of a conventional s -wave superconductor grown on the surface of a putative s ± proposed by Koshelev and Stanev , who showed thatthe signs of subdominant tunneling features could identify gap sign changes. Coupling between such systems can alsolead to controlled junctions, as shown by Linder et al. Nevertheless, such proposals are not trivial to realize; thus,controlled impurity experiments, which are also in principle phase sensitive, have acquired a new importance.
Spin resonance.
The observation of a neutron spin resonance in the magnetic susceptibility χ ”( q , ω ) as a test of an s ± state was proposed early on theoretically, and reported experimentally very quickly in all the FeSC understudy at that time (except LiFeAs, where it is weak and occurs at an incommensurate wave vector ). The familiar1111, 122 and 111 materials have strong magnetic fluctuations in the normal state at q = ( π, ), corresponding to theordered magnetic state of the parent compounds. I do not review this experimental situation here, nor the varioustheoretical approaches, but refer the reader to several excellent recent reviews, with focus on both experimental and theoretical issues.The neutron spin resonance phenomenon is generally understood to represent a paramagnon mode of the systemwhich sharpens only below T c because the corresponding pole in the magnetic susceptibility χ ( q , ω ) becomes undampeddue to the suppression of low-energy particle-hole excitations when the Fermi surface is gapped. The mode is of interestin the current context because the term in χ (cid:48)(cid:48) ( q , ω ) arising from the anomalous Green functions in the SC state isproportional to the coherence factor (cid:88) k (cid:20) − ∆ k ∆ k + q E k E k + q (cid:21) ... (1)where ... is the kernel of the BCS susceptibility. A cursory examination shows that this factor vanishes if ∆ k and∆ k + q have the same sign, and is maximized if the sign is changed. Thus the observation of a neutron resonancehas generally been taken as a “natural” indication of a sign change in the order parameter. It should be noted,however, that in situations where the quasiparticle relaxation rate in the SC state has a strong energy dependence, aconventional s ++ state may also have a peak that sharpens somewhat below T c . In a simple symmetric two-bandmodel, the peak is typically located above 2∆ in energy if generated by the scattering rate effect in an s ++ state,whereas in the s ± case, as for other unconventional pair states, it is always located below 2∆.Further information on recent developments is available in the review article by D. Inosov in this volume . Josephson tests.
An early experiment on a granular 1111 sample presented indirect evidence that Josephson π loops were present in the sample . In this experiment Chen et al. measured a very large number of randomly pairsof contacting grains with a Nb “fork”, in hope that some of them would accidentally fulfil the condition that thetwo contacts have the required phase shift. This can occur in a state where the order parameter changes sign if thenumbers that specify the Josephson current (orbital composition of the wave function, relative gap sizes, etc) worktogether in such a way that the current along one direction will be dominated by hole pockets and the other byelectron pockets. A thick tunneling barrier will for example preferentially favor tunneling from states at the zonecenter, i.e. hole states. So a small fraction of the loops containing the fork and two FeSC grains may contain a π phase shift; and this was in fact observed in the experiment . To my knowledge, no further experiments of this typehave been performed.A more promising technique that avoids the difficulties of in-plane tunneling was proposed in Ref. 55, a “sandwich”design whereby an epitaxial film of a hole-doped FeSC is grown on top of an electron-doped one (or vice versa). Ifthe epitaxial growth is very good, the parallel momentum is conserved at the interface and both types of carriersexperience coherent c-axis transport. An electrical contact to the hole-doped layer will then be dominated by the holecurrent, and the contact to the electron-doped layer by the electron current. If these contacts are now connected ina loop, there will be a phase difference between the two layers which will directly reflect on the sign change of theSC order parameter between the hole and electron sheets. Progress in fabricating such artificial heterostructures wasslow for some time, but success with interfaces with SrTiO (STO) (See e.g. Ref. 56 and citations in Sec. VI B 3)suggest that these problems are not insurmountable. Disorder effects.
In Sec. IX C below, I list some STM measurements performed in the past few years that haveobserved impurity bound states in situations where it appears unlikely that the impurity has a predominantly mag-netic character. This then suggests that at least this set of materials has an s ± character. Other reports of weak ornonexistent bound states in other situations can easily be understood as consistent with s ± in terms of simple modelsof impurity scattering in these systems, but there is no way to generate such a resonance in a s ++ state. In Secs.IX B and IX D, I discuss the qualitatively new insights provided by a new series of electron irradiation experimentson bulk properties of FeSC. These are important because they are the closest approximation to introducing purelypairbreaking random potentials to the system in a systematic way. Only in such a case can results be compareddirectly to theory. These experiments show a T c suppression rate relative to residual resistivity increase which is com-parable to the Abrikosov-Gor’kov rate for magnetic impurities in a single-band s -wave system, but for nonomagneticimpurities, implying s ± . Finally, in Sec. IX E, I discuss some issues that have arisen regarding the interpretation ofexperiments purporting to show from Fourier transform scanning tunneling microscopy (STM), commonly referred toas quasiparticle interference experiments, that the order parameter in some systems is s ± . The conclusion is probablystill correct, but there are subtleties in the analysis, and other ways of analyzing the data than used heretofore areprobably preferable. C. Fully gapped vs. nodal
As opposed to the d -wave case, where nodes are mandated on the hole pockets by symmetry, in an extended s -wavescheme they may appear on either type of pocket if amplitudes of higher harmonics in the angular expansion of theorder parameter are sufficiently large. As discussed in Section V, there are microscopic reasons why this may bethe case. Moreover, since nodeless s ± , nodeless s ++ , and an extended s with accidental nodes (either with averagegap switching sign between pockets (“nodal s ± ) or not (“nodal s ++ ”)) all belong to the same symmetry class, thedifference between them is only quantitative (but important).Information on the existence of nodes, or large gap anisotropy on individual Fermi surface sheets without nodes,is generally easier to obtain than clear indications of gap sign changes. This is because nodes or deep gap minimaresult in low-energy quasiparticle excitations, which can be detected in a number of thermodynamic and transportexperiments. Thermal conductivity measurements have played an important role in this debate, primarily becausethey can be extended to quite low temperatures and can thus often distinguish between true nodes and small gapminima. Penetration depth experiments have also been quite useful . A number of the low- T c systems, particularlythe phosphides, as well as KFe As , are found to be nodal. A simple explanation within the spin fluctuation theoryproposed by Kuroki and co-workers suggests that the larger Pn-Fe-Pn bond angle leads to a suppression of the d xy band at the Fermi surface, and thus the elimination of an important intraorbital stabilization of the isotropic s ± interaction (see Refs. 58,10 and Sec. VI A 1). FeSe crystals and thick films appear to be nodal as well, andconsistent with predictions of spin fluctuation theory .A higher T c system (optimal 30K) which has been established as nodal is BaFe (As,P) , thought to be doped bychemical pressure since As,P are isovalent . In this material, the d xy FS sheet is present, so alternative explanationsfor the strong anisotropy were required. Early 3D spin fluctuation calculations suggested that nodes might occuron hole pockets due to rapid changes in orbital content near the top of the Brillouin zone, and also exhibited deep gapminima on the electron sheets. It appears more likely from phenomenological fits to angle-dependent magnetothermalconductivity measurements, however, that these minima descend in real systems and become the primary nodes inthis system in the form of electron pocket loops .Finally, gap anisotropy has been investigated intensively in the context of BaFe As overdoped with both K andCo. In both cases, anisotropy increases signifcantly as one overdopes away from optimal doping, as predicted by spinfluctuation theory . The K-doped side is particularly interesting as the system must make a transition from a fullygapped to a nodal state that may be d -wave, a discussion that I postpone until Sec. VI A. The electron-doped sideis more anisotropic even at optimal doping, according to c -axis thermal conductivity measurements that indicate alow- T linear term, which reflects the presence of nodes, growing immediately away from optimal doping . Thesenodes have also been claimed from analysis of Raman data to be of the loop type .It is obviously extremely interesting to study the crossover between nodal and fully gapped systems. While thisis usually discussed in the context of doping or addition of disorder (see Secs. IX D), it can be driven by pressureor strain as well. Recently Guguchia et al. studied such a transition in Rb-doped Ba122 by measuring changesin low-temperature penetration depth. Deciding whether the nodal state in such a transition arises because of theonset of accidental nodes in an s ± state or a transition to d -wave is always a delicate issue, and probes with angularresolution are probably required to make clear statements. IV. MULTIBAND SUPERCONDUCTIVITY
To provide a background for a few theoretical topics on the frontier of the Fe-based superconductivity field, Ifirst review basic ideas about multiband superconductivity within a simple model where interactions are constant onand between bands. The general BCS gap equation for a multband system with intra- and interband interactions V µν ( k , k (cid:48) ) is ∆ µ ( k ) = − (cid:88) k (cid:48) ν (cid:48) V µν ( k , k (cid:48) ) ∆ ν ( k (cid:48) )2 E ν k (cid:48) tanh E ν k (cid:48) T (2) (cid:39) − (cid:88) ν Λ µν ∆ ν L (∆ ν , T ) , (3)where µ, ν are band indices, the prime on the sum indicates a restriction to states within a BCS cutoff energy ω D ofthe Fermi level, and the quasiparticle energy is E ν k = (cid:112) ξ ν k + ∆ ν k , with ξ k the single-particle dispersion measuredwith respect to the chemical potential. In BCS theory ∆ k is restricted to states on the Fermi surface, so I canperform the energy integration perpendicular to the Fermi surface. In the second line of (3), following Matthias,Suhl, and Walker and Moskalenko , constant densities of states N ν for each Fermi surface sheet ν and isotropicinteractions V µν are assumed. In this case the theory may be expressed in simple algebraic form as in (3), in terms ofthe dimensionless interaction matrix Λ with elements Λ µν = V µν N ν and L (∆ , T ) = (cid:82) ω D dξ tanh( √ ξ +∆ T ) / (cid:112) ξ + ∆ .In the BCS weak coupling limit, one may derive an expression for T c in such a system analogous to the singleband expression, T c = ω D exp( − /λ eff ) , where λ eff is the largest eigenvalue. The elements of the eigenvector arethen the ratios of the individual order parameters. Unless all V ij are the same, the temperature dependence ofindividual gaps ∆ µ ( T ) does not follow the usual BCS behavior. As an example, consider a two-band superconductorwhere Λ µµ (cid:29) Λ µ (cid:54) = ν : in the limit where the interband coupling is zero, there are two independent T c ’s for ∆ and∆ . However, even for very small interband coupling there can be only one thermodynamic transition, such thatthe smaller gap opens initially very slowly, and only at a T corresponding to the “bare” superconducting transition(Λ µ (cid:54) = ν = 0) does it start to increase rapidly. As the interband coupling increases, the two gaps begin to aquire asimilar T dependence.Equations (3) may have solutions even when all elements of the interaction matrices Λ are negative, (repulsiveaccording to the above convention) . The simplest case is a 2-band repulsive interband interaction: V = V = 0 ,V = V = − V < . In this case the solution reads: λ eff = √ Λ Λ = | V |√ N N , ∆ ( T c ) / ∆ ( T c ) = − (cid:112) N /N .The reader who wishes a more detailed account of multiband SC is urged to consult the reviews by Tanaka andStanev . V. MICROSCOPICS: THE “STANDARD MODEL” AND ITS VARIANTS
In the first year of study of the Fe-based superconductors, it became clear that the multiband, small pocket aspectof their electronic structure was crucial, and that repulsive Coulomb interactions among the various small pocketsfound both in DFT calculations and in ARPES was driving the superconductivity. From a modeling perspective, itwas desirable to begin with the simplest possible band structure, and then add local interactions. What were theessential features to include in the band modeling?First-principles calculations showed the electronic properties and the topology of Fermi surface consisting ofhole/electron pockets at the Brillouin zone center/corners and indicated the importance to superconductivity of Fe3d orbitals, which give the dominant density of states (DOS) at the Fermi level. The five Fe-3d orbitals split into the e g doublet ( d z , d x y ) and the t g triplet ( d xz , d yz , d xy ) under the tetrahedral crystal field , and it is the latterthat typically dominate the states at the Fermi energy. Several theoretical analyses of the superconductivity, aswell as speculations regarding the symmetry of the superconducting order parameters of LaO x F x FeAs based on two-orbital model of the electronic structure appeared very soon after its discovery. Three-orbital and more realisticfive-orbital Fe-only tight-binding models were used together with a spin-fluctuation interaction , predicting an s ± gap with various momentum dependences (anisotropy and gap nodes) which are qualitatively consistent withexperiments . In some special cases discussed below, 10- and even 16-orbital calculations have been performed.While faithful to the crystal symmetries of the DFT calculations from which they were derived, these calculationswere performed using a 1-Fe unit cell, appealing to an exact unfolding of the bandstructure in the k z = 0 plane asdiscussed by Eschrig . However, this treatment misses important 3D effects in some systems, particularly 122s, aswell as the effect of the nonperturbative band folding potential due to the out-of-plane As atoms on the quasiparticleweights in the unfolded zone. This has significant consequences for angle-resolved photoemission in the normal state,leading to the remarkable variation of spectral intensity around, e.g. the electron pockets . In principle, pairingcalculations involving all 10 Fe d -orbitals in the 2-Fe unit cell should capture these 1-particle effects properly; for afuller account of some subtleties, see Sec. VIII below. A. Spin fluctuation pairing theory
Virtually all calculations of this type have been performed using a model (“multiband Hubbard” or “Hubbard-Hund”) Hamiltonian consisting of a multiband tight-binding kinetic energy H as discussed above, plus an interaction H int containing all possible two-body on-site interactions between electrons in orbitals (cid:96) , H = H + ¯ U (cid:88) i,(cid:96) n i(cid:96) ↑ n i(cid:96) ↓ + ¯ U (cid:48) (cid:88) i,(cid:96) (cid:48) <(cid:96) n i(cid:96) n i(cid:96) (cid:48) (4)+ ¯ J (cid:88) i,(cid:96) (cid:48) <(cid:96) (cid:88) σ,σ (cid:48) c † i(cid:96)σ c † i(cid:96) (cid:48) σ (cid:48) c i(cid:96)σ (cid:48) c i(cid:96) (cid:48) σ + ¯ J (cid:48) (cid:88) i,(cid:96) (cid:48) (cid:54) = (cid:96) c † i(cid:96) ↑ c † i(cid:96) ↓ c i(cid:96) (cid:48) ↓ c i(cid:96) (cid:48) ↑ where n i(cid:96) = n i,(cid:96) ↑ + n i(cid:96) ↓ . The Coulomb parameters ¯ U , ¯ U (cid:48) , ¯ J , and ¯ J (cid:48) represent a Hubbard interaction, interorbitalHubbard interaction, Hund’s rule exchange and “pair hopping interaction”, respectively. The tight-binding model H used in H is typically downfolded from a full DFT band structure onto an Fe-only tight-binding model .In Equation (4), the intra- and inter-orbital Coulomb repulsion ¯ U and ¯ U (cid:48) are written as distinct, along with theHund’s rule exchange ¯ J and “pair hopping” interaction ¯ J (cid:48) for generality, but if they are generated from a singletwo-body term with spin rotational invariance they are related by ¯ U (cid:48) = ¯ U − J and ¯ J (cid:48) = ¯ J . In a real crystal, sucha local symmetry will not always hold. Work until now has focussed on generating tight-binding bandstructures for H from DFT. If Coulomb interactions are indeed local, the Hamiltonian (4) may be expected to provide a gooddescription of the physics of a given material, provided one knows or has a method of calculating the interactionparameters ¯ U , ... (see below). Attempts have been made to calculate these from first principles , but there aresome subtle issues associated with the different Wannier bases which have been implemented. Alternatively, thesecan be calculated within a GW framework .The generalization of simple 1-band Berk-Schrieffer spin fluctuation theory to the multiorbital case was performedby many authors . The effective pair interaction vertex Γ( k , k (cid:48) ) between bands i and j in the singlet channel isΓ ij ( k , k (cid:48) ) = Re (cid:34) (cid:88) (cid:96) (cid:96) (cid:96) (cid:96) a (cid:96) , ∗ i ( k ) a (cid:96) , ∗ i ( − k ) (5) × Γ (cid:96) (cid:96) (cid:96) (cid:96) ( k , k (cid:48) , ω = 0) a (cid:96) j ( k (cid:48) ) a (cid:96) j ( − k (cid:48) ) (cid:105) where the momenta k and k (cid:48) are confined to the various Fermi surface sheets with k ∈ C i and k (cid:48) ∈ C j , and a (cid:96)i ( k )are orbital-band matrix elements. The orbital pairing vertices Γ (cid:96) (cid:96) (cid:96) (cid:96) describe the particle-particle scattering ofelectrons in orbitals (cid:96) , (cid:96) into (cid:96) , (cid:96) and in the fluctuation exchange approximation are given byΓ (cid:96) (cid:96) (cid:96) (cid:96) ( k , k (cid:48) , ω ) = (cid:20)
32 ¯ U s χ RPA1 ( k − k (cid:48) , ω ) ¯ U s +12 ¯ U s −
12 ¯ U c χ RPA0 ( k − k (cid:48) , ω ) ¯ U c + 12 ¯ U c (cid:21) (cid:96) (cid:96) (cid:96) (cid:96) , (6)where each of ¯ U s , ¯ U c , χ , etc represent matrices in orbital space which depend on the interaction parameters. Here χ RPA1 describes the spin-fluctuation contribution and χ RPA0 the orbital (charge)-fluctuation contribution, determinedby Dyson-type equations as χ RPA1 (cid:96) (cid:96) (cid:96) (cid:96) ( q ) = (cid:110) χ ( q ) (cid:2) − ¯ U s χ ( q ) (cid:3) − (cid:111) (cid:96) (cid:96) (cid:96) (cid:96) , (7) χ RPA0 (cid:96) (cid:96) (cid:96) (cid:96) ( q ) = (cid:110) χ ( q ) (cid:2) U c χ ( q ) (cid:3) − (cid:111) (cid:96) (cid:96) (cid:96) (cid:96) . (8)0 FIG. 2. (Color online.) Gap functions on 3D Fermi surfaces of 122 FeSC from 10-orbital fluctuation exchange calculations. a)on electron pockets of BaFe (As,P) from Ref. 96; b),c) for LiFeAs from Ref. 97: b) Fermi surface with orbital content derivedfrom ARPES measurements; c) gap function on these sheets. where repeated indices are summed over. Here χ pqst is a generalized multiorbital susceptibility (see 10). Results of microscopic theory.
The simplest goal of the microscopic approach is to calculate the critical temperature T c via the instability equation and determine the symmetry of the pairing eigenfunction and the pairing eigenvalue(which determines T c ) there. If one writes the order parameter ∆( k ) as ∆ g ( k ), with g ( k ) a dimensionless functiondescribing the momentum dependence on the Fermi surface, then g ( k ) is given as the stationary solution of thedimensionless pairing strength functional λ [ g ( k )] = − (cid:80) ij (cid:72) C i dk (cid:107) v F ( k ) (cid:72) C j dk (cid:48)(cid:107) v F ( k (cid:48) ) g ( k )Γ ij ( k , k (cid:48) ) g ( k (cid:48) )(2 π ) (cid:80) i (cid:72) C i dk (cid:107) v F ( k ) [ g ( k )] (9)with the largest eigenvalue λ , a dimensionless measure of the pairing strength. Here k and k (cid:48) are restricted to thevarious Fermi surfaces k ∈ C i and k (cid:48) ∈ C j and v F ( k ) = |∇ k E ( k ) | is the Fermi velocity on a given Fermi sheet i . Notethe band indices on v F ( k ), etc. have been suppressed because specification of k also specifies the band.The most sophisticated calculations of this type are performed for the 122 systems, which are the most 3D ofthe FeSC due to the stronger overlap of the pnictogen/chalcogen between the layer containing Fe and the reservoirlayers , and a few other cases where 3D Fermi surface pockets exist, like LiFeAS. In these cases 2D calculations arenot sufficient, and one needs to account for 3D scattering processes. Two examples are shown in Fig. 2, indicatingthe rather complex gap structures g ( k ) that can be obtained. B. LiFeAs: challenge for theory
In earlier classes of unconventional superconductors, consensus on a particular pairing state was generally reachedonly after different classes of experiments, including a) bulk thermodynamic and transport measurements that probethe averaged quasiparticle spectrum, b) phase sensitive experiments, typically requiring surfaces or interfaces, whichprobe the order parameter sign changes directly, and c) ARPES, which in principle measures the momentum-resolvedgap structure | ∆ k | . In the Fe-based systems, designing and fabricating devices for phase sensitive measurements hasproven problematic. Furthermore, while ARPES experiments should in principle be the most direct measurementsof gap structure, early ARPES measurements indicated insignificant k -space anisotropy in the 122 systems, despitethe fact that thermodynamic probes all reported the presence of low-energy quasiparticles consistent with largeanisotropy and even nodes in several cases, particularly Ba(Fe − x Co x ) As . Poor surface quality may have limitedangular resolution in these cases.1 FIG. 3. (Color online.) (a) LiFeAs Fermi surface at k z = π for ARPES-derived band structure used by Refs. 97,107,108.(b) Functional renormalization group calculation of k z = π SC gaps from Ref. 109, where DFT Fermi surface was used. (c)Gaps at k z = π predicted by RPA spin fluctuation calculation of Ref. 97. Solid lines are theory; dashed lines are approximatesketches of experimental determination of same gaps105. (d) Comparison of gap predictions of Ref. 110 (LDA+DMFT-basedcalculation of the pairing interaction) with same experimental data. Thus a good deal of attention was focussed on the first high-quality crystals of LiFeAs, which were shown to haveexcellent nonpolar surfaces without significant reconstruction . Both detailed STM-quasiparticle interference andARPES measurements of the momentum dependence of the superconducting gaps on the various pockets werereported, and found to be qualitatively consistent with one another where comparisions were possible. Aspects of themeasured gap function in LiFeAs, according to these measurements, were a) the gap was nodeless; b) gap sizes variedfrom a minimum on the large Γ-centered hole pocket ( γ ) of about 3 meV to a maximum of about 6meV on the innerhole pockets ( α ); c) significant angular variation was reported on both the two electron pockets ( β ) and the outer holepocket. Note that the various ARPES groups disagreed somewhat on the details of the Fermi surface itself: accordingto Ref. 105, around the Γ point of the zone there was only a single hole pocket γ ; the α and α hole bands werebelow the Fermi level, whereas Ref. 41 found that α indeed crossed the Fermi level near Γ. Some of these data arereproduced in Fig. 3.This rough concordance of several experimental groups and methods on the detailed structure of the gap functionin one of the Fe-based materials led several theoretical groups to consider the challenge seriously. Inwhat follows, I give a somewhat detailed summary of these efforts, not because the details of the gap function inLiFeAs are fundamental to the physics of Fe-based superconductors, but because the agreements and discrepanciesillustrate the level at which the field of materials-specific calculation of superconducting properties has arrived forthese systems. The general reader can skip this discussion and glance at Fig. 3, where a subset of these resultsare presented, indicating that rather good agreement among the recent generation of calculations is obtained for thelarger Fermi surface pockets in the system, and discrepancies remain for those tiny, possibly incipient pockets wherethe interactions are harder to calculate.The earliest results of Thomale et al had been based on a 2D calculation using a tight-binding model derivedfrom a density functional theory calculation. However, the ARPES measurements had shown that the true Fermisurface in this material differed considerably from these DFT calculations, due to the small to negligible size of theinner α , α hole pockets observed by ARPES compared to the relatively large sizes found in DFT. Local-densityapproximation (LDA)+dynamical mean-field theory (DMFT) calculations had some success showing that strongerelectronic correlations in the 111 family lead to a shrinkage of the inner hole pockets but maintenance of the electron2pocket size and shape , but still did not really reproduce the ARPES Fermi surface accurately.An improved approach adopted by several theoretical groups was to use a tight-binding model for the low-energybands derived from ARPES . These calculations represented various versions of what may be called weak-coupling spin fluctuation theory, and all considered pairing only at the Fermi level. The remarkable thing wasthat, despite their differences in implementing this theory, all arrived at momentum-dependent gap functions for theelectron and outer hole pockets semiquantitatively in agreement with experiment, with the tuning of at most oneor two parameters within a reasonable range (Ref. 107 was more phenomenological in approach). They disagreed,however, on the size of the gaps on the rather small inner hole xz/yz Fermi surface pockets reported by ARPES, thosewhich undergo Lifshitz transitions first upon electron doping. Experiment reported these gaps to be the largest, bute.g. Ref. 97 found them to be intermediate to small among the various gaps in the system. In some works, betteragreement with the gap sizes on the smaller hole pockets was found, based on claims of improved formulation of theproblem in orbital space or inclusion of Aslamasov-Larkin vertex corrections to the RPA theory . In the onlyfully 3D spin fluctuation pairing calculation, Ref. 97, the small gaps found on these tiny pockets were attributed tothe neglect of states away from the Fermi level. This point is discussed further in Sec. VI C.A further point of contention among these theories was the distribution of gap signs over the various Fermi pockets.While Ref. 97 found a traditional s ± state, with all hole pockets one sign and all electron pockets the other, Yinet al. reported that their leading pair state had the signs of the electron and outer hole pockets equal (“orbitalantiphase s -state), while Saito et al. found both the electron pockets and the outer hole pockets with sign oppositethe “traditional” s ± . Recently, Nourafkan et al. reported a LDA+DMFT calculation, improved relative to Yin etal. , which also found a leading s ± state.Needless to say, experiments are not yet in a position to confirm any of these last predictions. They arise inthe theories because of relatively subtle differences in the treatment of the various interband interactions, which arebeyond the scope of this review, and may also to some extent depend on errors in the band structure adopted, sincethere is still some controversy about the small hole pockets. Nevertheless, one may be reasonably optimistic that,as experiments and crystal growth techniques improve, and some kind of consensus is reached about how to improvethe simplest spin-fluctuation theories, the role of these small pockets (and incipient bands) will be elucidated, andthe pairing state in the LiFeAs system will be understood. If this turns out to be the case, it will be a remarkabletriumph for theory, as this system is quite complicated, and (due to lack of nesting) relatively far from the “standard”situation in the FeSC where strong ( π,
0) magnetic fluctuations dominate the excitation spectrum. Of course, it ispossible that some ingredients, e.g. phonons, have been left out of these approaches, and may be essential to achievea quantitative description.
C. Alternative pairing theories
1. Orbital fluctuations
There are many theoretical approaches in addition to the weak-coupling spin fluctuation exchange approach de-scribed in Sec. V A, of which I treat only two here, because they have received the most attention, and (apparently)involve the inclusion of different physics. If one looks at Eq. 6, one sees that instabilities are in principle possibleboth in the spin χ and charge χ channel. Most of the community has noted the proximity of long range magnetismto superconductivity, as well as the strong magnetic fluctuations observed in NMR and inelastic neutron scattering,and concluded that spin fluctuations dominate the pairing in most FeSC. From a theoretical standpoint, if one takesEq. 4 with local interactions U, J, U (cid:48) , J (cid:48) as representing the essential physics, one can easily show for realistic modelsthat for sensible values of these parameters χ is singular and dominates over χ .The importance of orbital degrees of freedom was noted by several authors in the context of orbital ordering inthe Fe d states at the orthorhombic transition, usually in the context of localized models . In itinerant models,orbital order usually requires concomitant spin stripe order or sufficiently strong fluctuations. The competitionbetween theoretical models of “spin fluctuations” and “orbital fluctuations” is sometimes discussed as if it were aneither-or situation, but in fact all models used contain both types of fluctuations. The RPA treatment of Sec. Vyields the “fluctuation exchange” expression (4) for the effective interaction, which manifestly contains the enhancedcontribution of the charge/orbital fluctuations ( ∼ χ ), although it is sometimes referred to as “spin fluctuation pairingtheory”. The point is that for physical values of the bare interactions U, J, U (cid:48) , J (cid:48) , the spin fluctuation terms ( ∼ χ )are much larger since one is close to a magnetic instability. It is easy to check that if one artificially makes U (cid:48) > U in (4), the singularity in the charge, or orbital channel, can be favored as well. Ab initio calculations show, however,that ¯ U (cid:29) ¯ U (cid:48) , as one would expect on fundamental quantum mechanical grounds. Pairing by orbital fluctuations, asproposed by Kontani and co-workers , therefore focussed initially on finding ways of enhancing the effectiveinterorbital Coulomb interaction. They showed that in the event such an enhancement occurred, the ground state of3such a system would be of the s ++ type, since the efffective interaction due to orbital fluctuations is attractive.Initially, Kontani and Onari focussed on the role of certain in-plane Fe phonons that can in principle enhancethe interorbital scattering processes such that they dominate the spin fluctuation part of the interaction (the effect ofa different phonon was discussed by Yanagi ). The electron-phonon coupling to these phonons was included in anRPA-type calculation , suggesting that phonon-mediated enhancement of orbital interactions to drive s ++ pairingwas feasible. However a full first principles calculation of the same vertex by Nomura et al. found that this effectwas much too weak to overcome the dominant spin fluctuation interaction.More recently, the enhancement of the orbital fluctuation interaction has been sought in extensions of the electronicmodel to include additional vertex corrections beyond the RPA in the calculation of the Cooper vertex , inparticular Aslamasov-Larkin type diagrams. More accurate results are claimed, but the physical meaning of theseadditional processes is not transparent. Thus, to which extent this proposed orbital-fluctuation mechanism is at workin real materials remains open. As discussed in Sec. III B, there is some evidence in FeSC for sign-changing orderparameter behavior, but it is still weak and not all tests can be performed in all materials. For many of these tests,which have “natural” explanations in the context of spin fluctuation pairing, there are alternative explanations. Inmany cases, claims of weak impurity pairbreaking (see however Sec.IX), the damped inelastic neutron scatteringresonance observed in most FeSC , and a simple explanation of the Lee plot (maximum of T c within family attetrahedral Fe-As angle ) keep interest in the orbital fluctuation pairing idea alive, or at least muddy the waters.It is of course possible that, due to varying interactions strengths, orbital fluctuations dominate in some of theFeSC and spin fluctuation in the others. Particularly in systems like bulk FeSe (Sec. VI B 1), it is not clear thatspin fluctuations are strong enough to dominate pairing. Recent papers from the Kontani group indeed appear toemphasize that the properties of many FeSC, particularly the formation of nodal structures in superconducting gaps,depend on the interplay of both types of fluctuations .
2. “Strong coupling” theories
Interactions in the FeSC are not in the weak coupling limit; even the systems showing perhaps the weakest elec-tronic correlations, like LaFePO, manifest renormalization of the low-energy dispersions of factors of 1.5-2 relative toDFT, while other reported mass renormalizations are somewhat higher, and there are significant band shifts as well.Nevertheless FeSC are clearly also not Mott insulators, so a lively debate has existed almost since the beginning of thefield about which starting point is the most likely to capture the essential physical aspects of these systems. Whilemost of this discussion has played out in the arena of the normal state, together with the related debate regarding thedegree of itinerancy appropriate for models of the magnetism, the debate has also spilled over into the neighboringstage where the origin of superconductivity is discussed. I do not focus on issues of correlation strength, magnetism,nematicity, or Mott selectivity, although I will touch on them elsewhere in the text, and in the conclusions. Thesetopics are all reviewed in up-to-date accompanying articles by Bascones et al. , v. Roekeghem et al. , and B¨ohmerand Meingast .The so-called “strong-coupling” approach to pairing, based on the J − J generalized Heisenberg model for themagnetism in these systems begins from the premise that that magnetism is nearly localized. Since theFeSC are metals, one might ask whether or not a localized description is ruled out a priori, but in fact magneticexcitations in metals are reasonably well described by DFT, which for the FeSC gives a characteristic energy rangefor the magnetic interactions of order 100 meV or larger decaying in space as a power law, such that the moments areindeed quite localized. However actual band structure calculations also show that the magnetic excitationsin the FeSC cannot be mapped onto those of a generalized ( J − J − J ... ) Heisenberg Hamiltonian of any range.They can be mapped onto a Heisenberg Hamiltonian with biquadratic exchange , or possibly to a more complicatedHamiltonian (such as ring exchange), but not onto a pure Heisenberg model. The biquadratic exchange eliminatesthe need for the unphysical in-plane J a,b anisotropy of the nearest neighbor exchange which had been used to fitmagnon excitations in FeSC parent compounds .The strong-coupling approach assumes H = H + H int , where H is a multiband tight-binding model, and H int = (cid:80) ij J ij S i · S j , where typically only J , J J (1st-3rd neighbors) are kept, although recently the effect ofthe biquadratic term (cid:80) ij K ij ( S i · S j ) has also been considered . Such models are of course reminiscent of the t − J − J (cid:48) model used in the cuprates, but since (for the common pnictides and chalcogenide-based systems withnear 6 electrons/Fe) there is no proximity to a Mott insulator, there is no corresponding strong coupling limit of aHamiltonian with local Coulomb interactions. Also unlike the t − J or t − J − J (cid:48) models, there is no justification toproject the kinetic energy onto a reduced Hilbert space, and this is generally not done (when it is done, of courseit makes little difference). There are related models, equally ad hoc , where U and J , terms are treated as beingindependent before the mean field decoupling .The interaction is then decoupled in the pairing channel in mean field theory, such that the nearest neighbor4exchange J induces competing cos k x + cos k y and cos k x − cos k y ( s - and d x − y -wave) pairing harmonics, while thenext nearest neighbor exchange leads to cos k x cos k y and sin k x sin k y ( s - and d xy -wave). General phase diagrams for2- and 5-orbital band models were worked out in Ref. 140, who found that with J (cid:38) J , cos k x cos k y was the leadinginstability, leading to a nodeless s ± state, but that d -wave was a close competitor. The results of this approachintially showed an intriguing set of circumstantial agreement with the predictions of itinerant weak-coupling models,despite the lack of any fundamental connection between the two approaches. Indeed, the pairing symmetry in anyweak-coupling spin-fluctuation model is primarily determined by optimizing the structure of the spin fluctuations tothe given Fermi surface. The Fermi surfaces are the same or similar in the two approaches, and Wang et al haveshown that the low energy spin and charge excitations in the fRG treatment of the 5-orbital model (4) overlap verywell with those of the t − J − J model as well. Since both are rather similar in the two approaches, not surprisingly,the main results seem to agree.Despite these successes, I believe these approaches are less reliable, for the following reasons: • they unphysically separate the itinerant electrons and the local moments, as if the latter were coming from aseparate atomic species. Of course the moments are formed by exactly the same Fe d -electrons that form theband structure, which also mediate the magnetic interaction mapped onto the J − J Heisenberg Hamiltonian. • actual band structure calculations can never be mapped onto a Heisenberg Hamiltonian of any range.In some cases mapping onto a Heisenberg Hamiltonian with biquadratic exchange is possible, but in others evensuch mappings lead to large deviations from DFT (see, e.g. Ref. 142). • by including low-order Brillouin zone harmonics, “strong-coupling” models specify the shape of the spin-fluctuation induced interaction; therefore, the resulting solution to the gap equation specifies the structureof the gap nodes in momentum space, so that the amplitude, anisotropy and possible nodes on actual FSsdepends only on the proximity of these FSs to the nodal lines in the Brillouin zone. This result is quite differentfrom the weak coupling calculations and is likely unphysical. Another consequence of the low-order harmonicsis that it is impossible for such theory to capture the “antiphase” s -wave states that are close competitors insome systems such as LiFeAs, which in spin fluctuation theory arise due to small- q repulsive interactions. Asdiscussed in Sec. V B, these involve sign changes among gaps on Fermi surfaces that are in close proximity in k space, and it seems unlikely that the low-order harmonics will ever vary fast enough to drive this effect. Ina similar vein, antiphase d -wave states (opposite sign d -wave order parameters on two Γ-centered hole pockets,for example) which should be the ground state in a strongly overdoped system, are hard to understand in thestrong coupling picture.The partial agreement between fRG, RPA, FLEX and the t − J − J (+ more recent approaches including biquadraticcouplings) results is nevertheless an interesting open question that deserves further study, but is certainly related tothe fact that the spin excitations in all these models are peaked at roughly similar momenta. Essentially this questionmaps onto the larger question of understanding when band structure calculations, including those incorportatingcorrelations in various schemes, capture correctly the observed spin excitations in experiment, and deciding whetherthese excitations can ever be properly described by an effective localized model . VI. PAIRING AT THE EXTREMESA. s vs. d
1. Proximity in spin fluctuation theory
As discussed above, the “standard model” arising from elementary considerations on pairing by repulsive interactionssuggests an s ± pair state in systems with hole and electron pockets separated by wavevector ( π, s ± function, including orbitalweights on the Fermi surface, the frustrating effect of ( π, π ) pair scattering processes between electron pockets, andintraband Coulomb interactions . Obviously the electronic structure, and how it changes with doping, will be theprimary factor influencing the relative weights of these various effects. Kuroki et al., for example, famously noticedthat the pnictogen height controlled a d xy band near the Fermi level, whose existence bolstered the ( π,
0) interactionsand therefore made the superconducting gap more isotropic . However in cases of extreme overdoping, one type ofpocket must disappear, electron or hole, leaving one with a situation where the usual ( π,
0) pair scattering is stronglysuppressed or absent. In the strongly electron overdoped case, ( π, π ) scattering will be most prominent, favoring5
FIG. 4. (Color online.) Difference between superconducting and normal-state Raman spectra on Ba . K . Fe As in threedifferent Raman polarization channels. From Ref. 154. d -wave pairing (Fig. 1(e)). In the overdoped hole-pocket case, the leading pairing instability is not intuitivelyclear from such arguments.In fact, the d -wave pair channel was recognized early on as being a strong competitor for the s ± state . Graseret al. , using a multiband RPA calculation of the spin fluctuation pairing vertex, even found that the d -wave pairinginteraction dominated the s -wave for smaller values of the local Coulomb repulsion U in the case of a band suitablefor 1111 systems near 6 electrons/unit cell. Maiti et al. then mapped the model of Graser et al. onto a conceptuallysimpler one with low-order Fermi surface angular harmonics, and showed how the amplitudes of these harmonicsevolved with doping for the 1111 band structure. They concluded that the form of the interaction in the highlyoverdoped case was consistent with a strong tendency towards d -wave pairing in both the hole- and electron-dopedextremes.Interest in the competition between s and d states grew with measurements on the low- T c highly hole-dopedsuperconductor KFe As . Some experiments, in particular thermal transport, were interpreted in favor of d -wavesuperconductivity , and early functional renormalization group treatments of this system also predicted d -wave .Such a picture was also consistent with other experiments clearly indicating the existing of gap nodes . However, suchbulk experiments lack the resolution to determine easily the position of these nodes, and shortly thereafter ARPESmeasurements, in principle a more direct measurement of the gap, reported an s wave gap with four symmetricallyplaced pairs of nodes on the intermediate hole band . While there are no obvious problems with the ARPESidentification, reconciling the gap determined in this way with thermodynamics has proven difficult , and thissystem remains controversial. From the theoretical standpoint, a state with nodes of this type has been obtainedphenomenologically in Ref. 152, and was recently proposed to arise “naturally” from a non-standard representationof s ± states as “orbital triplet” structures .Perhaps even more interesting is the question of how, if the stoichiometric system KFe As is d -wave, it makes atransition when doped with Ba to the optimally doped Ba x K x Fe As with x ∼ .
4, which is generally agreed to befully gapped s ± . This question is discussed in Sec. VII .
2. Nematic order and superconductivity
In some of the situations where s - and d -wave interactions compete in the FeSC, electronic nematic order orfluctuations are also present. For the most part I have ignored the controversial and fascinating issues of nematicityin this article, primarily because excellent reasonably up-to-date reviews exist . The issues covered there aremostly related to the origin of nematicity; here I focus on a side question, namely the interplay between nematicand superconducting degrees of freedom. Ultimately it is probable that the same electronic states contribute to bothsuperconductivity and nematic interactions, but one may separate the two effects phenomenologically and ask howsymmetry determines their interplay near T c when s - and d -wave interactions compete. This was the approach adoptedby Fernandes and Millis , who studied a Ginzburg-Landau free energy F = F nem ( φ ) + F SC (∆ s , ∆ d , θ ) + F SC − nem ,6with the well-known s − d part with competing s - and d -wave orders F SC = t s s + t d d + β s s + β d d + 12 ∆ s ∆ d ( β sd + α cos 2 θ ) , (10)as well as a nematic part F nem , and a trilinear coupling between the two, F sc − nem ∝ φ ∆ s ∆ d cos θ . Here φ is thenematic order parameter, and θ is the relative phase of the s - and d - components.Taken by itself, the s − d SC free energy is well known to have minima for pure s , pure d , and T -breaking s + id mixtures (see Sec. VII), depending on coefficients, so one obtains the usual s − d phase diagram in the case of weaknematic order (Sec. VII). In the case where nematic order forms at high T , as in many of the Fe pnictides, the SCtransition takes place in the presence of C symmetry breaking and a robust expectation value of the nematic field φ . In this case the ground state is a real mixture s + d , i.e. ∆ k = ∆ s f s ( k ) + ∆ d f d ( k ), with f s,d real harmonics of s, d symmetry, and ∆ s,d real coefficients. Clearly C symmetry is broken simply because it is broken already beforethe SC condenses out of the normal state. A qualitatively different situation is found in a fluctuating nematic phase,where F nem = χ − nem φ accounts for the independent dynamics of the nematic field. This now provides a nontrivialrenormalization of the coupling coefficients β sd and α , leading to two new possibilities, a s + d state forming in thecompetition region between pure s and d phases, a spontaneous C symmetry-breaking in the sense that it condensesat high T from a tetragonal phase. When the nematic fluctuations become sufficiently strong, this phase disappearsand is replaced by a 1st-order transition between s - and d -phases. Thus the observation of properties (specific heat,elastic coefficients) of a nematic system like BaFe As or NaFeAs as a function of doping as d -correlations growin strength can provide information on the trilinear coupling between nematic and SC fields. Similar results wereobtained in microscopic weak coupling theory by Livanas et al. .Kang et al. considered how s + d states could be created and controlled starting from a tetragonal system byapplying uniaxial strain. Starting with a model where strain coupled via an orbital ordering ∆ oo ( n i,yz − n i,yz ), theyshowed how strain mixed a d -wave order parameter into a starting s -wave ground state. In principle, if the unstrained s -state was fully gapped, it is difficult to say if one could apply enough strain to create nodes by mixing in a d amplitude comparable to s . If one starts with a nodal s -state, however, addition of a small amount of d via strainsuffices to move the nodes around the Fermi surface, either separating or coming together in pairs. Nodes can also becreated or annihilated in pairs as well. They argued that one could in principle observe this process via the anisotropiccurrent response of the system.
3. Detection: Bardasis-Schrieffer mode
A recent experimental development of great interest has been the possible observation of a Bardasis-Schrieffer (BS)particle-particle exciton in several FeSC. This collective mode of the s -wave condensate in the presence of a d -wavesubdominant pair interaction was predicted by Bardasis and Schrieffer in 1961 . The frequency of the collective modeat q → s -wave system. This mode was never observed in conventionalsuperconductors, but may be present in FeSC due to the aforementioned near-degeneracy of the s and d channels.Devereaux and Scalapino proposed that Raman scattering would be a clear way to observe the mode, and provideda simple calculation of its frequency in the case of an s ± state. Recently, BS mode-like features were identified inBa − x K x Fe As and NaFe − x Co x As, in Raman scattering (Fig. 4). The exact identification of these featureswith a BS mode is hindered by the fact that these systems posses multiple gaps and the exact nature of possiblecollective modes and their evolution across a typical doping phase diagram is not clearly known. Collective modes inthe particle-hole channel can also arise and need to be separated from the particle-particle excitons . Nevertheless,these discoveries raise the prospect of systematic studies of the interaction strengths and collective modes in differentchannels in FeSC for the first time.
B. Missing hole pockets
1. Background: bulk FeSe
As I prepare to discuss three electron doped systems without hole pockets, all based on layers of FeSe, it is perhapsuseful to review briefly what is known about superconductivity of the bulk compound. FeSe is structurally rathersimple, displaying a tetragonal to orthorhombic structural phase transition at T s ∼
90 K, entering a low- T phaseexhibiting strong electronic anisotropy, but without SDW order. It makes a transition to a superconducting stateonly below a T c of about 9K. Early scanning tunneling microscopy (STM) studies of thick FeSe films on SiC or bilayer7graphene substrates found highly elongated vortices and impurity states, and also measured a nodal superconductinggap . Until recently, similar experiments on crystals were hindered by sample quality, but these problems wereovercome by T. Wolf and collaborators, who fabricated very clean samples, appropriate to study the details of thelow-energy properties , using cold vapor deposition. This discovery led to a small renaissance of studies on thishitherto neglected material, and it has now become one of the most intensively investigated FeSC.Central to the mystery of FeSe is the existence of strong nematic tendencies in the absence of obvious magnetism,leading to suggestions that nematicity in this system is driven by orbital order, unlike other FeSC. Recent Se NMRmeasurements have reported a splitting of the NMR line shape beginning at T s , with an order parameter-like T dependence below T S . Unlike the well-documented case of BaFe As , at high T , the spin-lattice relaxation rateis found to be unaffected by the structural transition, and the upturn at low T signaling the onset of a strongly spin-fluctuating state appears only close to the superconducting T c . The naive interpretation of these experimentsis that nematic order is driven by orbital fluctuations in FeSe, but despite the apparent weakness of local spinfluctuations near T s , the spin-nematic picture may still apply. Several proposals, including strongly frustrated long-range magnetic order due to competing magnetic ground states , quantum paramagnetic fluctuations of large Fespins , competition with charge-current density waves , or quadrupolar spin order have been discussed at thiswriting. Recently, the first inelastic neutron scattering measurements on these samples appeared, showing robustlow-energy ( π,
0) fluctuations very similar to the Fe pnictides persisiting up to T s , thus deepening the mystery of whythe system does not magnetize, and why NMR is apparently not sensitive to the spin fluctuations at higher T . I donot discuss these fascinating questions further here, as they are covered in more detail in the accompanying reviewby B¨ohmer and Meingast .Other remarkable properties of bulk FeSe include the significant enhancement of the superconducting critical tem-perature T c under pressure, , as well as the extremely small size of the Fermi surface pockets measured at low T , leading to suggestions of BEC-BCS condensation in these systems where the band extrema E F are of orderthe gap ∆. These large and T -dependent renormalizations of the Fermi surface needed as an input to pairing cal-culations have slowed theoretical understanding of the superconducting state and its properties. Although there arestill some disagreements among ARPES measurements on the new crystals , general aspects of recent studiesinclude a Fermi surface (FS) above T S consisting of two small hole pockets of primarily d xz /d yz character aroundthe Γ − Z line. The hole bands are split there above T s by a spin-orbit coupling (SO) of order λ Fe (cid:39)
20 meV, butbelow T S this evolves into a single hole surface . ARPES also finds an electron pocket at the M point of mainly d xz /d yz character. Here the d xz / d yz degeneracy is lifted by ∼
50 meV at low T , taken by several authors to indicatestrong orbital order in FeSe. As mentioned above, these results are very different from those obtained from DFTcalculations. For example, ARPES finds that the electronic bands in FeSe are renormalized compared to DFTcalculations by a factor of ∼ d xz /d yz bands and ∼ d xy band, and hole and electron pockets areshifted such that pockets are much smaller than calculated by DFT . Quantum oscillations (QO) at low T are consistent with the ARPES data in observing small, mostly 2D pockets, even though the amount of dispersionalong k z is still disputed. Recently it was suggested that the splittings at the high symmetry points were notconsistent with a simple momentum independent ferro orbital ordering , but were in fact of opposite signs at the Γand M points, leading to suggestions of bond-centered (sometimes referred to as d -wave since the d xz /d yz splittinghas opposite signs at M and ¯ M ).Recently, Mukherjee et al. proposed circumventing the theoretical issues associated with the band structure byconsidering the pairing problem using a tight binding model with site energies and hoppings renormalized with respectto DFT so as to reproduce the ARPES and QO data. They showed that adding phenomenological orbital orderingterms to the Hamiltonian allowed one to do so at both low and high T , and used the model to calculate NMRproperties in excellent agreement with experiment. They then took the tight-binding model and showed that it ledvia the conventional RPA spin-fluctuation pairing approach to an anisotropic s ± superconducting gap on the (stronglynematic) low T Fermi surface with nodes on the d xz /d yz electron bands, giving a penetration depth and V -shapedSTM spectrum quite similar to experiment. The nodes found were quite weak (shallow gap extrema), consistentwith a recent experiment on FeSe twin boundaries where the nodes were shown to be easily lifted . Recently, thisapproach was also shown to account well for inelastic neutron scattering data on FeSe .
2. Alkali-intercalated FeSe
The standard model for s ± pairing in Fe-based systems discussed above assumes the existence of a hole pocket andan electron pocket separated in the Brillouin zone by large q , as indeed found in most of the FeSC systems, includingbulk FeSe. Within this picture, repulsive interband Coulomb interactions yield a peak at this wavevector in themagnetic susceptibility, which in turn plays the essential role in a spin-fluctuation exchange pairing interaction. Thesystem can lower its energy however only if the superconducting order parameter changes sign between the two Fermi8 FIG. 5. (Color online.) Fermi surfaces of a) (Li . Fe . )OH FeSe ; b) FeSe monolayer on STO. ; c) (Tl,Rb)Fe Se measured byARPES. Note the apparent weight of the Γ-centered hole band in (c) is due to a broad band whose centroid disperses belowthe Fermi surface. From Ref. 186. surface sheets ? . The early expectation that all FeSC could be described by this single paradigm was graduallyeroded by the discovery of several Fe chalcogenide systems which are missing the characteristic Γ-centered Fermisurface holelike pockets altogether. It was immediately suggested that a new mechanism of pairing must be at work,or at least a new symmetry or structure of the order parameter must result.The discussion of a new paradigm for pairing in the FeSC began in 2011 with the discovery of relatively high-temperature superconductivity ( T c (cid:38) , which nominally correspondto the chemical formula AFe Se , with A=K,Rb,Cs. As of this writing, the superconducting samples of all materialsare available only in mixed-phase form and have relatively low superconducting volume fractions; thus the exactcomposition of the superconducting materials are unknown. They nevertheless excited considerable interest due totheir proximity to unusual high-moment block antiferromagnetic phases, and because ARPES measurements onKFe Se reported that there were no Γ-centered hole pockets at the Fermi level (although a small electron pocket isfound near the Z point near the top of the Brillouin zone); at Γ the hole band maximum is ∼
50 meV below theFermi surface. An example of one of the ARPES-determined Fermi surfaces of these materials is shown in Fig. 5c).Note that there is some spectral weight at the center of the zone, but that the centroid of the band is well beneaththe Fermi level.Several groups pointed out that despite the missing hole pockets, repulsive interactions at the Fermi level remainedamong the electron Fermi surface pockets, and could lead to d -wave pairing with significant critical temperatures .In addition, the expected hybridization of the two electron bands in the proper 122 body centered tetragonal crystalsymmetry leads to 2 hybridized electron Fermi surface pockets at the M point in the 2-Fe Brillouin zone, withconcomitant strong interactions between them. Mazin suggested that these interactions could lead to a newtype of s -wave state (“bonding-antibonding s -wave”) with sign change between inner and outer hybridized electronpockets. Such a state was not found to be competitive with the d -wave state in such a situation due to the weakhybridization in calculations based on the DFT bandstructure for KFe Se , but the bonding-antibonding s -waveremains an interesting candidate in part because this system is apparently intrinsically inhomogeneous and its exactelectronic structure is unknown.As mentioned briefly above, inelastic neutron scattering measurements agree rather well with the wave vector ∼ ( π, π/
2) for the inelastic neutron resonance predicted in Ref. 146, which correponds e.g. to the nesting of theinternal sides of the electron pockets at ( π,
0) and (0 , π ) in the 1-Fe zone. On the other hand, the d -wave state foundhere appears to disagree with the absence of nodes on the small Z -centered pocket reported by ARPES . Pandeyet al. argued that the bonding-antibonding s wave state would in fact support a resonance at roughly the wavevector observed in experiment. While this is natural in 2D, since the bonding-antibonding state is essentially a folded d -wave state stabilized by hybridization, it is less obvious in 3D, and again requires significant hybridization that isnot present, at least in DFT.The uncertainties associated with this system, including the materials issues, have left the question open. Recently,yet another solution was proposed by Nica et al as a way of reconciling neutron and ARPES experiments, byessentially incorporating both s and d -wave symmetry into different orbital channels, so that different symmetry9channels effectively dominated different Fermi surface sheets. The resulting mixed state was shown to be stablewithin a t − J − J mean field calculation over some range of parameters.
3. FeSe monolayers
The same states discussed in the alkali intercalates of FeSe have arisen in discussions of FeSe monolayers grownepitaxially on strontium titanate, STO . Of course the spectacular aspect of these films is their high T c , as muchas 70K in ARPES measurements , and 110K in in situ transport measurements . Fig. 6 (a) shows the epitaxialstructure of the film used originally to obtain a transport T c four times higher than bulk FeSe (8K), (b) the subsequentARPES gap closing temperature of 65K measured on similar samples, and (c) the more recent T c > T c in the last case is animportant hint, since a true 2D system would be expected to show the thermal broadening at a Kosterlitz-Thoulesstransition. In their annealed state, the STO films are conducting, and it is suspected that the high temperature3D current distribution becomes abruptly 2D as the SC transition is passed. This is one of many indications thatthe STO substrate is playing a crucial role. Others include the fact that FeSe on other substrates (graphene, forexample) produce superconductivity very similar to the bulk , and the remarkable observation that, while a onelayer FeSe/STO system is a high- T c superconductor, a 2-layer FeSe/STO system is not a superconductor at all . The ARPES measurements have carefully studied how the high- T c superconducting state evolves out of the as-grown sample, and how the requisite Fermi surface evolves with it. At the beginning of the annealing process, thecentral hole band imaged by ARPES is at the Fermi level, but by the end it has moved 80 meV below, and the electronpockets have enlarged. It is believed that this effective doping arises mostly from O vacancies in the STO , sinceSe vacancies in the film itself appear unlikely . The Fermi surface has only (apparently unhybridized ) electronpockets at the M points (Fig. 5(c)). Note that the Fermi surface obtained by standard DFT calculations is found tobe incorrect, even if the system is electron doped “by hand” using rigid band shift or virtual crystal methods, andeven accounting for the strained lattice constant imposed by the STO. So a pressing challenge is how to understandthe origin of the electronic structure in this case, presumably by incorporating O defects in a proper way includingdisorder effects and/or by properly including correlations.An important clue to the physics of these systems, and the influence of the substrate, was recently presented bynew ARPES measurements , which showed the existence of “replica bands” (nearly exact “shadow” copies of bandsboth at Γ and M shifted downward in energy by ∼
100 meV). These experiments indicate the presence of a largeelectron-phonon interaction, probably originating from the substrate . It has recently been argued in additionthat the electron-phonon interaction must be quite strongly peaked near momentum transfer q =0 to explain thisobservation .Both STM and ARPES agree that the monolayer system has a full superconducting gap, of order 20 meV. It isintriguing that STM reports two coherence-like peaks in the spectrum, although the hole band is presumed not toparticipate in superconductivity, and the two electron pockets at M in ARPES show no evidence of hybridization .It has been suggested that the two scales might indicate some substantial anisotropy (but no nodes) on the electronpockets, but, at least within BCS theory, the gap minimum does not lead to a singularity in the DOS. For this andother reasons, if one considers only the band structure at the Fermi surface, high- T c and the form of the SC gap arepuzzling. Despite the apparent role of phonons in the electronic structure, electron-phonon interactions in the FeSeare unlikely to be strong enough to explain a T c of 70K or above (a calculation that disagrees, and finds a muchhigher T phc than others under some rather generous assumptions is Ref. 206).Thus a “plain” s -wave from attractive interactions alone seems improbable, even if soft STO phonons play arole . The forward scattering character of the relevant phonon processes then implies that phonons cannotcontribute to the interband interaction. Taken by itself, however, the interband spin fluctuation interaction shouldlead to nodeless d -wave (since χ ( q , ω ) will be roughly peaked at the momentum connecting the electron pockets),similar to the arguments given for alkali-intercalates . On the other hand, there is some evidence that thesystem does not have a sign-changing gap. In STM measurements by Fan et al. T c and the gap were reported tobe suppressed only by magnetic impurities, as one might indeed expect from a “plain” s -wave SC. These arguments,if correct, would also rule out states of the “bonding-antibonding s -wave” type .A final possibility is the “incipient s ± ” state, which naiviely seems disfavored by energetic arguments. RecentlyBang and Chen et al. revisited these arguments and found that that in the presence of preexisting Fermi surfacesuperconductivity (e.g. a phonon attraction in the electron pockets of the monolayer) this state was quite stronglyfavored. D.H. Lee and co-workers have elaborated on a scenario for high- T c in FeSe monolayers in whichSTO phonons assist spin fluctuations, but the origin of the spin fluctuations near ( π,
0) was never clear due to theincipient hole pockets. With the current picture of Chen et al. this now becomes a clear candidate, along with0the d -wave. The bonding-antibonding s -wave state is also a possibility, but ARPES sees no hybridization of the twoelectron pockets that would stabilize this state . Some details of these arguments are reviewed in Sec. VI C.Of course, one still needs to understand the objections of Ref. 208 regarding the effect of impurities. All the abovecandidates involving spin fluctuations at either ( π,
0) or ( π, π ) exhibit a sign change. It may be, however, that theexpected pairbreaking effects in a sign-changing incipient s -wave states are weaker because impurity scattering iselastic . FIG. 6. (Color online.) (a) Resistivity of monolayer FeSe films on STO ; (b) Spectral gap measured by ARPES on suchfilms. From Ref.198; (c) Resistivity of newer films of FeSe/STO .
4. Other FeSe intercalates
Speculation on the origin of the higher T c in the alkali-intercalated FeSe centered on the intriguing possibilitythat enhancing the FeSe layer spacing improves the two-dimensionality of the band structure and therefore mightenhance Fermi surface nesting. To explore this effect, organic molecular complexes, including alkali atoms, weresuccessfully intercalated between the FeSe layers . The extremely air-sensitive powders synthesized had tran-sition temperatures up to 46 K. Until recently, the most intensively studied materials intercalated ammonia, e.g.Li . (NH ) . (NH ) . Fe Se with T c = 39 K with Li . (NH ) . (NH ) . Fe Se with T c = 44 K . Noji etal. correlated data on a wide variety of FeSe intercalates and noted a strong correlation of T c with inter- FeSe layerspacing, with a nearly linear increase between 5 to 9 ˚A, which then saturated between 9 to 12 ˚A. This tendencywas attributed to a combination of doping and changes in nesting with increasing two-dimensionality by Guterdinget al. .While the ammoniated FeSe intercalates are fascinating, their air sensitivity prevented many important experimentalprobes and limited their utility. Recently the discovery of a new class of air-stable FeSe intercalates, lithium ironselenide hydroxides, was reported in Ref. 222. Remarkably Li − x Fe x (OH)Fe − y Se has also been shown to have a Fermisurface without Γ-centered hole pockets. (Fig. 5 (a) ). It was recently pointed out that another similarity of thesesystems with the FeSe/STO monolayers is a double coherence peak in STM , with extremely large inferred gap- T c ratios of order 8. In addition, these authors reported detailed QPI measurements consistent with two hybridizedelectron pockets, which they associated with the two gaps. The further observation of an in-gap impurity resonanceat a native (Fe-centered) defect site suggests that the gap is sign changing. This is because the defect is either an Fevacancy or possibly a chemical substituent on this site, presumably not magnetic. The SC order parameter thereforeappears to have a bonding-antibonding s -wave structure. Further research on this system is certainly needed, andmay indeed provide insight into the other systems shown in Fig. 5. C. Incipient band pairing
The observation of high- T c in systems with missing hole pockets has been a challenge to the established s ± paradigm,which naively requires (at least) two Fermi surface pockets separated by large q . In their study of KFe Se , Wanget al. found however that pairing in an s ± channel, with gap sign change between a gap on the regular electronband at the Fermi surface and the incipient hole band ∼
80 meV below the Fermi level was surprisingly competitivewith the expected d -wave state. This ”incipient s ± ” state was treated as a candidate for pairing in these systems inRef. 5, along with the (quasi-nodeless) d -wave state and the “bonding-antibonding” s -wave state that changed signbetween two hybridized electron pockets taken in the 2-Fe zone. It did not receive a great deal of attention, however,1presumably because of the general prejudice that incipient bands (those that nearly cross the Fermi level) do not playa significant role in superconductivity.Interest in pairing in incipient bands in Fe-based superconductors was revived by the experiments on FeSe/STOmonloayers showing an absence of Γ-centered hole pockets , and later by the observation by He et al of thepersistence of the superconducting gap on one of the hole bands of LiFeAs as it sank below the Fermi level withelectron doping by Co. The gap on this band was only weakly suppressed in this process, at least up to bandextremum values of E g ∼
10 meV (Fig. 8), and probably significantly further. Moreover, there was no indication ofany abrupt change in either T c or the gap on the incipient band through the Lifshitz transition, and the gap on theincipient band was the largest gap on the Fermi surface. FIG. 7. ((Color online.) a) Case(I)-A: Four examples of a single hole band corresponding to (1) regular band (2) shallowband (3) incipient band (4) vegetable band depending on position of band maximum with respect to Fermi level and rangeof attractive energies between ± ω D (blue region). (b) Case (I)-B: Spin fluctuation driven pairing on incipient band (cut-off ω sf -yellow region) SC. (c) Case (II)-A: SC is driven by spin fluctuations in the regular (blue) bands, and SC in the incipientband is induced by the same interaction. (d) Case(II)-B: SC driven by phonons in the regular (blue) band, and induced viaspin fluctuations in the incipient band. Bang had earlier pointed out that T c could remain large in a situation where superconductivity was stabilized byan intraband attractive interaction . In an attempt to explain the results in LiFeAs, Hu et al. calculated theeffect of the incipient band using a 3-orbital model with next-nearest neighbor BCS pairing ansatz, and found thatthe qualitative aspects of the experiment could be understood, but only if the pairing interaction exceeded a criticalvalue. Chen et al. criticized this approach as a nongeneric formlation of the usual multiband problem, and arguedthat weak coupling – without a threshold interaction – was sufficient to explain the experiment. Within Eliashbergtheory, Leong and Phillips studied a 5-band model for LiFeAs where the incipient band was coupled to the othersonly by Coulomb interactions, and other Coulomb interactions were neglected; using this approach, they showed thatthe energy dependence of the Coulomb interactions gave rise to a large gap on the incipient band. However, generalconclusions were hard to draw due to the complexity of the model.Here I follow the simple approach of Ref. 210 and consider (Fig. 7) several different situations with a minimalnumber of bands, showing how they differ qualitatively. The first, depicted in Fig. 7(a), is the well-known case of asingle incipient band with band extremum E g within the BCS cutoff energy ω D , with gap equation1 = − mV ph π (cid:34)(cid:90) E g − ω D d(cid:15) E tanh E T (cid:35) , (11)where V ph is a putative attractive interaction due to phonons. The single band case also has a critical value of E g where SC disappears before the lower cutoff − ω D is reached, given by E crit g = − ω D e vph , (12)with v ph = mV ph / π . In fact, since the spacing between pairs in the 1-band case becomes larger than the Cooper pairradius at some point approaching the Lifschitz transition in case (I)A, one enters the BCS-BEC crossover regime inthe low density limit. If this occurs, T c should vanish at the Lifschitz point if the problem is treated properly .Cases (I)B and (II)A are shown in Fig. 7(b,c) respectively, and can be discussed with a simple multiband model ofthe sort discussed in section IV: ∆ e = − V sf ∆ h L h − V sf ∆ h L h , (13)∆ h = − V sf ∆ e L e , (14)∆ h = − V sf ∆ e L e . (15)2where L h = (cid:90) ω sf − ω sf d εN h tanh E h T E h ,L e = (cid:90) ω sf − ω sf d εN e tanh E e T E e ,L h = (cid:90) E g − ω sf d ε m π tanh E h T E h , (16) m is the mass of the incipient band h . Here V sfα with α = 1 , e and the two hole bands h and h of Fig. 7(c). Note that the definition of L ν here includes thedensity of states N ν . The case depicted in Fig. 7(b) can be obtained from these equations by dropping the hole band h . Case (II)B shown in Fig. 8(d) is slightly more complicated since in principle it involves two energy scales forpairing, which may be thought of for convenience as a spin fluctuation cutoff and phonon cutoff. Ref. 210 obtainedanalytical and numerical solutions for these equations and showed that • in the single band incipient case T c is strongly suppressed as soon as E g falls below the Fermi level. There is acritical value of E g below which no superconductivity is possible. • these thresholds do not exist in other cases. However, in case (I)B, where superconductivity is driven by aninterband repulsion connected to the incipient band, T c is still strongly suppressed as the band falls below theFermi level. This is the intuition that seems to have guided the conclusions of Ref. 226. • when superconductivity existed before the incipient band was considered, either due to a finite q spin-fluctuationinteraction ((II)A), or due to an intraband phonon-like attraction ((II)B), the introduction of the incipient bandsubstantially assisted T c , and induced a large gap on the incipient band. • In the latter case, there is no abrupt change in T c or gaps as the Lifshitz transition is crossed, and the inducedgap on the incipient band can easily be of the same magnitude or larger than gaps on regular bands at the Fermisurface, depending on details of interactions.In Figure 8, the essential data from the experiment Ref. 226 are plotted alongside the T c suppression for different E g as the Lifshitz transition is crossed found from the equations of Ref. 210 for cases (I)A, (I)B, and (II)A. FIG. 8. (Color online.) (a)-(c) adapted from Ref. 226. (a,b) Band measured by ARPES near Γ point in LiFe − x As at high(30K) and low (8K) for 1% (a) and 3% Co. (c) Plot of gap measured by pullback of leading edge of ARPES EDC intensityvs. Co doping, together with energy of band maximum below the Fermi level E g vs. doping. Line labelled “weak coupling”is apparently sketched under the assumption that incipient gap should behave as in case (I) in Fig. 7. (d) adapted from Ref.210. Plot of T c /T c for interactions v ph = 0 . v sf = 0 . v sf = 0.2; v sf =0.3 (green). Inset: gap on both Fermisurface pockets ( h and e ), together with gap on incipient band ( h ), vs. E g / Λ. proposedthat an s ± state of this type might represent a possible candidate for the ground state of FeSe monolayers on STO,with a large gap on both the electron pockets at the Fermi surface band and the incipient hole band well below it. Thespin fluctuations were shown capable of significantly (factor 10, depending on ratios of phonon and spin fluctuationinteractions and cutoffs) enhancing a weak phonon T c . Due to the sign change, the s ± state naively has a difficultywith the results of Ref. 208. However, Chen et al. anticipated that since impurities scatter elastically, pairbreakingeffects in an incipient band might not be so effective. VII. MULTIPHASE GAPS AND TIME REVERSAL SYMMETRY BREAKING
The close proximity of s and d pairing channels in spin fluctuation theory led Lee et al. to propose that a mixedsymmetry ground state, a so-called s + id state, could be stablilized, reviving a proposal that had arisen in the cupratecontext some years before, but had never been realized. Note that such a state is a mixture of two distinct irreduciblerepresentations of the symmetry group of the system, and as such should lead generically to two T c ’s except at asingle point of doping or whatever parameter tunes the transition. The Ginzburg-Landau free energy studied by Leeet al., identical to the the earlier works, was shown to have a minimum with 90 ◦ phase difference between s and d , depending on the sign of the β quartic mixing coefficient. And of course the same interesting phenomena wereidentified: locally C symmetry-breaking spontaneous current patterns near impurities and edges, and the predictionof a resonance mode corresponding to the oscillation of the relative phase of s and d , predicted to appear in the B g polarization of a Raman scattering experiment. The spontaneous currents are difficult to measure, because theyare not globally chiral and there is no direct coupling to observables measurable by STM. On the other hand, theRaman mode should be observable, and is in fact exactly analogous to the mode predicted by Balatsky et al. in the cuprate context. Neither of these calculations went beyond the simplest GL approximation to work in thelow- T phase where s + id states are actually stable, however. The microscopic origin of the phase stability and theRaman-active mode itself were therefore unclear. Using a realistic five-orbital model appropriate for 1111 systems,and local Hubbard-Hund interactions treated with the functional renormalization group, Platt et al. showed thatan s + id state could be obtained as the ground state for intermediate electron doping range close to where the holepockets disappear. There appears to be no analogous calculation for the strongly overdoped hole case, where, asdiscussed above, there is evidence from Raman scattering of a strong d -wave interaction implied by the observationof a subgap Bardasis-Schrieffer type mode.These studies considered as the “ s ” component of s + id the standard s ± state found in standard spin fluctuationcalculations for systems with hole and electron pockets. However, from the symmetry point of view any s -typerepresentation is possible, and Khodas and Chubukov studied the combination with the bonding-antibonding s ± state arising from the hybridization of two electron pockets in the context of KFe Se . They found the bonding-antibonding s ± stabilized with sufficently large hybridization, which they considered as a parameter. In the weakhybridization limit, they also found that the d -wave was stable; and at the border between these two phases, an s + id state was favored. The identification of the symmetry of this and related systems is still controversial. On the onehand, inelastic neutron scattering measurements appear to give a resonance peak very close to the incommensuratewave vector predicted by the spin fluctuation theory calculations giving d -wave ; on the other, the nodes on thetiny Z -centered electron pocket expected for the d -wave case were not observed by ARPES .Finally, again following the cuprate example, the notion that an s + id state can be induced as a result of gradientterms in the free energy that mix the two representations even if only one is stable in the homogeneous case, has beenproposed. High quality FeSe crystals, orthorhombic at low T , exhibit twin boundaries that can be imaged in STM.Recently Watashige et al. measured the STM spectrum in the superconducting state as a function of distanceto the twin boundary. They found a V -shaped gap in the bulk, indicative of nodes, but a U -shaped full gap thatopened smoothly as the twin boundary was approached. This full gap was largest when measurements were madebetween two parallel twin boundaries; the simultaneous proximity to both evidently magnified the effect. On the otherhand, this interference of the two twin boundaries took place over a length scale corresponding to nearly 7 times thelow-temperature bulk coherence length, as measured by standard means. Under normal circumstances, perturbationsto the electronic structure should influence superconductivity over a few coherence lengths at most. Watashige et al.appealed to an idea put forward in the cuprate context by Sigrist et al. , that a new length scale can appear closeto the critical point where an s + id state is created from the pure s or d state, corresponding to the coherence lengthof the subdominant order parameter component which appears at the 2nd order transition, and argued that theirresults were indirect evidence for T -breaking.It is interesting to pose the question how best to identify the transition from an s -state to an s + id state, bothof which are fully gapped, as might be realized, e.g. in K − x Ba x Fe As . The quasiparticle spectrum will not changein any distinctive way at the transition, the impurity signatures are weak , and there are no edge states or other4 FIG. 9. (Color online.) (a) Phase diagram of a 3-pocket model appropriate for hole-overdoped FeSC, from Ref. 233. (b)Evolution of the gap components with fixed u s and varying u d in the 3-pocket model at T = 0. Red is for the hole pocket andblue is for the electron pocket(s). The solid line is the s − wave part and dashed is the d − wave part. The grey line at zero isthe size of error in the numerical calculation due to the the choice of grid and resolution parameters. (c): The (undamped)collective modes across the phase diagram in different channels. “MSBS”= mixed symmetry Bardasis Schrieffer mode; “BAG”= Bogoliubov-Anderson-Goldstone mode (neutral analog system). The dashed black line is 2∆ min in the system. The dotsindicate the s + id phase boundaries at T = 0. In all the figures z = 1 / u s ν D = 0 . chiral phenomena predicted. Perhaps the best method is that proposed in Ref. 233, to look in a Raman scatteringexperiment for the softening of the Bardasis-Schrieffer mode at the phase boundary. In the s -ground state, the BSmode is a pure d -exciton, and should show up only in the B g Raman polarization; however, because it is a mixedsymmetry mode in the s + id phase, it should begin to show up in both A g and B g polarizations as the s + id phaseis entered, and subsequently harden (See Fig. 9).In parallel with the discussion regarding s + id , it was realized that the multiband aspects of the Fe-based systemallowed for a new type of T -breaking state. Neglecting all angular dependence of the effective interband repulsions,Stanev and Tesanovic considered a simple model with three bands, and showed that it had, in addition to “conven-tional” s -wave generalizations of the two-band s ± , a possible ground state which minimized the frustration associatedwith the additional pocket by acquiring an internal nontrivial phases between the different gaps. A possible “ s + is ”phase of this type is shown in Fig. 1 (c), where the gaps on the three bands are ∆ ∼ e iφ/ , ∆ ∼ e − iφ/ are complex,but ∆ is real. Here φ is a phase angle that depends on interactions, and evolves between 0 and π as a functionof these interactions, or doping. Such a state was proposed as a natural “intermediary” between a standard s ± state for the optimally doped Ba − x K x Fe s As , where both central hole pockets have the same sign, and the statefound from from spin fluctuation phenomenology (low-order harmonic parametrization of the interband interactions)for KFe As , where the two gaps on the hole pockets have opposite sign . In Fig. 10, their conclusions are sketchedin the form of a generic phase diagram for a simple multiband model. This state was also one of several consideredby Ahn et al. in the context of LiFeAs (Sec. V B).Since the expected local current signatures of a spatial perturbation in an s + is state are even weaker than in s + id (the net current vanishes in both states since they are not chiral, but it vanishes even locally in the s + is case unless orthorhombic symmetry is broken), distinguishing such states will be difficult. The best possibility atthe present time appears to be measuring characteristic collective modes. Since the system is by assumption at least3-band, there are several possible Leggett-type modes that soften at the phase boundaries to the pure (real) s -wavestates . It has also been proposed that s + is states would have magnetic signatures at domain walls that couldbe detected by scanning SQUID probes .In a realistic system with several hole bands at the Fermi surface like LiFeAs, the ground state is a subject ofconsiderable controversy even among groups that share basic assumptions about its electronic structure and thefundamental origin of the pairing. This disagreement is a good illustration of the sensitivity of theory in a multibandsystem to small changes in electronic structure. VIII. η -PAIRING Although much theoretical work on the pairing state of FeSC has been done using simplified models based on a 1-Feunit cell containing 5 d orbitals, it has not always been clear why this procedure works quite well. In fact, there areseveral claims in the literature that the reconstruction of the near-Fermi level states by the As(P,Se...) potential whichbreaks the translational symmetry of the Fe lattice has important consequences for the pairing of electrons, despite thefact that the As states are nominally 2eV below. These works often imply that the early pairing calculations missed5 FIG. 10. (Color online.) Qualitative phase diagram for Ba x K x Fe As , from Ref. 237. (a) phase diagram as function of ratio U he /U hh of interband to intraband interactions, for case where two hole pockets h and h are degenerate. “TSRB” indicatestime-reversal symmetry broken s + is phase. Arrows indicate phases of ∆ (red), ∆ (blue), and ∆ (green). (b) Phase diagramwith nondegenerate hole pockets. an essential symmetry aspect of the problem because they did not explicitly account for the glide plane symmetry ofthe Fe-As layer (space group P4/nmm) . In particular, it was suggested by Hu et al. that the “standard” spinfluctuation approach misses a spin singlet odd-parity pairing state, which may represent the ground state of FeSC ingeneral.A pure version of a state of this kind, as discussed in Sec. II, would indeed be quite different from any statesobserved in unconventional superconductors to this point, and would be of great interest if actually realized. InHu’s work, such states are shown to occur naturally if one properly respects the glide symmetry, allowing nonzerocenter of mass crystal momentum Cooper pairs (called η -pairing in the cuprate context by C.N. Yang , see alsoRefs. 244,245). Casula and Sorella pointed out that the two-orbital models used in Hu’s work were insufficient torecover the full generality of pairing states formally possible in the P /nmm symmetry of a single FeAs plane, sincethese models neglected pairing of orbitals of unlike z-parity under reflection through the Fe plane. Including suchstates in a variational Monte Carlo calculation led to a linear combination of “planar” d - and s − states, i.e. thosecorresponding to the irreducible representations of the tetragonal group conventionally used; such admixtures provide,in their picture, a natural explanation for so-called “nematic” C symmetry-breaking in the superconducting state, asseen in STM on FeSe (see however Sec. VI A 2). Finally, there are even recent theoretical suggestions suggestingthat the odd-parity singlet state can explain the so-called “ARPES paradox” , the apparent disagreement betweenARPES and a variety of thermodynamic and transport probes concerning the anisotropy of the superconducting gapin 122 systems, and may in addition break time reversal symmetry .The implications of the glide plane symmetry of a single Fe-pnictogen/chalcogen plane for various normal stateproperties as well as for pairing was discussed in several works : The Fe lattice is symmetric underthe glide plane operation P z = T r σ z , i.e., a single translation along the x - or y -direction T r by 1 Fe-Fe distance,together with a reflection σ z along z . As a result, the intra-sub-lattice hopping between d orbitals “even” under P z ( xy, x − y , z − r ) and “odd” orbitals ( xz, yz ) changes sign betwen the A- and B- Fe sublattices. If onelabels states by the “physical” 1-Fe crystal momentum k , it forces a mixing between momenta k and k + Q with Q = ( π, π ) of the type (cid:80) k ,σ (cid:104) t xz,xy ( k ) c † xz,σ, k + Q c xy,σ, k + h . c . (cid:105) , and similar terms, between even and odd (with respectto z -parity) orbitals. As a result, one is forced to consider off-diagonal propagators with both even and odd orbitalstates, evaluated at momenta k and k + Q . At first glance, this is an important difference in the representation of pairstates compared to the usual BCS ansatz, because one now has nonzero total momentum η -pairs (cid:104) c (cid:96) , ↑ , k c (cid:96) , ↓ , − k + Q (cid:105) for (cid:96) even, (cid:96) odd or vice versa, in addition to the standard zero center of mass momentum pairs (cid:104) c (cid:96) , ↑ , k c (cid:96) , ↓ , − k (cid:105) for (cid:96) , (cid:96) either both even or both odd. . However, this mixing is absent if one uses the eigenvalues of P z , i.e., thepseudo-crystal momentum ˜ k to classify the states , which simply amounts to shifting the momentum ofeither the even or the odd orbitals by Q .In Ref. 252, Wang et al. studied the problem of η -pairing in models of a FeAs layer with both a 5 and 10 d -orbitalbasis, assuming spin-fluctuation driven superconductivity with pairing interaction (5). They chose the shift Q in theeven orbitals, so that states labelled by pseudo-crystal momentum ˜ k are related to the states labelled by physical6crystal momentum k via ˜ c (cid:96),σ, ˜ k = (cid:40) c (cid:96),σ, k , if (cid:96) odd, c (cid:96),σ, k + Q , if (cid:96) even. (17)The tilde (˜) basis is now identical to that used to perform pairing calculations in the 5-orbital model, where theexact unfolding symmetry was assumed. The Hamiltonian automatically accounts for the additional terms arisingfrom the mixing between k and k + Q in the physical 1-Fe crystal momentum k space.One now simply uses the basis transformation to relate the pairing condensate eigenfunction g ( k ) in the tworepresentations: g N(cid:96) (cid:96) ( k ) = (cid:104) c (cid:96) ↑ , k c (cid:96) ↓ , − k − c (cid:96) ↓ , k c (cid:96) ↑ , − k (cid:105) = (18) ˜ a (cid:96) ν, k ˜ a (cid:96) ν, − k ˜ g ν ( k ) , (cid:96) , (cid:96) odd,˜ a (cid:96) ν, k − Q ˜ a (cid:96) ν, − k + Q ˜ g ν ( k − Q ) , (cid:96) , (cid:96) even,0 , otherwise,and η -pairing terms shifted by center of mass momentum Q , g η(cid:96) (cid:96) ( k ) = (cid:104) c (cid:96) ↑ , k c (cid:96) ↓ , − k + Q − c (cid:96) ↓ , k c (cid:96) ↑ , − k + Q (cid:105) = (19) ˜ a (cid:96) ν, k ˜ a (cid:96) ν, − k ˜ g ν ( k ) , (cid:96) odd, (cid:96) even,˜ a (cid:96) ν, k − Q ˜ a (cid:96) ν, − k + Q ˜ g ν ( k − Q ) , (cid:96) even, (cid:96) odd,0 , otherwise.Using (18) and (19) Wang et al. showed that the gap function ˜∆ ν ( k ) ∝ g ν ( k ) obtained in the 5-orbital modelin the pseudo-crystal momentum representation contains both normal and η -pairing terms in the physical crystalmomentum space, i.e., ˜∆ ν ( k ) = ∆ N odd ( k ) + ∆ N even ( k + Q )+ ∆ η odd-even ( k ) + ∆ η even-odd ( k + Q ) . (20)Thus the η -pairs are seen to be indeed present in FeSC, but they form a natural part of the usual gap function thathas been discussed thus far in the literature. Closer examination of the η terms show that they are imaginary and oddparity singlet states only when viewed in the orbital basis ; they are real and have the usual even parity when viewedin band space.To see explicitly that there are no direct observable consequences beyond the models discussed already in Sec. V,one can construct the one-particle spectral function in the physical momentum space, as observed by ARPES , A ( k , ω ) = (cid:88) ν (cid:34) (cid:88) (cid:96) odd | ˜ a (cid:96)ν, k | ˜ A ν ( k , ω )+ (cid:88) (cid:96) even | ˜ a (cid:96)ν, k − Q | ˜ A ν ( k − Q , ω ) (cid:35) , (21)where ˜ A is the spectral function corresponding to the pseudomomentum basis, i.e. containing poles at the energies E ν (˜ k ) = (cid:113) (cid:15) ν (˜ k ) + ˜∆ ν (˜ k ). Therefore, the superconducting gap that enters A ( k , ω ), as measured in ARPES experi-ments is given by the gap function ˜∆ ν (˜ k ) calculated in the 5-orbital 1-Fe zone in pseudo-crystal momentum space.˜∆ ν (˜ k ) thus accounts completely for the symmetry breaking potential caused by the pnictogen/chalcogen atom .Thus Wang et al. concluded that η -pairing is indeed an important ingredient in the superconducting condensate,when properly defined. These terms occur as a natural part of the gap in the standard 5-orbital theories, whichcoincide identically (for such single layer models) with 10-orbital calculations . They contribute, however, with thestandard even parity spin singlet symmetry in band space , and time reversal symmetry is not broken.7 IX. DISORDER
In principle, disorder is a phase-sensitive probe of unconventional superconductivity, because nonmagnetic impuritiesare pairbreaking only if they scatter electrons between portions of the Fermi surface with gaps of different sign. Boundstates of electrons near strong nonmagnetic impurities (Andreev bound states) can similarly be formed only in thesign-changing case. This subject has a long history in the context of d -wave superconductivity in the cuprates whichhas been reviewed several times . Many of the ideas worked out in this context have found interesting extensionsin the FeSC field, because of the added aspect of multiband physics. A. Intra- vs. interband scattering
In 2-band superconductors, nonmagnetic impurities can either scatter quasiparticles between bands or within thesame band. Intraband processes tend to average the gaps and can thus lead to some initial T c suppression, buteventually T c will saturate, according to Anderson’s theorem . However interband scattering is pairbreaking in asystem with two gaps of different signs , just as it is in single-band systems where the order parameter has nodes,e.g. the d -wave case. In the presence of any nonzero interband scattering amplitude, T c will eventually be suppressedto zero at some critical concentration, just as in the more familiar theory of scattering by magnetic impurities in a1-band s -wave superconductor . The effect of interband scattering in the FeSC was discussed early on by severalgroups .The simplest toy model of a chemical impurity in 2-band system characterizes the scatterer crudely by an effectiveinterband potential u and an intraband potential v , such that results for various quantities in the superconductingstate will depend crucially on the size and relative strengths of these two quantities. Most calculations are performedusing the T -matrix approximation to the disorder self-energy for pointlike scatterers ˆΣ imp ( ω ),ˆΣ imp ( ω ) = n imp ˆ U + ˆ U ˆ G ( ω ) ˆΣ imp ( ω ) , (22)where ˆ U = U ⊗ ˆ τ , n imp is the impurity concentration, and the τ i are Pauli matrices in particle-hole space. Here U is a matrix in band space ( U ) αβ = ( v − u ) δ αβ + u , and α, β = 1 ,
2. With the expression for the k -integrated Green’sfunction ˆ G ( ω ) = (cid:80) k ˆ G ( k , ω ), this completes the specification of the equations which determine the Green’s functionsˆ G ( k , ω ) − = ˆ G ( k , ω ) − − ˆΣ imp ( k , ω ) , (23)where ˆ G is the Green’s function for the pure system. The self-consistent T -matrix approximation includes thosescattering processes corresponding to multiple scattering from a single impurity, and may be considered correct fordilute impurities (except possibly for nodal systems in 2D, where it has some singularities at low energies ). B. T c suppression Several groups have pointed out what is claimed to be a “slow” decrease of T c in systematic chemical sub-stitution measurements , which is then claimed to be related to the natural robustness of an s ++ superconductorto disorder. The meaning of “fast” and ”slow” T c suppression can be deceptive, however, especially in a situationwhen only large momentum scattering is pairbreaking, as expected for an s ± state. Note that chemical substitutioncan cause pairbreaking, but also affect the electronic structure of the system by doping with carriers or by chemicalpressure. It is therefore useful to calculate a measure of disorder that is simultaneously observable along with T c , suchas the residual resistivity, against which to compare T c suppression rates. This problem was addressed particularlyby Wang et al. , who argued that almost any rate of T c suppression was possible in an s ± system, depending onthe ratio of intra- to interband scattering. This conclusion is implicit in the discussion by Golubov and Mazin whodiscussed the general case of pairbreaking in a multiband superconductor.To see this, I note that near T c , the gap equation can be linearized,∆ µ ( k ) = − T ω n = ω c (cid:88) k (cid:48) ,ν,ω n > V µν kk (cid:48) ˜∆ ν ( k (cid:48) )˜ ω ν + ξ ν , (24)where ξ ν is the dispersion of band ν near the Fermi level, and the renormalized gaps and frequencies are ˜∆ ν ( k (cid:48) ) =∆ ν ( k (cid:48) ) + Σ ( imp, ν and ˜ ω ν = ω n + i Σ ( imp, ν .8To investigate both the effects of sign-changing gaps, and that of possible gap anisotropy, let us consider a modelsimilar to that obtained from spin fluctuation theories, where the gap on pocket a is isotropic, ∆ a , and the gap on thepocket b may be anisotropic, ∆ b = ∆ b + ∆ b ( θ ), where θ is the angle of k around the b pocket, with (cid:82) dθ ∆ b ( θ ) = 0.The pairing interaction is then assumed to be V µν kk (cid:48) = V µν φ µ ( k ) φ ν ( k (cid:48) ), with φ µ = 1 + rδ µ,b cos 2 θ . Note that thechoice of cos 2 θ harmonic implicitly means that b is identified with the electron pockets at X and Y, but this is easilygeneralizable. The parameter r > b , and yields gap nodes if r >
1. This ansatz thengives coupled equations for (∆ a , ∆ b , ∆ b ) T ≡ ∆ which may be written as∆ = ln (cid:16) . ω c T c (cid:17) M ∆ ≡ L M ∆ , (25)where the matrix M = (1 + V R − X R ) − V and the constant L = ln (cid:0) . ω c T c (cid:1) were introduced. V is the interactionmatrix in the above basis, and R is the unitary matrix which diagonalizesΦ = πn imp D N N b u − N b u − N a u N a u
00 0 N b v + N a u . (26)Here N µ is the density of states of band µ at the Fermi energy, D N is the determinant of the T -matrix in the normalstate, and X is a diagonal matrix, X ii = L − (cid:20) Ψ (cid:16)
12 + ω c πT c + λ i πT c (cid:17) − Ψ (cid:16)
12 + λ i πT c (cid:17)(cid:21) , (27)where Ψ is the digamma function and λ i are the eigenvalues of matrix Φ. The critical temperature is now determinedby finding the largest ( λ max ( T c )) of the matrix M , T c = 1 . ω c exp [ − /λ max ( T c )].These expressions now suffice to determine T c at any given impurity concentration n i , with intra- and interbandscattering strengths v and u given. However, since u and v are not known in general, it is not particularly usefulwhen comparing to experiment to present calculations of T c vs. n i , nor vs. some scattering rate component, simple tocalculate but difficult to measure directly. Claims in the literature that T c suppression is particularly slow, implyingno gap sign change, or particularly fast, implying one, abound. They are mostly not relevant, however; in order todraw conclusions in multiband systems, it is crucial to present results for T c vs. the residual resistivity ∆ ρ due todisorder. The total DC conductivity is the sum of the Drude conductivities of the two bands, σ = σ a + σ b , with σ µ = 2 e N µ (cid:104) v µ,x (cid:105) τ µ , where v µ,x is the Fermi velocity component in the x -direction and τ µ the corresponding one-particle relaxation time, τ − µ = − ( imp, µ . The transport time and single-particle lifetime are identical if thescatterers are δ -function like, such that corrections to the current vertex vanish.In Fig. 11 (a),(b), T c vs. residual resistivity is plotted for various gap structures and ratios of intra- to interbandscattering. The extreme cases include i) purely isotropic s ± gaps with intraband scattering, where Anderson’s theoremimplies that there can be no pairbreaking until localization begins to occur; ii) α ≡ u/v = 1, where the rapid T c suppression follows the universal curve predicted by Abrikosov and Gor’kov for magnetic impurities in an isotropic s -wave system. It is obvious that essentially any other rate of T c suppression can be obtained for some particular valueof the ratio α = u/v , such that the observed rate will depend on the range (and orbital character) of the impuritypotential itself. The role of anisotropy is also interesting, if not surprising. If one of the gaps is anisotropic, thisspeeds the initial rate of T c suppression even in the case of dominant intraband scattering, as the system averages outthe gap by scattering until it is roughly isotropic. Subsequent pairbreaking is due to the interband component of thescattering, which is sensitive to the gap sign change between the two pockets. This sequence will be discussed furtherin Sec. IX D below.In this context, the data on T c suppression by chemical substitution, e.g. those of Li et al (Fig. 11 (c)) have tobe treated with caution. While the critical resistivity for which T c vanishes is indeed large compared to the small ∆ ρ scale one might find for an s ± state and large interband scattering, the impurities are certainly doping the system, atleast in some cases. It is also somewhat bizarre to find in a putative s ++ state that Mn, certainly a magnetic impurity,suppresses T c at the same rate as Co, certainly not; such behavior is rather characteristic of an s ± state, where themagnetic character of the impurity is generically irrelevant. What is needed are measurements of T c as a function ofsome quantity tied directly to pairbreaking. This test is in fact provided by electron irradiation , since theseworks have shown that the main effect of low-energy electron irradiation is the creation of pointlike Frenkel-typepairs of Fe vacancies and interstitials, which act as nonmagnetic scatterers that do not dope the system. This is,incidentally, not the case with proton or heavy ion irradiation .In Figure 11(d), I show the T c data from the Ames group on Ru-doped BaFe As under electron irradiation, plottedvs. the residual resistivity ∆ ρ . The data clearly show a much more rapid suppression of T c with a much smaller9 α =1 α =0.7 α =0.5 α =0.2 α =0 . . . .
81 00 .
51 0 500 1000 T c / T c ∆ ρ ( µ Ω · cm) . . . .
81 0 20 40 60 8000 .
51 0 500 1000 T c / T c ∆ ρ ( µ Ω · cm) T c / T c T c / T c ∆ ρ ( µ Ω · cm)(a)(b) FIG. 11. (Color online.) a) Normalized critical temperature T c /T c vs. disorder-induced resistivity change ∆ ρ for isotropic s ± waveparing for various values of the inter/intraband scattering ratio α ≡ u/v . Insert: same quantity plotted over a larger ∆ ρ scale. b) Same as a) but for anisotropic (nodal) gap with anisotropy parameter r = 1 .
3. From Ref. 270. (c) T c suppression vs.residual resistivity in m Ω-cm for various transition metal impurities . (d) T c suppression vs. residual resistivity inRu-doped Ba122 irradiated with 2.5meV electrons . critical resistivity for the destruction of the superconducting state, roughly 40 µ Ω-cm, in fact very close to the valuepredicted but not measured by Ref. 266 for the “symmetric” s ± state under optimal pairbreaking conditions. Whilean s ± state can show a much slower rate of T c suppression due to large intraband scattering, there is no way an s ++ state can show such a rapid rate of suppression unless the impurities are purely magnetic. Ref. 271 was able to ruleout this possibility by measurements of the low- T magnetic penetration depth, which showed no indication of theupturns that would be caused by localized magnetic states.Here I have mostly discussed the role of nonmagnetic impurities, but I close this the subsection with a remarkor two about some novel phenomena associated with magnetic ones. A rather general summary of the propertiesof magnetic impurities in multiband superconductors has recently been given by Korshunov et al. , who foundsomewhat surprisingly that there are several cases where one’s intuition based on standard AG theory breaksdown, for example situations in which T c saturates in the limit of large disorder, instead of falling to zero. Theseeffects may be responsible for the weak suppression of T c by Mn along with nonmagnetic transition metal impuritiesin BaFe As (Fig. 11(c)), but do not seem useful in understanding why all transition metal impurities (nominallymagnetic and nonmagnetic) suppress T c at essentially the same rate. While this result, taken in isolation, suggeststhat the system must have a sign changing gap, in none but the simplest toy models could this result for differentimpurities with different potentials give such similar T c suppressions.Another interesting aspect of Mn in the BaFe As system was pointed out by Ref. 275, namely that Mn has theeffect of actually enhancing the Ne´el temperature in parts of the phase diagram. By combining neutron and µ SR data,Inosov et al observed that this impurity-induced high-temperature magnetic state was actually an inhomogeneous( π,
0) stripe state, and proposed that each Mn nucleated a small stripe region around it. A plausible model versionof this effect based on the effect of antiferromagnetic correlations on the impurity state was developed by Gastiasoroand Andersen , generalizing an “order by disorder” scenario developed for the cuprates .Whereas Mn suppressed T c rather slowly in BaFe As , Hammerath et al. reported the remarkable result thatin optimally F-doped La-1111, a concentration of Mn of only 0.2% sufficed to drive T c from about 30K to zero.0Simultaneously, spin fluctuations measured by NMR were observed to increase with Mn concentration (indicating,among other things, that the Mn impurities were not simply introducing a magnetic pairbreaking potential) until astate with quasi-long range magnetic order was reached at a concentration of around 1%. The authors attributed thisextreme sensitivity of T c to disorder to the proximity of this sample to the AF critical point. While the theoreticalproblem of the behavior of the local spin fluctuation spectrum around an impurity embedded in a system near aquantum critical point has been studied briefly in the cuprate context , where there is little hard evidence ofquantum critical behavior near the low temperature AF transition, it has received relatively little attention in theFeSC field where critical points are much better established. C. Impurity bound states
The observation of a localized resonance near a nonmagnetic impurity in STM is one of the clearest indications of asign-changing order parameter. Establishing that a given impurity is not magnetic is not always trivial, particularlysince in the presence of electronic correlations, a bare nonmagnetic potential can induce a local moment . Howeverin the case of chemical substituents or irradiation, creation of such moments should lead to other observable signals,such as Schottky anomalies in specific heat or upturns in the London penetration depth, so induced moments ofthis type can usually be excluded. On the other hand, if no bound state is seen, one may not conclude that thesuperconducting gap does not change sign. This is because in the multiband system the existence of an impuritybound state requires a fine tuning of the impurity potential, as pointed out in Ref. 5, and discussed extensively inRef. 281. This applies not only to weak potentials, which never give rise to bound states even in single band systems,but also to strong potentials in general. The bound state can also be obscured if the gap function of the system inquestion has deep minima or nodes, in which case the impurity pole may occur at an energy less than the coherencepeaks but still be so broadened by its coupling to extended states at energies greater than the gap minimum that itis not observable in a practical sense. In the simplest case, that of a 2-band isotropic s ± superconductor, it is clearthat the intraband scattering does not contribute to bound state formation. This point will be relevant in the nextsection.There are a few relatively clear observations of subgap nonmagnetic impurity bound states in the FeSC by STM,the only tool allowing direct measurement. Grothe et al. observed clear in-gap bound states at native defect siteson the surface of LiFeAs, some with remarkable shapes which broke tetragonal symmetry. H. Yang et al. foundsuch a state for Cu in another 111 material, Na(Fe . − x Co . )As, and Song et al. reported a bound state around anative defect site and Fe adatom in FeSe . These impurity resonances do constitute strong evidence in favor of s ± pairing, but careful bulk studies are still needed to rule out magnetic character, although this seems unlikely.These observations reported above all have the qualitative form more or less expected for standard theories ofmultiband impurity resonances in these systems : the state occurs at nonzero energy ± Ω somewhere in thegap, reflecting the fine-tuning of the impurity potential relative to the given band structure, and tends to have verydifferent weights at positive and negative energies, such that it is often visible only at one or the other. Often suchbound state energies are found to be slightly different depending on the local disorder environment, an effect expectedon the basis of the splitting of the states due to interference processes. On the other hand, a rather unusual impuritystate that does not seem to behave in this way was reported recently by Yin et al. , around what was believed tobe an Fe interstitial in Fe(Se,Te) ( T c =12K). This state was located at exactly zero energy, to within experimentalresolution, and was unsplit and unmodified by a magnetic field up to 8T, by disorder environment, and by dopingover some range. This smells like some kind of protection of the state by some kind of energy barrier, and the authorsof Ref. 287 speculated that it might be a Majorana fermion, without providing further details. D. Gap evolution vs. disorder
As disorder is added to an s ± superconductor, various qualitative evolutions of the gaps are possible, dependingboth on the intra- vs. interband character of the impurity scattering, but also on the intra- and interband characterof the interactions. Remarkably, in the case of an isotropic s ± system with two gaps ∆ a and ∆ b , one can classifythese evolutions in a rather simple way. Efremov et al. make use of the average interaction strength (cid:104) Λ (cid:105) ≡ [(Λ aa + Λ ab ) N a /N + (Λ ba + Λ bb ) N b /N ] with N = N a + N b the total density of states . Depending on (cid:104) Λ (cid:105) , thereturn out to be two classes of s ± superconductivity. The first, for (cid:104) Λ (cid:105) <
0, is the usual case, for which T c is suppressedas disorder, including some amount of interband potential scattering, is increased. In this case T c decreases until itvanishes at a critical scattering rate. There is however also a second class with (cid:104) Λ (cid:105) > T c tends to a nonzerovalue as disorder increases. At the same time, intraband scattering has no pairbreaking effect on the gaps(becausethey are assumed isotropic), and the interband scattering mixes the + and - gaps. In this case, the smaller of the two1 FIG. 12. (Color online.) a), Schematic of possible s ± and s ++ states. Large circles and small ellipsoids (black lines) are holeand electron Fermi surfaces, respectively. Red and blue represents the superconducting order parameter with different signs;b), Schematic of s ± order parameter vs. azimuthal angle φ (top row) and density of states vs. energy ω (bottom row) withincreasing irradiation dosage. Dotted lines (top row) represent zero gap. From Ref.273. in magnitude changes its sign, leading to an s ++ state. This transition, which should be accompanied by a passagethrough a gapless regime, has never been observed to my knowledge.One may ask what should happen to a gap with accidental nodes upon adding disorder. Here again two possiblescenarios obtain: the first, in the case that interband scattering dominates, leads to the formation of an impurityband that overlaps the Fermi level, with concomitant increase of the Fermi level density of states with disorder.This evolution is then similar to the d -wave one-band case, as in cuprates . On the other hand, if intrabandscattering dominates (as is usually the case for screened Coulomb impurities), the effect will be primarily to averagethe anisotropy in the gaps themselves. In such a case the averaging can lead to the lifting of the nodes altogether .Note that node lifting can occur either in nodal s ± or s ++ states (see Fig. 12a)), so one may ask how they are tobe distinguished. If the gap averaging continues as disorder is increased, eventually both gaps will be isotropic, andintraband scattering will no longer play a role. In such a case, the remaining smaller interband scattering potential willbegin to dominate. Then – only in an s ± state – it may create bound states and an impurity band within the full gapcreated when the initial nodes in the clean system were lifted, as shown in Fig. 12b). As complicated as this scenariomay seem, this effect was apparently observed in penetration depth measurements on electron-irradiated P-dopedBaFe As by Mizukami et al. , shown in Fig. 13. These authors concluded that this system must have a nodal s ± -wave ground state due to the apparent observation of the impurity band appearance at the highest irradiationdoses. Note that there should be various signatures of this effect in a nodal s ± system , e.g. the analogous evolutionfor NMR spin lattice relaxation 1 / ( T T ) should be T → T → exp( − Ω G /T ) → T . Similarly, the residual linear T term in the thermal conductivity κ ( T → /T should initially be nonzero, disappear when the nodes are lifted, andthen reappear with increasing disorder. In the s ++ case, the last step in each sequence is missing, because interbandscattering cannot give rise to low-energy bound states. E. Quasiparticle interference
Quasiparticle interference (QPI), or Fourier transform STM spectroscopy , is sensitive to the wavelengths ofFriedel oscillations caused by defects present in a metal or superconductor. In the absence of disorder, the quasiparticlesignal disappears; yet if disorder is not too strong, information on the electronic structure of the pure system can beobtained. The wavelengths of the Friedel oscillations appear in the form of peaks in the Fourier transform conductancemaps at particular wavevectors q ( ω ) which disperse as a function of STM bias V = ω/e . The interplay between theoryand experiment in this niche field has a checkered history . The usual method has been to calculate a QPI patternat a particular bias, given a theory of the superconducting state, and then compare with the expermental maps.However typically the sources of disorder and exact impurity potentials are unknown, so the weights of the various2 FIG. 13. (Color online.) Effect of electron irradiation on the low-temperature penetration depth in BaFe (As − x P x ) singlecrystals: change in the magnetic penetration depth ∆ λ plotted against ( T /T c ) for the T c0 = 28 K sample. Lines are the T dependence fits at high temperatures. From Ref. 273. q are imperfectly reproduced, and the process of comparison turns out to be somewhat subjective, to say the least.In addition, the STM tip measures the LDOS at the tip position several ˚A above the surface, not necessarily wheresuperconductivity occurs . Fortunately, the positions of the q do not depend on these effects and are related onlyto the band structure, including the superconducting gap function, so considerable information can be semi-empiricallyextracted without depending on detailed theoretical calculations.Can QPI information be used to give information on the sign changes of the gap over the Fermi surface? Someinformation can be gleaned by noting that subsets of these q connecting gaps of equal or opposite sign on the Fermisurface can be enhanced or not, according to the type of scatterer . In Ref. 296 it was suggested that a disorderedvortex lattice should act like a scattering center of the Andreev type for quasiparticles (scattering from order parametergradients); increasing the field was therefore in this sense equivalent to introducing disorder in a controlled way. Thisanalysis was first performed on the cuprate NaCCOC by Hanaguri et al. , and more recently on the Fe-basedsuperconductor Fe(Se,Te) . These authors identified three q related to hole-electron scattering and two associatedwith two different electron-electron q . The last two peaks were enhanced by field with respect to the first one, whichled them to conclude that the hole and electron pockets have opposite signs of the order parameter, consistent with a s ± state . Recently Chi et al. did a QPI analysis of their data on LiFeAs in zero external field. They identifiedtwo q peaks, a large one corresponding to interband and a small one corresponding to intraband scattering (see Fig.14). They observed that with decreasing bias, one peak was suppressed while the other enhanced, and argued thatthis was possible only with s ± pairing (Fig. 15(a),(b)).To make their claims, both experiments used reasoning based on heuristic arguments by Maltseva and Coleman stating that the scattering processes between wave vectors k , k (cid:48) were weighted in the Fourier transform density ofstates by the BCS coherence factors, ( u k u k (cid:48) ± v k v k (cid:48) ) , where u k , v k are the usual Bogoliubov variational coefficients.This elegant argument is unfortunately not strictly correct, as shown in Ref. 292, which gave examples where thisnotion can in fact be quite misleading. For example, by analogy with coherence factors in BCS, one would expect the3 FIG. 14. (Color online.) Model Fermi surface and scattering processes for analysis of QPI data in LiFeAs. From Chi et al. interband and intraband QPI weights, measured as a function of temperature, to increase or decrease just below T c ,by analogy with NMR relaxation or ultrasound attenuation . This is not the case . However, a) the argumentthat the different sign slopes of the conductance weights at the upper gap edge implied an order parameter sign changeprobably is correct, as shown in Fig. 15(c). Still the claim should be treated with caution, and a much more definitivecriterion proposed by Ref. 292, namely the temperature dependence of the intraband antisymmetrized conductanceweight (Fig. 15 (d)), has not yet been tested. X. CONCLUSIONS
This review has been targeted at experimental and theoretical developments related to the iron-based supercon-ducting state, mostly since 2011 when a previous review examined the state of the art. Here I’ve focussed on a set ofphenomena which seem to me the most interesting and important currently. In 2011, the the familiar 1111, 111 and122 systems possessing both hole and electron Fermi surface pockets dominated the discussion, and the picture of s ± superconducting states formed by ( π,
0) spin fluctuations was both intuitive and appealing. The pressing questionsat that time seemed to be deciding if some or all FeSC could be shown to have sign-changing s -wave ground states,and understanding the origin of gap nodes in some systems.At the present writing, the field seems just as vigorous, in fact infused with enthusiasm to answer a new set ofpressing questions. These are mostly related to a new class of systems which seem to belong to a different paradigmthan those discussed in 2011, namely those lacking either electron or hole pockets. Does the existence of such systemswith high critical temperatures imply that a completely new physical mechanism for superconductivity is at work,or is the “standard model” capable of subsuming these materials into a different subclass, also driven by repulsive,primarily interband interactions? Is there any evidence for an important role for d -wave interactions in stabilizingthese systems, and how can one best detect this? What is the role of phonons in “bootstrapping” the spin fluctuationinteraction, particular in the fascinating FeSe/STO monolayers? Under what circumstances can phonons help or hurtthe critical temperature in a strongly spin-fluctuating system? All these questions are under intense debate.There are other questions which affect the more familiar systems with both hole and electron pockets as well. Canone find a system where a transition between two channels of superconducting order, e.g. s and d takes place as afunction of doping or pressure? What is the nature of this transition, does it involve time-reversal symmetry breakingstates and if so, how does one detect them? What new types of pairing are possible utilizing the orbital degreesof freedom of the Fe d -electrons, and are there systems with interactions that can stabilize such exotic states? Dosuch states necessarily involve pairing of electrons at different energies, and if so is this necessarily fatal, given newunderstanding of the role of incipient bands in pairing? Is nematicity a passive bystander in superconductivity, or adriver? I have described some important contributions to these questions, but they are not settled.In the introduction I pointed to the fact that the lack of universality in many of the properties of the Fe-basedsuperconductors, while disturbing to a certain mindset reared on cuprate physics, can be viewed as an advantage.The hope is that, since electronic correlations are less pronounced in these systems than the cuprates, a quantitativetheory of superconductivity is not out of reach. In this context, the diversity of these systems can therefore be thought4 FIG. 15. (Color online.) (a) q-scan indicating area integrated to define weight in intraband (red) and interband(blue) peaks.(b) Low temperature bias scan of weights defined as in (a). From Chi et al. (c) Calculation of intra- (red) and interband(black) weights from Ref. 292; (d) Proposal of Ref. 292 to distinguish s ± (solid) and s ++ (dashed) by measuring temperaturedependence of antisymmetrized interband LDOS (cid:104) δρ − inter (Ω; T ) (cid:105) vs. T for Ω between two gap scales | ∆ | and | ∆ | . of as a challenge to build a theory explaining this diversity using first principles input, ultimately helping in the searchfor new superconducting materials. It should be clear from this article, for example the description of the situationregarding LiFeAs, that theoretical methods are not quite refined enough to assist in this effort.My own view is that theoretical progress is needed in two directions. First, the tight-binding models taken fromDFT are not adequate for the description of the most interesting systems, which tend to be more strongly correlated.Methods are needed to accurately reproduce the correlated electronic structure over a few hundred meV near theFermi surface accurately. While LDA+DMFT has done well enough to reproduce some qualitative trends, it stillproduces inaccurate Fermi surfaces for LiFeAs and FeSe, to name two. It is clear, however, that electronic structuremethodology is advancing rapidly , and I have little doubt that this situation will be vastly improved within a fewyears. The second, more difficult, question is whether the treatment of electronic excitations involved in pairing isadequate. It is possible that the simple Fermi-surface based RPA treatment of spin fluctuations will be sufficientto predict semiquantitative trends in pairing correctly once adequate tight-binding models for the more correlatedsystems are available. More likely is that the current machinery will need to be improved at least by accounting morecarefully for the contribution of the near-Fermi level states to pairing, and possibly to include further electronic vertexcorrections and, in some cases, phonons. All these developments will require significant increase in computationalpower.Looking carefully at cases like LiFeAs where various theoretical approaches have been brought to bear, however, Iwould say that there is considerable reason for optimism. It is remarkable that the detailed gap structure over much5of the Fermi surface, and the relative sizes of gaps, can be obtained accurately without fine tuning. Experimentalprogress since 2011, particularly the discovery of new systems with nonpolar surfaces accessible to high resolutionSTM and ARPES, has allowed theory and experiment to follow each other in a tight circle, an ideal scientific situationwhich seems destined to lead to a solution to many of these fascinating problems in the not-too-distant future. XI. ACKNOWLEDGEMENTS
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