# Gap symmetry and structure of Fe-based superconductors

P.J. Hirschfeld  Department of Physics, University of Florida, Gainesville,
Florida 32611, USA<EMAIL_ADDRESS>M.M. Korshunov  Department of Physics,
University of Florida, Gainesville, Florida 32611, USA  L.V. Kirensky
Institute of Physics, Siberian Branch of Russian Academy of Sciences, 660036
Krasnoyarsk, Russia<EMAIL_ADDRESS>I.I. Mazin  Code 6393, Naval
Research Laboratory, Washington, D.C. 20375, USA<EMAIL_ADDRESS>
###### Abstract

The recently discovered Fe pnictide and chalcogenide superconductors display
low temperature properties suggesting superconducting gap structures which
appear to vary substantially from family to family, and even within family as
a function of doping or pressure. We propose that this apparent
nonuniversality can actually be understood by considering the predictions of
spin fluctuation theory and accounting for the peculiar electronic structure
of these systems, coupled with the likely “sign-changing $s$-wave” ($s_{\pm}$)
symmetry. We review theoretical aspects, materials properties, and
experimental evidence relevant to this suggestion, and discuss which further
measurements would be useful to settle these issues.

###### type:

Review Article

###### Contents

  1. 1 Introduction.
    1. 1.1 Aim and scope of this article.
    2. 1.2 Fe-based superconductors.
      1. 1.2.1 Comparison with cuprates.
      2. 1.2.2 Comparison with MgB2.
      3. 1.2.3 Conceptual importance.
      4. 1.2.4 Gap symmetry and structure.
  2. 2 Electronic structure.
    1. 2.1 First principles.
      1. 2.1.1 General properties of FeBS electronic structure.
      2. 2.1.2 Limitations of DFT calculations.
    2. 2.2 Minimal band models.
  3. 3 Theoretical background.
    1. 3.1 Spin fluctuation pairing.
    2. 3.2 Alternative approaches.
    3. 3.3 Multiband BCS theory.
    4. 3.4 Disorder in multiband superconductors.
      1. 3.4.1 Intra- vs. interband scattering.
      2. 3.4.2 Effect on $T_{c}$.
      3. 3.4.3 Anisotropic states.
      4. 3.4.4 Single impurity problem.
      5. 3.4.5 Magnetic impurities.
      6. 3.4.6 Orbital effects.
    5. 3.5 Dimensionality.
  4. 4 Gap symmetry.
    1. 4.1 Triplet or singlet?
    2. 4.2 Chiral or not?
    3. 4.3 $d$ or $s$?
  5. 5 Gap structure.
    1. 5.1 Does the gap in FeBS change sign?
      1. 5.1.1 Spin-resonance peak.
      2. 5.1.2 Josephson junctions.
      3. 5.1.3 Quasiparticle interference.
      4. 5.1.4 Coexistence of magnetism and superconductivity.
    2. 5.2 Evidence for low-energy subgap excitations.
      1. 5.2.1 Penetration depth.
      2. 5.2.2 Specific heat.
      3. 5.2.3 Thermal conductivity.
      4. 5.2.4 The ARPES “paradox”.
      5. 5.2.5 NMR $1/T_{1}$.
      6. 5.2.6 Electronic Raman scattering.
    3. 5.3 Alkali-intercalated iron selenide
    4. 5.4 Differences among materials: summary.
  6. 6 Conclusions.

  * Satisfactoriness has to be measured by a multitude of standards, of which some, for aught we know, may fail in any given case; and what is more satisfactory than any alternative in sight, may to the end be a sum of pluses and minuses, concerning which we can only trust that by ulterior corrections and improvements a maximum of the one and a minimum of the other may some day be approached.
William James, Meaning of Truth

##  1 Introduction.

###  1.1 Aim and scope of this article.

The iron arsenide superconductor LaFeAsO with critical temperature 26K was
discovered in 2008 by Hideo Hosono and collaborators [1]. Within two months,
materials based on substitution of La with other rare earths had been
synthesized, raising the critical temperature of Fe-based superconductors
(FeBSs) to 55K. This rapid sequence of discoveries captured the attention of
the high-temperature superconductivity community. The following three years
saw the discovery of several related families of materials, the rapid
calculation of their electronic structure within density functional theory
(DFT), and the development of microscopic models for superconductivity largely
based on these DFT calculations. The existence of a second class of high-
temperature superconductors is generally agreed to be important not only for
the possible existence of materials with even higher $T_{c}$’s within the same
class of Fe-based materials, but because the comparison with the cuprates can
allow one to potentially understand the essential ingredients of high-
temperature superconductivity. Because of the extremely rapidly advancing
nature of the field, the perils of writing a review are obvious. Several
authors have nevertheless recently attempted to summarize the status of
research in this area, and we have benefited greatly from the existence of
these works [2, 3, 4, 5, 6, 7, 8, 9, 10].

We intend in this smaller scope review to focus on one particular question
among the several fascinating issues surrounding the Fe-based superconductors,
namely the symmetry and structure of the superconducting gap. In the study of
cuprate superconductors, the $d_{x^{2}-y^{2}}$ symmetry of the gap, with $\cos
k_{x}-\cos k_{y}$ structure, was empirically established soon after high
quality samples were prepared, by penetration depth, ARPES, NMR and phase
sensitive Josephson tunneling experiments. After three years of intensive
research on the Fe-based superconductors, no similar consensus on any
universal gap structure has been reached, and there is strong evidence that
small differences in electronic structure can lead to strong diversity in
superconducting gap structures, including gaps with nodes in some and full
gaps in other materials. The actual symmetry class of most of the materials
may be the same, of generalized $A_{1g}$ ($s$-wave) type, probably involving a
sign change of the order parameter between Fermi surface sheets in most
materials. In addition, there have been recent suggestions that some related
materials, furthest from the nearly compensated semimetal band structure of
the originally discovered compounds, may have $d$-wave symmetry. Understanding
both the symmetry character of the superconducting ground states and the
detailed structure should provide clues to the microscopic pairing mechanism
in the pnictides and thereby a deeper understanding of the phenomenon of high-
temperature superconductivity.

A complete review of the literature of even the more focussed problem of gap
structure we consider here is beyond the scope of our paper. We have attempted
to select those works we find most relevant (or at least sufficiently
representative) to the questions we believe to be important:

  * •

What do experiments tell us about the pairing symmetry and gap structure of
the Fe-based materials, and what systematic trends can one identify?

  * •

How do changes in electronic structure drive changes in gap symmetry and
structure?

  * •

What physical effect drives pairing in Fe-based materials? Is more than one
important, at least in some materials?

  * •

What role does disorder play, and how is it related to changes in carrier
densities, electronic structure, and pairing interactions?

If any works in this category have been omitted or slighted, we beg the
authors will attribute it to ignorance or haste rather than malice!

###  1.2 Fe-based superconductors.

####  1.2.1 Comparison with cuprates.

High-$T_{c}$ cuprates are known for their high critical temperature,
unconventional superconducting state, and unusual normal state properties. The
Fe-based superconductors (FeBS), with $T_{c}$ up to 55K in SmFeAsO1-xFx stand
in second place after cuprates and 15K above MgB2. When superconductivity in
the FeBS was discovered, the question immediately arose: how similar are they
to cuprates? Let us compare some of their properties.

Both cuprates and FeBS have 2-dimensional lattices of $3d$ transition metal
ions as the building blocks. In both cases orthorhombic distortions can be
present at small doping. The main structural difference in these planes is
that the $2p$-ligands lie very nearly in the plane with the Cu in cuprates,
while in FeBS As, P, Se, or Te lie in nearly tetrahedral positions above and
below the Fe plane. The in-plane subset of Cu $d$-orbitals $e_{g}$ are both
present near the Fermi level, and of these, the planar $d_{x^{2}-y^{2}}$ is
quite dominant (see, however, [11]), allowing in principle the reduction of
the multiband electronic structure to a low-energy effective one-band model.
In FeBS, on the other hand, out-of-plane As hybridize well with the $t_{2g}$
Fe $d$-orbitals and all three of them have weight at the Fermi surface. In
addition (as opposed to the cuprates), there is substantial overlap between
the $d$-orbitals. The minimal model is then essentially multiband and that
makes FeBS in this respect more similar, e.g. to ruthenates than to cuprates.

At first glance, the phase diagrams of cuprates and many FeBS are similar. In
both cases the undoped compounds exhibit antiferromagnetism, which vanishes
with doping; superconductivity appears at some nonzero doping and then
disappears, such that $T_{c}$ forms a “dome”. While in cuprates the long range
ordered Néel phase vanishes before superconductivity appears, in FeBS the
competition between these orders can take several forms. In LaFeAsO, for
example, there appears to be a first order transition between the magnetic and
superconducting states at a critical doping value, whereas in the 122 systems
the superconducting phase coexists with magnetism over a finite range and then
persists to higher doping. It is tempting to conclude that the two classes of
superconducting materials show generally very similar behavior, but there are
profound differences as well. The first striking difference is that the
undoped cuprates are Mott insulators, but FeBS are metals. This suggests that
the Mott-Hubbard physics of a half-filled Hubbard model is not a good starting
point for pnictides, although some authors have pursued strong-coupling
approaches. It does not of course exclude effects of correlations in FeBS, but
they may be moderate or small. In any case, DFT-based approaches describe the
observed Fermi surface and band structure reasonably well for the whole phase
diagram, contrary to the situation in cuprates, especially in undoped and
underdoped regimes.

The second important difference pertains to normal state properties.
Underdoped cuprates manifest pseudogap behavior in both one-particle and two-
particle charge and spin observables, as well as a variety of competing
orders. At least for hole-doped cuprates, a strange metal phase near optimal
doping is characterized by linear-$T$ resistivity over a wide range of
temperatures. In FeBS, different temperature power laws for the resistivity,
including linear $T$-dependence of the resistivity for some materials, have
been observed near optimal doping and interpreted as being due to multiband
physics and interband scattering [12]. The FeBS do not manifest a robust
pseudogap behavior in a wide variety of observable properties.

The mechanism of doping deserves additional discussion. Doping in cuprates is
accomplished by replacing one of the spacer ions with another one with
different valence or adding extra out-of-plane oxygen, e.g. La2-xSrxCuO2,
Nd2-xCexCuO2, and YBa2Cu3O6+δ. The additional electron or hole is then assumed
to dope the plane in an itinerant state. In FeBS, the nature of doping is not
completely understood: similar phase diagrams are obtained by replacing the
spacer ion as in LaFeAsO1-xFx and Sr1-xKxFe2As2, or by in-plane substitution
of Fe with Co or Ni as in Ba(Fe1-xCox)2As2 and Ba(Fe1-xNix)2As2, or by
replacing Ba with K, Ba1-xKxFe2As2. Whether these heterovalent substitutions
dope the FeAs or FeP plane as in the cuprates was not initially clear [13],
but now it is well established that they affect the Fermi surface consistent
with the formal electron count doping [14, 15]. Another mechanism to vary
electronic and magnetic properties is via the possibility of isovalent doping
with phosphorous in BaFe2(As1-xPx)2 or ruthenium in BaFe2(As1-xRux)2.
‘Dopants’ can act as potential scatterers and change the electronic structure
because of differences in ionic sizes or simply by diluting the magnetic ions
with nonmagnetic ones. But crudely the phase diagrams of all FeBS are quite
similar, challenging workers in the field to seek a systematic structural
observable which correlates with the variation of $T_{c}$. Among several
proposals, the height of the pnictogen or chalcogenide above the Fe plane has
frequently been noted as playing some role in the overall doping dependence
[16, 17, 18].

It is well established that the superconducting state in the cuprates is
universally $d$-wave. By contrast, we review evidence below that the gap
symmetry and/or structure of the FeBS can be quite different from material to
material. Nevertheless, it seems quite possible that the ultimate source of
the pairing interaction in both systems is fundamentally similar, although
essential details such as pairing symmetry and the gap structure in the FeBS
depend on the FS geometry, orbital character, and degree of correlations.

####  1.2.2 Comparison with MgB2.

MgB2 was the first example of multigap superconductivity (or at least, the
first one recognized as such). There is little doubt that this property is
shared by FeBS, as discussed below; therefore it is instructive to see which
multiband features have been discovered in MgB2 and what similarities can be
found in FeBS.

The thermodynamic properties of MgB2 show very characteristic behavior which
can easily be understood within multiband BCS theory (see Section 3.3)
assuming a weak coupling between bands. At the critical temperature the larger
gap is clearly visible in thermodynamics, and at a lower temperature, roughly
corresponding to the critical temperature of the weaker band alone, the second
gap becomes manifest in thermodynamic properties. The second gap, while
formally appearing at the same temperature as the first gap, remains very
small until a much lower temperature. Such considerations of course lead one
to examine also the opposite situation where the bands are strongly coupled
(as in essentially all theories of superconductivity in FeBS). In this case
both gaps gradually diminish as T is raised, but one or both may show non-BCS
behavior, and the thermodynamic properties cannot be accurately described by
one gap; the sum of two gaps, on the other hand, can provide a realistic
description. This is indeed the case in many FeBS, as probed by specific heat,
penetration depth, NMR relaxation rate etc. (see Section 5). On the other
hand, the picture is additionally clouded, as compared to MgB2, because of
presumably larger gap anisotropy and pair-breaking effects of impurity
scattering, as discussed below.

Another manifestation of multiband superconductivity is found in the thermal
conductivity. The reduced thermal conductivity $\kappa/T$ is, generally
speaking, zero at $T=0$, if the Fermi surface is fully gapped in the
superconducting state (although pair breaking effects due to magnetic
impurities may, in principle, create mobile quasiparticle states with zero
energy [19]). In MgB2, as well as in many FeBS, this is the case. Upon
applying magnetic field, Abrikosov vortices form in the system. As soon as
these vortices begin to overlap, the thermal conductivity starts growing. This
happens at field on the order of $H_{c2}/3$. Now, if there are two gaps in the
system, one substantially smaller than the other, one may think that the
vortex overlap will start at much smaller fields. Indeed, the distance between
the vortices is proportional to $H^{-1/2}$, while their size is defined by the
coherence length and thus inversely proportional to $\Delta$. So, the critical
field where the “weaker” band will be smaller than that for the “stronger”
band by a factor of, roughly. $(\Delta_{1}/\Delta_{2})^{2}$, which is, for
MgB2, about 10. So, the argument goes [20], one cannot observe the flat low-
field part of $\kappa(H)/T$, and experimentally the dependence looks linear at
the smallest accessible $H$. Of course, one must see flattening at $H\lesssim
H_{c2}/30$, but so far nobody has observed this. We only emphasize here that
many FeBS studied by this technique show a linear increase of
$\kappa/T|_{T\rightarrow 0}$ with $H$ at small $H$, which, in a traditional
multiband interpretation, suggest a considerable disparity between the largest
and the smallest gaps, or possibly strong gap anisotropy.

Another interesting lesson that one can derive from the MgB2 studies is
negative. One of the reasons why a number of theorists were initially
reluctant to accept the two-gap scenario for this material is the fact that
nonmagnetic impurities, in the Abrikosov-Gor’kov theory, should suppress
$T_{c}$ linearly with fairly large slope until the gaps are averaged. This
effect has not been observed in MgB2. While there have been reasonable
explanations of why particular impurities may have little effect on $T_{c}$
[21], in retrospect it is clear that the impurity effect is weaker than that
expected from the theory in many different cases.

Finally, it is worth looking back at the normal state of MgB2. Detailed
quantum oscillation studies [22] prove unambiguously that the two band systems
($\pi$ and $\sigma$) are shifted with respect to each other by up to 100 meV.
Similarly, in FeBS quantum oscillations shows that the hole bands and the
electron bands are shifted with respect to each other by up to 70 meV, so that
the hole and the electron Fermi surface become smaller relative to DFT
predictions. This holds both in magnetic [23] and nonmagnetic [24] cases. It
has been ascribed to correlation effects [25], but the comparison with MgB2
demonstrates that these effects beyond LDA are, if anything, less severe than
in MgB${}_{2},$ which is not generally considered to be a correlated metal.

####  1.2.3 Conceptual importance.

While the FeBS may not signify a particular advance in terms of practical
applications—their $T_{c}$ is only 15K higher than that of MgB2, and, just as
the cuprates, they are expensive to make and difficult to work on—their
conceptual value is hard to overestimate. Indeed, fullerides and MgB2 clearly
belong to a different class than the cuprates, being in certain respects
exotic, but still phonon-driven superconductors. Not surprisingly, there had
been a growing feeling among physicists that phonon superconductivity will
probably never grow past 50-60K, while true high-temperature superconductivity
is a strong-correlation phenomenon limited to the unique family of layered
cuprates. It had been justly pointed out that the CuO2 layers have many unique
properties, largely coming from the fact that Cu is the last $3d$ transition
metal and as such is by far the most strongly correlated of all, yet its
simple one-orbital electronic structure provided for a simple and large Fermi
surface when doped. One can point to many aspects in which cuprates are
unique, and many people did.

What the discovery of the FeBS brought onto the table was the understanding
that however unique cuprates may be, these features are not prerequisites for
non-phonon, high-temperature superconductivity. And, if that is true, there
are likely many other crystallochemical families to be discovered, some of
which may have higher critical temperature or be better suited for
applications than cuprates and FeBS.

In a twisted way, we are lucky that FeBS and cuprate are so different in so
many aspects. This makes it more reasonable to look for those few
commonalities which exist and to assume, even without profound theoretical
insight, that these commonalities are important for high $T_{c}$. Some of
these obviously include proximity to magnetism and quantum criticality, or
substantial anisotropy of the Fermi surface (quasi-2D) and it is has already
been argued by many that one should look for a combination of these factors to
search for novel superconductors [26].

####  1.2.4 Gap symmetry and structure.

  *
Figure 1: Cartoon of order parameters under discussion in the Fe-pnictide
superconductors represented in the 2-dimensional, 1-Fe Brillouin zone (see
Section 2). Different colors stands for different signs of the gap.

The group theoretical classification of gap structures in unconventional
superconductors is somewhat arcane and has been amply reviewed elsewhere [27].
Here we present the simplest notions relevant to the discussion of symmetry
and structure of the order parameters under discussion in the Fe-based
superconductors at present. In the absence of spin-orbit coupling, the total
spin of the Cooper pair is well-defined and can be either $S=1$ or $S=0$.
Experimental data appear to rule out spin triplet states (see Section 4), so
we focus on the spin singlet case. We focus first on simple tetragonal point
group symmetry. In a 3D tetragonal system, group theory allows only for five
one-dimensional irreducible representations: $A_{1g}$ (“$s$-wave”), $B_{1g}$
(“$d$-wave” [$x^{2}-y^{2}$]), $B_{2g}$ (“$d$-wave” [$xy$]), $A_{2g}$
(“$g$-wave” [$xy(x^{2}-y^{2})$]), and $E_{g}$ (“$d$-wave” [$xz,yz$]) according
to how the order parameter transforms under rotations by 90∘ and other
operations of the tetragonal group. In figure 1 we have illustrated two of
these symmetries, namely $s$-wave and $d_{x^{2}-y^{2}}$-wave. Note that the
$s_{++}$ state and $s_{\pm}$ states represented all have the same symmetry,
i.e. neither changes sign if the crystal axes are rotated by 90∘. By contrast,
the $d$-wave state changes sign under a 90∘ rotation. Note further that the
mere existence of the single hole and single electron pocket shown lead to new
ambiguities in the sign structure of the various states. In addition to a
global change of sign, which is equivalent to a gauge transformation, one can
have individual rotations on single pockets and still preserve symmetry; for
example, if one rotates the gap on the hole pocket for the $d$-wave case in
figure 1 by 90∘ but keeps the electron pocket signs fixed, it still represents
a $B_{1g}$ state. Note also that $B_{2g}$ states, while not shown in the
figure, are also possible by symmetry and would have nodes on the electron
pockets. Further, more complicated, gap functions with differing relative
phases on the different pockets become possible when more pockets are present,
and/or when 3D effects are included (See Section 3.5).

These symmetry properties are distinct from gap structure, a term we use to
designate the $k$-dependent variation of an order parameter within a given
symmetry class. Gaps with the same symmetry may have very different
structures, as also illustrated in the figure, where three different types of
$s$-wave states are shown. The isotropic, fully gapped $s_{++}$ and $s_{\pm}$
states differ only by a relative phase of $\pi$ in the latter case between the
hole and electron pockets. On the other hand, in the nodal $s$ case, the gap
is shown vanishing at certain points on the electron pockets. This particular
case shows a case we will sometimes refer to as “nodal $s_{\pm}$”, in that the
sign on the hole pockets is still opposite the average sign on the electron
pockets. Nodes of this type are sometimes described as “accidental”, since
their existence is not dictated by symmetry, but rather by the details of the
pair interaction. As such, they can be removed continuously, resulting in
either an $s_{++}$ or an $s_{\pm}$ state.

##  2 Electronic structure.

###  2.1 First principles.

####  2.1.1 General properties of FeBS electronic structure.

The basic crystallographic element of the FeBS is the FeAs (where instead of
As one can also have P, Se or Te) plane with an ${a}\times{a}$ square plane of
Fe ions, and two $\tilde{a}\times\tilde{a}$ square planes of As above and
below (where $\tilde{a}={a}\sqrt{2}).$ The minimal unit cell of the entire
FeAs plane is, therefore, also $\tilde{a}\times\tilde{a}$ and includes two
formula units. In some, but not all cases the low-energy part of the
electronic structure can be “unfolded” into a Brillouin zone (BZ) which is
twice as large, corresponding to the ${a}\times{a}$ unit cell, so that the
real band structure can be recovered by folding the 2D Brillouin zone in such
a way that the “unfolded” $\mathrm{X}=(\pi/a,0)$ and $\mathrm{Y}=(0,\pi/a)$
points fold on top of each other, forming the
$\tilde{\mathrm{M}}=(\pi/\tilde{a},\pi/\tilde{a})$ point in the small
Brillouin zone. Here and throughout the article we always use the untilded
notations in the one-Fe unit cell and the “unfolded” Brillouin zone, and the
tilded notations in the crystallographic unit cells and the corresponding
Brillouin zone.

Despite the large variety of crystal structure and chemical compositions, all
FeBS share the same gross features of the electronic structure. These can be
listed as follows:

1\. In the nonmagnetic state, the band structure is of semimetallic nature,
with two or more hole band crossing the Fermi level near the $\Gamma$ point
and two electron bands crossing the Fermi level near the $\tilde{M}$ point.

2\. Two hole bands, universally present in all superconducting compositions,
are formed by the $xz$ and $yz$ derived Fe band, which are degenerate (without
the spin-orbit effects) at the $\Gamma$ point, but split apart (and acquire
some $z^{2}$ character) in a relatively uniform manner away from it. The two
electron bands take their origin from the downfolding effect described above,
and are formed mostly by the ${x}{z}$ and ${y}{z}$ orbitals, respectively,
plus the ${x}{y}$ orbital.

3\. As a result, there are always at least two hole Fermi surfaces and two
electron Fermi surfaces, which are well separated in the reciprocal space.
Moreover, their respective centers are removed from each other exactly by
$\bi{Q}=\widetilde{(\pi/\tilde{a},\pi/\tilde{a})}=(\pi/a,0)$. In general, the
Fermi surfaces have sufficiently different shapes so that one cannot speak of
a good nesting here, only of a quasi-nesting.

On the other hand, many aspects of the electronic structure vary from material
to material. For instance, in some materials another hole band appears, which
may be either of $z^{2}$ character, in which case it is substantially 3D, or
${x}{y},$ which is even more 2D than the $xz/yz$ bands.

Different FeBS may have different degree of charge doping, of either sign,
ranging from 0.5 h to 0.15 e per Fe. It appears though that in all this
compositional range, the general structure of the FS almost always survives.
That is, while either the electron or the hole FS shrinks, they never entirely
disappear in the superconducting range of dopings, even though the nesting
conditions may have drastically deteriorated. It worth remembering that in
strongly anisotropic systems the size of a FS has little correlation with its
density of states at the Fermi level. In addition, recently two FeBS systems
have been discussed that are superconducting but which may lie outside this
“typical” range; in KFe2As2 the Fermi surface pockets near the $\mathrm{X}$
point nearly disappear, while the hole pockets around $\Gamma$ are greatly
expanded, and KFe2Se2 may (or may not; see Section 5.3) be completely lacking
the hole pockets.

By contrast, the $k_{z}$ dispersion can vary substantially from material to
material. This depends mostly on two factors: the thickness of the “filler”
layer between the FeAs or FeSe layer, and on crystallographic symmetry.
Obviously, materials with the 1111 structure are more anisotropic, in fact
nearly 2D, than those with the same $P4/nmm$ symmetry but no filler species,
that is, with the 11 structure. Less obviously, materials with a body center
symmetry, such as 122 ($I4/mmm$), have an additional reason for a 3D
character: the downfolding procedure in that case projects the electronic
states near the $\left({\pi/a},{0},{0}\right)$ point onto the states near
$\left(0,{\pi/a},{\pi/c}\right)$. Crossings occur at general $k$-point, and
therefore hybridization between these states is not forbidden, but depends on
$k_{z}$. As a result these materials tend to be even more 3D than the 11
compounds, despite having a filling monolayer in the former. We explore the
consequences of $k_{z}$ dispersion for superconductivity in Section 3.5.

####  2.1.2 Limitations of DFT calculations.

First principles calculations were very important in the beginning of the FeBS
era, and they have informed the emerging understanding of the physics of the
FeBS much more than in the case of cuprate superconductors, although due to
somewhat stronger correlations and complexity of materials they have not yet
proven as definitively useful as in MgB2. In FeBS, they successfully predicted
the right topology of the FS [28], as well as the correct magnetic ordering in
the normal state [29]. The most successful proposal regarding the pairing
symmetry so far has been made based on band structure calculations [29]. These
facts have been instrumental boosting the reputation of the band theory in the
superconducting context, but one should remember that while DFT is not a snake
oil, it is not a panacea either, and nor are any of its generalizations such
as DFT+DMFT etc. Let us list below the most important shortcomings and
limitations of the DFT calculations as regards FeBS.

1\. The DFT is, by construction, a mean-field theory (but not a low-energy
theory, as is sometimes incorrectly asserted). It is a more sophisticated mean
field theory than many, for it includes in the energy functional (and thus in
the mean field potential), all correlation effects and integrates in all
fluctuations. On the other hand, the actual implementations of the DFT, such
as the local density approximation or the generalized gradient approximation,
by construction only include those correlations and those fluctuations that
are present in the reference system, the uniform electron gas, at densities
comparable with the electron densities in real solids. Remember that the
uniform electron gas at such densities is very far from magnetism, and even
farther from the electron localization (Wigner crystallization). The
corresponding physics is, therefore, largely missing when DFT is used in a
“local” approximation. This belongs to two major classes: (i) on-site Coulomb
fluctuations, also called Hubbard correlations, which are included in a very
limited way on the level of the Stoner magnetic interaction (reflecting the
first Hund’s rule), and (ii) quantum critical fluctuations; examples of such
are long range ferromagnetic fluctuations in nearly ferromagnetic metals [30].
The hallmark of the former is underestimation of the tendency to magnetism in
a DFT calculation, of the latter - overestimation.

In cuprates, the DFT calculations suffer from the former problem, in FeBS
mostly from the latter. From the density functional point of view, it is
rather curious that despite the fact that the calculated magnetic moments are
large, as opposed to such known cases of near-quantum-critical materials ZrZn2
or Fe3Al, yet the effect of such long-range fluctuations appears to be strong.
The explanation is that magnetic moments in this system are quite soft (in the
calculations they can change from nearly zero to more than a Bohr magneton
depending on the magnetic pattern), and on top of the transverse spin
fluctuations typical for strong antiferromagnets, there are longitudinal
fluctuations characteristic of itinerant magnets and quite efficient in
reducing the ordered magnetic moment. Moreover, there is a possibility that
other, so-called “nematic” fluctuations, may play an additional role in
reducing the ordered moment.

This does not mean that the first DFT problem, underestimation of on-site
Coulomb correlations, is nonexistent in these materials. It is relatively
mild, and secondary compared to the other deficiency, yet it exists and it
manifests itself, for instance, in the bandwidth renormalization. From the
point of view of this Section, the important corollary of the above is that
superconductivity in FeBS develops not on the background of a non-magnetic
state, but of a paramagnetic state which still has fluctuating local
magnetization of Fe ions. Therefore, the calculated bands and the Fermi
surfaces are true only as long as averaging over these fluctuating quasi-local
magnetic moments is equivalent to dropping the spin-dependent part of the
crystal potential entirely. Experimental evidence so far has been
inconclusive. De Haas - van Alphen experiments generally agree well with the
DFT calculation, up to some uniform shift of different bands with respect to
each other and overall mass renormalization. ARPES derived Fermi surfaces,
while conforming with the general topology, predicted by DFT, differ in
details substantially. It is fair to say that that the DFT bands are a
reasonable, but not exceedingly good approximation of the actual band
structure, even after accounting for the bandwidth renormalization and the
band shifts. They appear to be renormalized by a factor up to three, and may
be even larger for some systems, and the renormalization appears to be
stronger as the system approaches the AFM quantum critical point [24]. Also,
additional repulsive interaction between the holes and the electrons seems to
be operative, pushing the (mostly occupied) hole bands down, and (mostly
empty) electron bands up. The last effect is responsible for shrinking all
Fermi surfaces compared to DFT, but this effect is weak (but is, again,
stronger near the quantum critical point) [25].

The band renormalization comes from both the on-site and long-range
fluctuations. Existing DMFT calculations, while qualitatively agreeing among
themselves, disagree on the exact share of the total mass renormalization
provided by the on-site vs. long range fluctuations. Indeed, all groups find
that the effect comes predominantly from Hund’s $J$, and not Hubbard’s $U$,
and that the 11 family is substantially more correlated than other families.
At the same time, the Rutgers group has obtained mass renormalizations closely
matching the experiment, leaving basically no room for the long-range
fluctuations, while the other DMFT groups’ results suggest that both effects
provide comparable contributions to the total renormalization in pnictides
[31, 32, 33].

###  2.2 Minimal band models.

On the basis of the DFT band structure one can make a simplified model which
then can be studied by sophisticated theoretical methods like a Green’s
function formalism. There is always a trade off between complexity of a model
and physical effects captured by it. Here we discuss several popular models
with increasing levels of complexity.

  *
Figure 2: (a) FeAs lattice indicating As above and below the Fe plane. Dashed
green and solid blue squares indicate 1- and 2-Fe unit cells, respectively.
(b) Schematic 2-dimensional Fermi surface in the 1-Fe Brillouin zone whose
boundaries are indicated by a green dashed square. Arrow indicates folding
wave vector $\bi{Q}_{F}$. (c) Fermi sheets in the folded Brillouin zone whose
boundaries are now shown by a solid blue square.

According to DFT, FeBS are essentially multiband systems and a minimal model
must include both hole and electron bands. The first complication comes from
the As, which forms square lattice planes between the lattice sites of, but
also above and below, the square lattice of Fe. This alternating pattern of As
makes the correct real space unit cell twice the one-Fe unit cell. The
corresponding 2-Fe BZ is twice as small as the 1-Fe one and called the “folded
BZ”, see figure 2. For the simplest case of single-layer FeBS the folding wave
vector is 2-dimensional and equal to $\bi{Q}_{F}=(\pi,\pi)$. Most experimental
results and DFT band structures are reported in the folded BZ since
crystallographically it is the correct one. However, some experiments
sensitive to the Fe positions, like neutron scattering on Fe moments, may have
more meaning in the 1-Fe (“unfolded”) zone. Theoretically, the virtue of using
this zone is its simplicity, since the number of bands is smaller by a factor
of two. From the point of view of symmetry, one might think of the As height
as a perturbation: if $z_{As}=0$, the electronic structure is correctly
reproduced in the unfolded BZ, and when $z_{As}$ increases, the procedure of
unfolding becomes less and less justified from the structural point of view.
Despite the fact that the As displacements from the Fe plane are not small,
use of the Fe-only band structure in the 1-Fe zone is frequently a good
approximation (see, however, Section 5.3) since DFT calculations predict the
band structure near the Fermi level to be mostly due to Fe $d$-bands, while
the As $p$-bands are about 2eV below [28].

The simplest model accounting for distinct electron and hole Fermi surfaces
would be a model in the 1-Fe zone with parabolic dispersions [34, 35],

$H=\sum_{\bi{k},\sigma,i=\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}}\varepsilon^{i}_{\bi{k}}c_{i\bi{k}\sigma}^{\dagger}c_{i\bi{k}\sigma}.$
(1)

Here, $c_{i\bi{k}\sigma}$ is the annihilation operator for an electron with
momentum $\bi{k}$, spin $\sigma$, and band index $i$,
$\varepsilon^{\alpha_{1,2}}_{\bi{k}}=-\frac{\bi{k}^{2}}{2m_{1,2}}+\mu$,
$\varepsilon^{\beta_{1}}_{\bi{k}}=\frac{(k_{x}-\pi/a)^{2}}{2m_{x}}+\frac{k_{y}^{2}}{2m_{y}}-\mu$,
and
$\varepsilon^{\beta_{2}}_{\bi{k}}=\frac{k_{x}^{2}}{2m_{y}}+\frac{(k_{y}-\pi/a)^{2}}{2m_{x}}-\mu$
are the dispersions of hole $\alpha_{i}$ and electron $\beta_{i}$ bands.

The electron pockets have to be distinct since they are located in the
different points of the Brillouin zone, but since the two hole pockets around
$\Gamma$ point are close in size, some physics can be captured by
approximating them as one band. After the folding procedure, this model
produces a Fermi surface topology which agrees with the results of DFT
calculations, see figure 2. This is useful for qualitative analysis of the
physics near the Fermi level, like magnetic susceptibility and formation of
the SDW state. The simplest extensions are the tight-binding models which can
be formulated in either the folded [36] or the unfolded BZ [37]. These can
reproduce correctly the Fermi surface and Fermi velocities, but neglect the
orbital character of the bands. The orbital character of electrons in
different bands is important e.g. for a correct analysis of scattering in
particle-particle and particle-hole channels. Furthermore, local Coulomb
interactions like Hubbard $U$ and Hund’s exchange $J$ are momentum-independent
only in the orbital representation.

According to DFT analysis, the band structure near the Fermi level consists
mainly of Fe $d$-orbitals, since the out-of-plane As $p$ orbitals hybridize
most effectively with Fe orbitals with both out-of-plane and in-plane
components. These conditions are satisfied for $d_{xz,yz}$ orbitals, so their
contribution at the Fermi level is dominant. The second largest contribution
comes from the $d_{xy}$ orbitals. The other two $d$ orbitals,
$d_{x^{2}-y^{2}}$ and $d_{3z^{2}-r^{2}}$, also contribute at low energies, but
their weight at the Fermi surface is minimal except in some materials near the
top of the Brillouin zone.

A possible minimal model for the FeBS is then one which includes two orbitals,
$d_{xz}$ and $d_{yz}$ [38]. This has the virtue of being simple while having
mostly correct orbital character along the Fermi surface – $(0,\pi)$ and
$(\pi,0)$ pockets have dominant $d_{xz}$ and $d_{yz}$ contributions,
respectively, and hole pockets with a mixture of these two orbitals. This
model has several significant disadvantages [39, 40], however. The first one
is that the Fermi velocities are incorrect, leading to incorrect tendencies
towards superconductivity and SDW formation. Second, it is missing small
patches of $d_{xy}$ character at the tips of the electron pockets, which can
be important for node formation [41] and transport [42]. Finally, a serious
flaw is the position of the larger hole pocket which is located at $(\pi,\pi)$
point of the 1-Fe zone. After the folding, this Fermi surface sheet is
centered at the $\Gamma$ point and resembles DFT results for the Fermi
surface. But the fact that the two bands forming hole sheets are not
degenerate at the $(0,0)$ point of the unfolded zone contradicts the symmetry
of the DFT wave functions. This problem can be adjusted by adding a $d_{xy}$
orbital to the model [43], but this three-orbital model has other pathologies
and fails to reproduce the peak at the nesting wave vector $\bi{Q}$ [44],
which is established both experimentally and theoretically. Although the
origin of the problem is not obvious, it is related to the matrix elements of
transformation from the orbital to the band basis.

The next step in the direction of more realistic models is to include four or
all five Fe $t_{2g}$ orbitals. Models of this type [40, 45] work well in
reproducing the DFT Fermi surface and band structure, and they are free from
the disadvantages described above. The kinetic energy in [40] is then given by
the Hamiltonian

$H_{0}=\sum_{\bi{k}\sigma}\sum_{\ell\ell^{\prime}}\left(\xi_{\ell\ell^{\prime}}(\bi{k})+\epsilon_{\ell}\delta_{\ell\ell^{\prime}}\right)d_{\ell\bi{k}\sigma}^{\dagger}d_{\ell^{\prime}\bi{k}\sigma},$
(2)

where $d_{\ell\bi{k}\sigma}^{\dagger}$ creates a particle with momentum
$\bi{k}$ and spin $\sigma$ in the orbital $\ell$,
$\xi_{\ell\ell^{\prime}}(\bi{k})$ are the hoppings, and $\epsilon_{\ell}$ are
the single-site level energies. This model with the parameters obtained by
Wannier fits to $k_{z}=0$ cuts of the 1111 band structure of Cao et al [46]
gives rise to the Fermi surfaces shown in figure 3 displayed in the unfolded
BZ [40, 47]. The undoped material has completely filled $d^{6}$ orbitals
corresponding to electron number $n=6$. Note for the hole doped case $n=5.95$
shown in figure 3 there is an extra hole FS $\gamma$ around the $(\pi,\pi)$
point. An important role is played by the orbital matrix elements
$a^{\ell}_{\nu}(\bi{k})=\langle\ell|\nu\bi{k}\rangle$ which relate the orbital
and band states. The dominant ($>50\%$) orbital weights
$|a^{\ell}_{\nu}(\bi{k})|^{2}$ on the Fermi surfaces are illustrated in figure
3 by the colors indicated.

  *
Figure 3: Fermi sheets of the five-band model in the unfolded BZ for $n=6.03$
(top) and $n=5.95$ (bottom) with colors indicating majority orbital character
(red=$d_{xz}$, green=$d_{yz}$, blue=$d_{xy}$). Note the $\gamma$ Fermi surface
sheet is a hole pocket which appears for $\sim 1\%$ hole doping [47].

The orbital matrix elements and the small patches of the $d_{xy}$ contribution
to the electron sheets play an important role in the formation of nodes in the
superconducting order parameter as will be discussed in Section 3.1. Natural
generalizations of the multiorbital models are 1) to include dispersion along
the $k_{z}$ direction; and 2) to include the proper effects of the 122 body
centered cubic symmetry. These effects appear to have important consequences
for the pairing in Ba1-xKxFe2As2, where effects of three-dimensionality are
more important than in 1111 systems [48, 49].

##  3 Theoretical background.

###  3.1 Spin fluctuation pairing.

It has become commonplace when a new class of superconductors is discovered to
discuss electronic pairing mechanisms as soon as there is some evidence that
the electron phonon mechanism is not strong enough to produce observed
critical temperatures; this was the case in both the cuprates and Fe-based
superconductors. Among many candidates for electronic pairing, Berk-Schrieffer
[50] type spin fluctuation theories are popular because they are relatively
simple to express and give some qualitatively correct results in the well-
known cases of 3He and the cuprates. The interesting history of the
development of this theory in the one-band case has been reviewed by Scalapino
[51]. It is important to keep in mind that this type of description cannot be
regarded as the complete answer even in superfluid 3He, where the true pairing
interaction contains a significant density fluctuation component, while in the
cuprates it is controversial whether the full pairing interaction can be
described by a simple boson exchange theory at all. Nevertheless spin
fluctuation theories can explain the symmetry of the order parameter in both
systems quite well, in part because other interaction channels are projected
out in the ground state. For example, in the cuprates, the $d$-wave nature of
the pair wave function follows from the strongly peaked spin susceptibility at
$(\pi,\pi)$, characteristic of repulsive local interactions between electrons
hopping on a square lattice. In the Fe-based superconductors, the early
realization that the Fermi surface consisted of small, nearly nested electron
and hole pockets led to the analogous anticipation of a strongly peaked
susceptibility near $\bi{Q}=(\pi,0)$, and a corresponding pairing instability
with sign change between electron and hole sheets. Below we illustrate the
basic equations leading to the canonical $d$-wave case within the one-band
Berk-Schrieffer approach, and sketch the generalization to the
multiorbital/multiband case. Many authors have obtained similar results with a
variety of related methods (see Section 3.1, “similar approaches” below).

It is important to emphasize that while spin fluctuation theories come in many
varieties and flavors, they share more commonalities than differences. Indeed,
as will be discussed later in this Section, in the singlet channel spin
fluctuations exchange always leads to a repulsive interaction, and therefore
can only realize sign-changing superconducting states. If this interaction is
sufficiently strong at some particular momentum it will necessarily result in
superconductivity. In case of a single Fermi surface this superconductivity
will necessarily be nodal, usually of a $d$-wave symmetry. Examples of this
situation are high-$T_{c}$ cuprates and, possibly, overdoped KFe2As2 FeBS. On
the other hand, in a multiband system there may be a possibility to avoid
nodes, while still preserving a sign-changing structure. Examples of this are:
$d$-wave superconductivity that can develop in a cubic system with Fermi
surface pockets around the $\mathrm{X}$ points and in a hexagonal system with
pockets around the $\mathrm{M}$ points [52], $d$-wave superconductivity in a
tetragonal system with FS pockets near $\mathrm{X}$/$\mathrm{Y}$ points [45],
$s_{\pm}$ superconductivity proposed for bilayer cuprates (where the bonding
and the antibonding band have opposite signs of the order parameter) [53, 54],
and the electron-hole $s_{\pm}$ superconductivity that can develop in
semimetals [55].

While all these options had been discussed theoretically many years ago, all
of them have been revisited in connection to FeBS. The last option is now the
leading contender for the majority of pnictides and selenides, while $d$-wave
superconductivity has been proposed for KFe2As2 that is on the verge of losing
its semimetallic character (see Section 5.4), a version of the nodeless
$d$-wave state first discussed in [45] has been proposed for Se-based 122
materials (see Section 5.3), and the “bonding-antibonding” nodeless $s_{\pm}$
state, analogous to that discussed in [53, 54] has been also proposed for
these selenides. What is important, however, is that as long as the spin
fluctuations are strong and nonuniform, some superconducting state will
unavoidably form, and the details of the electronic structure and of the
pairing interaction will decide which particular symmetry will form, often in
close competition with other symmetries.

##### Historical: ferromagnetic spin fluctuations.

The original proposal of superconducting pairing arising from magnetic
interactions was put forward by Emery [56] and by Berk and Schrieffer [50],
who were interested primarily in transition metal elements and nearly
ferromagnetic metals. Such systems are considered to be close to a
ferromagnetic ordering transition in the Stoner sense, i.e. their
susceptibility may be approximated by $\chi=\chi_{0}/(1-U\chi_{0})$, where $U$
is a local Hubbard-like Coulomb matrix element assumed to be large since
$U\chi_{0}\simeq 1$ ($\chi_{0}$ is the susceptibility in the absence of
interactions). Physically this means a spin up electron traveling through the
medium polarizes the spins around it ferromagnetically, lowering the system’s
energy as illustrated in figure 4.

  *
Figure 4: Heuristic snapshot of pairing of two test spins by ferromagnetic
spin fluctuations.

The excitations being “exchanged” in such a picture are not well- defined
collective modes of the system such a phonons or magnons, but rather
”paramagnons”, defined by the existence of a peak-like structure in the the
imaginary (absorptive) part of the small q susceptibility. Diagramatically,
only the ring-type diagrams shown in figure 5 contribute to the equal-spin
channel $\Gamma_{\uparrow\uparrow}$, whereas both ring and ladder-type
diagrams contribute to the effective pairing vertex in the opposite-spin
channel $\Gamma_{\uparrow\downarrow}$. These series may be summed in the usual
way to give:

$\displaystyle\Gamma_{\uparrow\uparrow}$ $\displaystyle=$
$\displaystyle{-U^{2}\chi_{0}(\bi{k}^{\prime}-\bi{k})\over
1-U^{2}{\chi_{0}}^{2}(k^{\prime}-k)},$ (3)
$\displaystyle\Gamma_{\uparrow\downarrow}$ $\displaystyle=$
$\displaystyle{U\over
1-U^{2}{\chi_{0}}^{2}(\bi{k}^{\prime}-\bi{k})}+{U^{2}\chi_{0}(\bi{k}^{\prime}+\bi{k})\over
1-U^{2}{\chi_{0}}^{2}(\bi{k}^{\prime}+\bi{k})}$ (4) $\displaystyle=$
$\displaystyle U^{2}\left({3\over 2}\chi^{s}-{1\over 2}\chi^{c}\right)+U,$ (5)

where we have defined $\chi^{s}\equiv\chi_{0}/(1-U\chi_{0})$ and
$\chi^{c}=\chi_{0}/(1+U\chi_{0})$, and in the last step we have changed
$-\bi{k}$ to $\bi{k}$ in the second term of $\Gamma_{\uparrow\downarrow}$
because we assume we work in the even parity (singlet pairing) channel. The
total pairing vertex in the triplet (singlet) channel is $\Gamma_{t}={1\over
2}\Gamma_{\uparrow\uparrow}$ ($\Gamma_{s}={1\over
2}(2\Gamma_{\uparrow\downarrow}-\Gamma_{\uparrow\uparrow})$). In the original
paramagnon theory, $\chi_{0}(\bi{q})$ is the noninteracting susceptibility of
the (continuum) Fermi gas, i.e. the Lindhard function. This function at small
frequency has a maximum at q=0, meaning correlations are indeed ferromagnetic.
Thus due to the negative sign in the equation for $\Gamma_{\uparrow\uparrow}$
(note $\chi_{0}>0$ and $U\chi_{0}<1$ to prevent a magnetic instability),
pairing is attractive in the triplet channel and singlet superconductivity is
suppressed, which explains why Pd, for example, does not superconduct [50].

  *
Figure 5: Effective pairing interaction between (a) equal spins and (b)
opposite spins. Solid lines are electron $G$’s, dashed lines Hubbard $U$’s,
i.e. interactions between electrons of opposite spin only.

##### Antiferromagnetic spin fluctuations.

In the context of heavy fermion systems it was realized [57, 58] that strong
antiferromagnetic spin fluctuations in either the weak or strong coupling
limit lead naturally to spin singlet, $d$-wave pairing. The weak coupling
argument has been elegantly reviewed by Scalapino [59]. Suppose the
susceptibility is strongly peaked near some wave vector $\bi{Q}$. The form of
the singlet interaction is

$\displaystyle\Gamma_{s}(\bi{k},\bi{k}^{\prime})=\frac{3}{2}U^{2}\frac{\chi_{0}(\bi{q})}{1-U\chi_{0}(\bi{q})}$
(6)

if we neglect terms which are small near the RPA instability
$U\chi_{0}(\bi{q})\rightarrow 0$ [60]. This now implies that
$\Gamma_{s}(\bi{q})$ is also peaked at this wavevector, but is always
repulsive. Nevertheless, if one examines the BCS gap equation for this
interaction

$\Delta{k}=-{\sum_{\bi{k}^{\prime}}}^{\prime}\Gamma_{s}(\bi{k},\bi{k}^{\prime}){\Delta{k}^{\prime}\over
2E{k}^{\prime}}{\rm tanh}{E{k}^{\prime}\over 2T},$ (7)

one sees immediately that an isotropic state cannot be a solution, but that if
the state changes sign,

$\displaystyle\Delta{k}=-\Delta_{\bi{k}+\bi{Q}},$ (8)

a solution will be allowed.

In the cuprates, $\chi$ is peaked at $\bi{Q}\simeq(\pi,\pi)$, and the two
possible states of this type which involve pairing on nearest neighbor bonds
only are

$\displaystyle\Delta{k}^{d,s}$ $\displaystyle=$ $\displaystyle\Delta_{0}(\cos
k_{x}a\mp\cos k_{y}a).$ (9)

Which state will be stabilized then depends on the Fermi surface in question.
So we need to use the fact that the states close to the Fermi surface are the
most important in Equation (7), and examine the pairing kernel for these
momenta. For example, for a $(\pi/a,0)\rightarrow(0,\pi/a)$ scattering,
$\Delta{k}^{s}$ satisfies Equation (8) by being zero, whereas $\Delta{k}^{d}$
is nonzero and changes sign, contributing to the condensation energy. It
should therefore not be surprising that the end result of a complete numerical
evaluation of Equation (7) over a cuprate Fermi surface gives $d$-wave
pairing.

An alternative way to approach the question of how a purely repulsive
interaction allows for pair formation is to examine the interaction Fourier
transformed back to real space, where it shows regions (in the cuprates with
$\bi{Q}=(\pi,\pi)$, on nearest neighbor sites), where the pair potential
becomes attractive [59] if the interaction is sufficiently nonuniform in
momentum space.

##### Effect of 2 bands.

The discussion in Section 2 described the unusual, fully compensated Fermi
surface of the parent Fe-pnictide materials. In the Fe-based superconductors,
the proximity of the Fe $d$-states to the Fermi level has led many authors to
consider a Hamiltonian which takes a kinetic energy $H_{0}$ consisting of the
bands derived from the 5 Fe $d$-orbitals found in DFT, approximated within
some tight-binding or other scheme. The 2D Fermi surface in the 1-Fe zone thus
obtained is shown in figure 3. Like DFT, this model Fermi surface is
characterized by small concentric hole pockets around the $\Gamma$ point and
slightly elliptical electron pockets around the $\mathrm{M}$ points. Mazin and
collaborators [29] pointed out that modeling these pockets in the simplest
possible way, allowing for one hole and one electron pocket, led to a very
simple and elegant generalization of the “standard” argument for $d$-wave
pairing in the cuprates (a similar result was reached in a strong-coupling
approach by Seo et al [61]). In a weak-coupling approach, the near-nesting of
the hole and electron pockets suggested the existence of a peak in the spin
susceptibility at $\bi{Q}=(\pi,0)$ in the 1-Fe zone. The gap equation (7) then
admits a solution with the property (8) if there is a sign change of
$\Delta_{k}$ between electron and hole pockets. In the simplest version of
this theory, the anisotropy on each electron sheet is neglected, with the
argument that the pockets are small. This leads to the so-called isotropic
“$s_{\pm}$” state (figure 1) which has become the leading candidate for the
discussion of many of the superconducting properties of these materials. Note
that such a state has the full symmetry of the crystal lattice and is
therefore formally an $A_{1g}$ or “$s$-wave” state, but with fundamentally
different gap structure which leads to many nontrivial superconducting
properties.

##### Spin fluctuation pairing in multiorbital systems.

More realistic analyses of pairing in these systems by electronic interactions
soon followed. Many authors began with a Hamiltonian consisting of a kinetic
energy $H_{0}$ for the effective Fe bands as described above, plus an
interaction $H_{int}$ containing all possible two-body on-site interactions
between electrons in Fe orbitals as a good starting point for a microscopic
description of this system,

$\displaystyle H$ $\displaystyle=$ $\displaystyle
H_{0}+\bar{U}\sum_{i,\ell}n_{i\ell\uparrow}n_{i\ell\downarrow}+\bar{U}^{\prime}\sum_{i,\ell^{\prime}<\ell}n_{i\ell}n_{i\ell^{\prime}}$
$\displaystyle+\bar{J}\sum_{i,\ell^{\prime}<\ell}\sum_{\sigma,\sigma^{\prime}}c_{i\ell\sigma}^{\dagger}c_{i\ell^{\prime}\sigma^{\prime}}^{\dagger}c_{i\ell\sigma^{\prime}}c_{i\ell^{\prime}\sigma}$
$\displaystyle+\bar{J}^{\prime}\sum_{i,\ell^{\prime}\neq\ell}c_{i\ell\uparrow}^{\dagger}c_{i\ell\downarrow}^{\dagger}c_{i\ell^{\prime}\downarrow}c_{i\ell^{\prime}\uparrow}$

where $n_{i\ell}=n_{i,\ell\uparrow}+n_{i\ell\downarrow}$. The Coulomb
parameters ${\bar{U}}$, ${\bar{U}}^{\prime}$, ${\bar{J}}$, and
${\bar{J}}^{\prime}$ are in the notation of Kuroki et al [45], and are related
to those used by Graser et al [40] by $\bar{U}=U$, $\bar{J}=J/2$,
$\bar{U}^{\prime}=V+J/4$, and $\bar{J}^{\prime}=J^{\prime}$. The
noninteracting $H_{0}$ is given by a tight-binding model spanned by the 5 Fe
$d$-orbitals, Equation (2), which give rise to the Fermi surfaces shown in
figure 3.

In Equation (3.1), we have distinguished the intra- and inter-orbital Coulomb
repulsion, as well as the Hund’s rule exchange $\bar{J}$ and “pair hopping”
term $\bar{J}^{\prime}$ for generality, but if they are generated from a
single two-body term with spin rotational invariance they are related by
$\bar{U}^{\prime}=\bar{U}-2\bar{J}$ and $\bar{J}^{\prime}=\bar{J}$. In a real
crystal, such a local symmetry will not always hold.

The generalization of the simple 1-band Berk-Schrieffer spin fluctuation
theory to the multiorbital case was discussed by many authors [62, 63]. The
effective pair scattering vertex $\Gamma(\bi{k},\bi{k}^{\prime})$ between
bands $i$ and $j$ in the singlet channel is

$\displaystyle{\Gamma}_{ij}(\bi{k},\bi{k}^{\prime})$ $\displaystyle=$
$\displaystyle\mathrm{Re}\left[\sum_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}}a_{\nu_{i}}^{\ell_{2},*}(\bi{k})a_{\nu_{i}}^{\ell_{3},*}(-\bi{k})\right.$
$\displaystyle\left.\times{\Gamma}_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}}(\bi{k},\bi{k}^{\prime},\omega=0)a_{\nu_{j}}^{\ell_{1}}(\bi{k}^{\prime})a_{\nu_{j}}^{\ell_{4}}(-\bi{k}^{\prime})\right]$

where the momenta $\bi{k}$ and $\bi{k}^{\prime}$ are confined to the various
Fermi surface sheets with $\bi{k}\in C_{i}$ and $\bi{k}^{\prime}\in C_{j}$.
The orbital vertex functions $\Gamma_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}}$
describe the particle-particle scattering of electrons in orbitals
$\ell_{1},\ell_{4}$ into $\ell_{2},\ell_{3}$ (see figure 6) and in the
fluctuation exchange formulation [64, 63] are given by

$\displaystyle{\Gamma}_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}}(\bi{k},\bi{k}^{\prime},\omega)=\left[\frac{3}{2}\bar{U}^{s}\chi_{1}^{\rm
RPA}(\bi{k}-\bi{k}^{\prime},\omega)\bar{U}^{s}+\right.\,~{}~{}~{}~{}~{}~{}\,$
$\displaystyle\,~{}~{}~{}~{}~{}\left.\frac{1}{2}\bar{U}^{s}-\frac{1}{2}\bar{U}^{c}\chi_{0}^{\rm
RPA}(\bi{k}-\bi{k}^{\prime},\omega)\bar{U}^{c}+\frac{1}{2}\bar{U}^{c}\right]_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}},$
(12)

where each of the quantities $\bar{U}^{s}$, $\bar{U}^{c}$, $\chi_{1}$, etc
represent matrices in orbital space which depend on the interaction
parameters. This is the multiorbital generalization of Eq. (5). Here
$\chi_{1}^{\rm RPA}$ describes the spin-fluctuation contribution and
$\chi_{0}^{\rm RPA}$ the orbital (charge)-fluctuation contribution, determined
by Dyson-type equations as

$(\chi_{0}^{RPA})_{st}^{pq}=\chi_{st}^{pq}-(\chi_{0}^{RPA})_{uv}^{pq}(U^{c})_{wz}^{uv}\chi_{st}^{wz}$
(13)

and

$(\chi_{1}^{RPA})_{st}^{pq}=\chi_{st}^{pq}+(\chi_{1}^{RPA})_{uv}^{pq}(U^{s})_{wz}^{uv}\chi_{st}^{wz},$
(14)

where repeated indices are summed over. Here $\chi_{st}^{pq}$ is a generalized
multiorbital susceptibility (see [47]).

  *
Figure 6: Top: pairing vertex $\Gamma_{\ell_{1},\ell_{2},\ell_{3},\ell_{4}}$
defined in terms of orbital states $\ell_{i}$ of incoming and outgoing
electrons. Bottom: representative examples of classes of orbital vertices
referred to in the text: intra-, inter- and mixed orbital vertices. From [47].

##### Results of microscopic theory.

The simplest goal of the microscopic approach is to calculate the critical
temperature $T_{c}$ via the linearized gap equation and determine the symmetry
of the pairing instability there. If one writes the superconducting order
parameter $\Delta(\bi{k})$ as $\Delta g(\bi{k})$, with $g(\bi{k})$ a
dimensionless function describing the momentum dependence on the Fermi
surface, then $g(\bi{k})$ is given as the stationary solution of the
dimensionless pairing strength functional [40]

$\lambda[g(\bi{k})]=-\frac{\sum_{ij}\oint_{C_{i}}\frac{dk_{\parallel}}{v_{F}(\bi{k})}\oint_{C_{j}}\frac{dk_{\parallel}^{\prime}}{v_{F}(\bi{k}^{\prime})}g(\bi{k}){\Gamma}_{ij}(\bi{k},\bi{k}^{\prime})g(\bi{k}^{\prime})}{(2\pi)^{2}\sum_{i}\oint_{C_{i}}\frac{dk_{\parallel}}{v_{F}(\bi{k})}[g(\bi{k})]^{2}}$
(15)

with the largest eigenvalue $\lambda$, which provides a dimensionless measure
of the pairing strength. Here $\bi{k}$ and $\bi{k}^{\prime}$ are restricted to
the various Fermi surfaces $\bi{k}\in C_{i}$ and $\bi{k}^{\prime}\in C_{j}$
and $v_{F,\nu}(\bi{k})=|\nabla{k}E_{\nu}(\bi{k})|$ is the Fermi velocity on a
given Fermi sheet.

In figure 7, we plot the leading dimensionless gap function $g(k)$ derived
from the RPA theory around the electron $\beta_{1}$ sheet for two different
values of the doping, for spin-rotationally invariant parameters $U=1.3$ and
$J=0.2$. The gap on the hole sheets is seen to be essentially isotropic, while
on the electron sheets the average of the gap is of opposite sign compared to
the hole sheets, and is highly anisotropic, with nodes in the case of electron
doping. One would like to understand the origin of the anisotropy and its
doping dependence.

  *
Figure 7: Top: false color plots of dimensionless gap function $g(k)$ on
various Fermi surface sheets for electron doped $n=6.03$ (left) and hole doped
$n=5.95$ (right). Bottom: detail of $g(k)$ on $\beta_{1}$ pocket for $U=1.3$
and $n=5.95,\bar{J}=0.2$ (red squares) and $n=6.01,\bar{J}=0.2$ (blue
circles). Here the angle $\phi$ is measured from the $k_{x}$-axis. From Kemper
et al [47].

##### Physical origins of anisotropy of pair state and node formation.

An unusual element which emerges from the spin fluctuation pairing analysis
based on (3.1) is that the orbital structure of the Fermi surface can have a
significant impact on the anisotropy of the pair state. The intra-orbital
scattering of $d_{xz}$ and $d_{yz}$ pairs between the $\alpha$ and $\beta$
Fermi surfaces by $(\pi,0)$ and $(0,\pi)$ spin fluctuations (figure 6) leads
naturally to a gap which changes sign between the $d_{xz}/d_{yz}$ parts of the
$\alpha$ Fermi surfaces and the $d_{yz}$ and $d_{xz}$ parts of the $\beta_{1}$
and $\beta_{2}$ electron pockets, just as in the early proposal of Mazin et al
[29], see figure 3. However, as discussed by Maier et al [41], Kuroki et al
[16], and Kemper et al [47], scattering between the $\beta_{1}$ and
$\beta_{2}$ Fermi surfaces frustrates the formation of an isotropic $s_{\pm}$
gap there. Furthermore, this anisotropy can also be driven by the effect of
the intraband Coulomb interaction. Finally, inter-orbital pair scattering can
also occur, depending upon $\bar{U}^{\prime}$ and $\bar{J}^{\prime}$. The
contributions to the total pairing interaction arising from mixed and
interorbital processes may be shown explicitly to be subdominant to the
intraorbital processes but important for nodal formation [47].

On the other hand, the $\beta_{1}-\beta_{2}$ $d_{xy}$ orbital frustration is
weaker or does not occur if an additional hole pocket $\gamma$ of $d_{xy}$
character is present (see figure 3); these scattering processes are at
($\pi,0$) in the unfolded zone and therefore support isotropic $s_{\pm}$
pairing (see figure 7). Kuroki et al [16] took the important step of relating
the crystal structure, electronic structure, and pairing, by noting that the
As height above the Fe plane in the 1111 family was a crucial variable
controlling the appearance of the $\gamma$ pocket and thus driving the
isotropy of the $s_{\pm}$ state. It is important to note that the transition
between nodal and nodeless $A_{1g}$ gap structures, investigated by a number
of authors [16, 65, 47, 66, 67, 68], does not involve any symmetry change, and
relies only on the details of the pairing interaction.

##### Similar approaches.

In this discussion we have presented “spin fluctuation theory” in terms of an
RPA approximation to the pair scattering vertices $\Gamma$, which also
includes subleading charge/orbital contributions via Equation (3.1). It is
important to note that other approaches have obtained very similar results for
the Hamiltonian (3.1). The most closely related technique is the FLEX
(fluctuation exchange) approximation [64], which is a conserving approximation
to the Luttinger-Ward functional and the self-energy. Several authors have
applied this approach to the pairing problem [69, 70, 71, 72, 73], employing a
five-band model for FeBS based on Wannier fits to DFT results. Qualitatively,
results are similar for the leading pairing instabilities, including the
doping dependence [72] (see figure 8), although node formation or strong
angular anisotropy have not been observed in FLEX so far. In addition, FLEX
has certain well-known peculiarities which need to be handled carefully. In
particular, the real part of the self-energy, which shifts the band structure
in a momentum-dependent way, includes some of the correlations already
included in DFT; thus the use of a kinetic energy $H_{0}$ fit to DFT plus the
self-consistent FLEX self energy tends to overcount these correlations. This
problem is traditionally handled by subtracting the real part of the static
self-energy [73].

  *
Figure 8: Phase diagram in FLEX treatment of SDW and SC instabilities (from
[72]).

A further popular approach is the functional renormalization group [74] (fRG),
which has the advantage that it is capable of studying various instabilities
of the system on an equal footing, which is of particular relevance for
studying the competition between SDW and superconductivity in these systems in
realistic models. Numerical RG equations are derived by dividing the Brillouin
zone into $N$ patches, and then summing at each RG step over the five one-loop
Feynman diagrams to compute the renormalized 4-point vertex function. This
technique was pioneered in connection with FeBS by Fang et al [75],
generalizing numerical work on 1-band systems [76], and has been extended and
applied to various FeBS [66, 67, 68, 77, 78]. These works are the intellectual
offspring of earlier analytical (logarithmic) RG calculations in two-band
models [79, 80] which were later generalized to include low-order angular
harmonics [65] to describe gap anisotropy. Recently this approach has been
extended and compared directly to the RPA results, such that the doping
evolution of the fundamental band interactions could be obtained [81].

It is remarkable that rather dissimilar techniques, none of which are
controlled in the usual perturbative sense, give such similar results. To
illustrate this, in figure 9 we plot results for a FeBS Fermi surface for
LaFePO or LaFeAsO including only the two inner hole pockets, obtained by fRG
and RPA techniques. Although the scale of the interactions are quite
different, the ratio $J/U$ is $\ll 1$ in both cases. Note that the order
parameter on both hole pockets is quite isotropic, but that on the electron
sheets nodes appear, but consistent with the overall average $s_{\pm}$
character of the state.

  *
Figure 9: Comparison of fRG and RPA results for the gap function $g(\theta)$
vs. $\theta$, with $\theta$ parametrizing each Fermi pocket, for 4-Fermi
pocket model of LaFePO/ LaFeAsO. Left: FRG results from [67] with
$\bar{U}=3.5$, $\bar{U}^{\prime}=2.0$, $\bar{J}=\bar{J}^{\prime}=0.7$. Right:
RPA results from [41] with $\bar{U}=1.67$, $\bar{U}^{\prime}=1.46$,
$\bar{J}=\bar{J}^{\prime}=0.10$. Bands $\alpha_{1}$, $\alpha_{2}$, $\beta_{1}$
and $\beta_{2}$ correspond to inner and outer hole pockets, and two electron
pockets, respectively. Energy units for parameters are in eV, units on
vertical axes of both figures are arbitrary.

Finally, we comment on so-called “strong-coupling” approaches to pairing,
based on the $J_{1}-J_{2}$ multiple competing exchange model of the magnetism
in these systems [82, 83, 84]. Note that this term frequently appears in
reference to the large size of the Hubbard interaction relative to the
bandwidth, which in the 1-band case at half-filling allows for a description
in terms of the spin degrees of freedom as the charge degrees of freedom
become localized. The spin dynamics can become more localized in situations
other than the canonical Mott-Hubbard case, however, and are in fact well
described by DFT, which for the FeBS gives a typical energy scale for the
magnetic interactions of order 100 meV or larger. Thus the moments are large
and quite localized. This is not a contradiction; even in genuinely itinerant
systems (elemental 3D metals are a good example) magnetic interactions are
essentially local, decaying with distance as a power law.

This magnetic model leads to a competition between the Neel and stripe-
collinear orders, also present in the (itinerant) DFT calculation,
corresponding to the same ground state magnetic pattern and to a similar
structure of spin fluctuations in the reciprocal state [a maximum near
$(\pi,0)$]. In [61], a $t-J_{1}-J_{2}$ model with two bands was studied, and
the exchange terms were decoupled in mean field in the pairing channel. In
this procedure, the nearest neighbor exchange $J_{1}$ induces competing $\cos
k_{x}+\cos k_{y}$ and $\cos k_{x}-\cos k_{y}$ ($s$\- and
$d_{x^{2}-y^{2}}$-wave) pairing harmonics, while the next nearest neighbor
exchange leads to $\cos k_{x}\cos k_{y}$ and $\sin k_{x}\sin k_{y}$ ($s$\- and
$d_{xy}$-wave). In the region of the general phase diagram with $J_{2}\gtrsim
J_{1}$, $\cos k_{x}\cos k_{y}$ was the leading instability for the 2-band
Fermi surface, leading to a nodeless $s_{\pm}$ state, and a similar ground
state was also found for a 5-orbital model [85].

Mean field theories of the strong coupling type (see also models where $U$ and
$J_{1,2}$-type terms are treated as independent before the mean field step
[86, 43]) show an intriguing set of circumstantial agreement with the
predictions of itinerant weak-coupling models, despite the lack of logical
continuity between the two types of models. Yet it is not that surprising.
Indeed, the pairing symmetry in any spin-fluctuation model is mainly defined
by matching the structure of the spin fluctuations and the FS geometry. Since
both are rather similar in the two approaches, not surprisingly, the main
results agree. In fact, Wang et al have shown that in the case with 5 Fermi
surface pockets, the low energy spin and charge excitations in the fRG
treatment of the 5-orbital model (3.1) overlap very well with those of the
$t-J_{1}-J_{2}$ model. At the Hamiltonian level, however, there is no way to
derive these particular strong coupling models (with unrenormalized kinetic
term) from the general model with on-site interactions.

An advantage of such strong coupling approaches is that they capture the local
character and large amplitude of Fe magnetic moments, in agreement with the
DFT calculations. They also have some attractive conceptual simplicity. On the
other hand, they have several uncontrollable shortcomings, which make proper
application of this approach tricky.

First and foremost, such theories artificially separate the itinerant
electrons and the local moments, as if the latter were coming from a separate
atomic species. Yet, the moments are formed by exactly the same electrons that
form the band structure, which also mediate the magnetic interaction mapped
onto the $J_{1}-J_{2}$ Heisenberg Hamiltonian. In some papers an attempt to
account for this fact is made by adding spin susceptibility of itinerant
electrons to the aforementioned Heisenberg Hamiltonian, which essentially
amounts to a double counting.

Second, actual band structure calculations [87, 88, 89] cannot be mapped onto
a Heisenberg Hamiltonian of any range. They can be mapped onto a Heisenberg
Hamiltonian with biquadratic exchange, or possibly to a more complicated
Hamiltonian (such as ring exchange), but not onto a pure Heisenberg model.

Third, “strong-coupling” models essentially fix the shape of the spin-
fluctuation induced interaction; therefore, the resulting solution fixes the
structure of the gap nodes in momentum space, so that the amplitude,
anisotropy and possible nodes on actual FSs depends only on the proximity of
these FSs to the imaginary nodal lines. This result is not corroborated by the
weak coupling calculations and is likely unphysical.

The partial agreement between fRG and the $t-J_{1}-J_{2}$ results will
therefore remain a curiosity until a more concrete understanding why the
similarity of the low energy sectors of the two theories is observed.

##### Pairing in multiorbital systems from DFT perspective.

RPA calculations, as well as other approaches discussed above, use the same
model Hamiltonian (3.1), but resort to different approximate methods to solve
the problem of superconductivity emerging from this Hamiltonian. Comparing
results of such different approaches one can get an idea of how accurate these
solutions are. Yet this Hamiltonian itself is a rather uncontrollable
approximation, and one can legitimately ask the question, to what extent the
simplifications introduced when constructing this Hamiltonian are justified.
Indeed, some of the qualitative results discussed above, such as anisotropy
(and possible nodes) of the order parameter, are intimately related to the
details of the model that may or may not be sufficiently universal.
Specifically, there are two aspects of the model that appear to be
qualitatively significant. First, the leading spin-dependent term in this
Hamiltonian, $\bar{U}\sum_{i,\ell}n_{i\ell\uparrow}n_{i\ell\downarrow},$ is
orbital diagonal (there are two other spin-dependent terms, the Hund’s term,
proportional to $\bar{J},$ and pair hopping term proportional to
$\bar{J}^{\prime},$ but these are smaller). That makes the pairing interaction
rather sensitive to the orbital composition, with the parts of the Fermi
surface whose orbital content is not matched by that of the rest of the Fermi
surface effectively decoupled from the rest. This is true in RPA, FLEX, fRG or
any other method based on the same Hamiltonian. On the other hand, there is
growing belief [90] that physics of these materials is controlled by the Hund
rule’s coupling rather than by the direct Coulomb repulsion, as implied for
instance in the original expressions (3), (3.1). The second aspect of the
model is that it does not include the retardation effects, at least on the RPA
level. The large Coulomb repulsion that in real life is logarithmically
reduced in the calculations needs to be avoided, which can be done, for an on-
site local repulsion, by tuning the order parameter in such a way that it
integrates to zero. These two problems, characteristic for this model, suggest
that the tendency towards gap anisotropy is probably somewhat overestimated in
this approach.

It is instructive to compare the effect of direct Coulomb repulsion in
conventional superconductors, such as V, to another $3d$ transition metal with
the interaction parameters in the charge channel not much different from Fe
based superconductors. There the Coulomb repulsion, obviously strong, is
renormalized as

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle UN_{0}$ (16)
$\displaystyle\mu^{*}$ $\displaystyle=$
$\displaystyle\mu/[1+\mu\ln\frac{E_{C}}{E_{b}}],$ (17)

where $N_{0}$ is the density of states at the Fermi level, $E_{C}$ is a
characteristic electronic energy (it may be the total band width, or the
plasmon energy, or a combination thereof), and $E_{b}$ is the energy of the
pairing bosons. Given that $\mu\ln\frac{E_{C}}{E_{b}}\gg 1,$ $\mu^{*}\approx
1/\ln\frac{E_{C}}{E_{b}}\sim 0.1-0.15$ for typical conventional
superconductors. For most Fe superconductors $T_{c}$ is only a factor of two
or three larger than for the best transition metal superconductors (niobum’s
$T_{c}$ is 10 K, that of some binary alloys reaches 23 K), and the full d-band
widths are comparable, so if the Coulomb repulsion were uniform one could
apply similar reasoning and conclude that for these systems $\mu^{*}\sim
0.15-0.2$. If, on the other hand, the unrenormalized Coulomb repulsion is
different in the interband and intraband channels, the renormalization
equation should be written differently, namely

$\mu_{ij}^{*}=\mu_{ij}-\sum_{n}\mu_{in}\ln\frac{E_{C}}{E_{b}}\mu_{nj}^{*}$
(18)

where $i$, $j$, and $n$ are band indices. The solution in the limit
$|\mu_{i\neq j}-\mu_{ii}|\ln\frac{E_{C}}{E_{b}}\gg 1$ (which may or may not
fold for FeBS) reads: $\mu_{ii}\approx 1/\ln\frac{E_{C}}{E_{b}}$, $\mu_{i\neq
j}\approx const/\ln^{2}\frac{E_{C}}{E_{b}}$. Thus the renormalization in a
multiband system has two effects: (1) it strongly reduces the effect of
Coulomb repulsion in general and (2) it suppresses interband repulsion
compared with intraband. The latter effect has important implication – effect
of ‘Coulomb avoidance’. Indeed, if the Coulomb repulsion does not depend on
the wave vector, the condition for it to cancel out of the equations on
$T_{c}$ is that the order parameter, when averaged over the entire FS and all
its sheets, integrates to zero. If the Coulomb repulsion is only present in
the intraband channel, than to avoid its effect on $T_{c}$ entirely, the order
parameter must integrate to zero in each band separately.

It should be kept in mind that the preceding argument implicitly relies upon
an analogy with the Eliashberg theory, assuming that all relevant interactions
can be separated into groups: a pairing interaction (boson exchange) that is
restricted to sufficiently low energies (recall that the spin resonance in
FeBS occurs at the energies around 30-40 meV), and is also subject to the
Migdal theorem, and a direct Coulomb repulsion that exists at all energies,
and gets renormalized after the ladder diagrams are summed. Most existing
approaches explicitly or implicitly make this very assumption, with the
notable exception of the renormalization group techniques. However, in reality
for electronic (as opposed to a phonon) superconductivity all interactions
emerge from electrons themselves, and such separation is not always possible.
Nor is it always possible to limit the approximation to the ladder diagrams.
In more complex cases, formally involving parquet diagrams, the solution may
depend on whether the Coulomb interaction is first screened and then
renormalized or first renormalized and then screened, etc [91]. On the other
hand, in the renormalization group approach interactions are not separated
into these channels, and in principle all retardation effects are supposed to
be included. While it is very difficult to make a one-to-one correspondence
between such approaches and RPA (and similar) calculations, is quite obvious
that they are based on substantially different physical assumptions. Why,
nevertheless, results obtained in RPA and in functional renormalization group
approaches are quantitatively similar, is unclear and needs to be understood.

Given these uncertainties involved in RPA calculations described in the
previous sections, as well as other similar approaches, it is interesting to
look at the problem at hand not from the Hubbard model point of view, but from
the opposite, density functional one. Indeed many believe that, as opposed to
the high $T_{c}$ cuprates, DFT is a reasonable starting point for these
materials, and in some sense it is more consistent to use DFT for the full
susceptibility as long as we use the DFT band structure for the noninteracting
one. For instance, in classical semiconductors, such as Si, DFT (even the
exact DFT) underestimates the fundamental gap, and thus yields an incorrect
noninteracting susceptibility, yet the full susceptibility calculated entirely
within DFT is by definition exact [92]. Note that within DFT there are only
two electronic interactions: the charge interaction is defined as
$\delta^{2}E_{tot}/\delta n(\bi{r})\delta n(\bi{r}^{\prime})$, which includes
the Coulomb (Hartree) and the exchange-correlation interactions, and the spin
interaction is defined as
$I_{xc}(\bi{r},\bi{r}^{\prime})=\delta^{2}E_{tot}/\delta
n_{\uparrow}(\bi{r})\delta n_{\downarrow}(\bi{r}^{\prime})$. Therefore, RPA is
exact in DFT. Obviously, LDA and GGA approximations to DFT are not exact, yet
one may think that the full spin susceptibility calculated, say, within LDA-
DFT is not a bad approximation. The only caveat is that in a quantum critical
material LDA deviates from the exact DFT in one systematic way: the reduction
of the Hund rule coupling due to long range spin fluctuations is not included.
Emprically, it can be accounted for by scaling $I_{xc}$ down by 15-20%. After
that, one can use Equations (3), (4) using
$I_{xc}(\bi{r},\bi{r}^{\prime})=I_{xc}(\bi{r})\delta(\bi{r}-\bi{r}^{\prime})$
instead of $U.$

One can work in the orbital basis again, just as one does in the Hubbard
model, except that the spin-dependent term now reads
$\sum_{i,\ell,\ell^{\prime}}I_{\ell\ell^{\prime}}n_{i\ell\uparrow}n_{i\ell^{\prime}\downarrow}.$
It is easy to show that $I_{\ell\neq\ell^{\prime}}=I_{\ell\ell}/3,$ so some
orbital dependence remains. Whether it will be enough to provide for nodal
lines, is unclear. The charge channel should also be revisited in this case.
The Coulomb repulsion term should be added in form of a matrix in band
indices, $\mu_{ij}^{\ast}$, as defined in Equation (18). The diagonal elements
of this matrix should be taken as 0.15-0.20, and nondiagonal between 0 and the
diagonal ones, depending on how different are the matrix elements of $U$ in
the band representation.

Although the course of action outlined above is straightforward, so far it has
not been tried yet. In principle, such calculations could be very useful for
understanding the stability of the qualitative results, such as nodal
structure, with respect to principal approximation, because the Hubbard model
and the DFT in many aspects represent two opposite approximations.

###  3.2 Alternative approaches.

Historically, many other pairing agents have been proposed proposed as
mediators for unconventional (non-phonon) superconductivity [93]. Arguably,
historically the first suggestions were those of Little [94] and Ginzburg
[95], who proposed, respectively, quasi-1D metal chains and quasi 2D metal
planes embedded in highly polarizable media. This was dubbed “excitonic
superconductivity”, after the simple physical picture of Cooper pairs living
in the metallic subsystem, and the intermediate bosons being excitons
localized in the surrounding nonmetalic media. In the context of FeBS, this
proposition was recently brought back into limelight by Sawatzky et al, who
pointed out that As and Se are large ions and thus have large polarizability
[13, 96]. This model is still being discussed; the arguments usually brought
up against it include the fact that Fe-As hybridization is not small, as the
model requires, that ion-ion interaction should also be subject to screening
by polarizable As ions, but no anomalous phonon softening is observed, and,
finally, that superconductivity seems to be always adjacent in the phase
diagram to antiferromagnetism.

Other proposals rely on long wavelength electron charge fluctuations known as
plasmons, particularly on acoustic plasmons. Theories of this type were
attempted for cuprates, for MgB2, and, most recently, for intercalated
graphites. So far this model, however, has not had any confirmed success. One
shortcoming that plagues the papers advocating this mechanism is that they
hardly ever address the lattice stability, yet any sort of overscreening (and
essentially any attractive interactions in the charge channel can be
considered as a sophisticated overscreening) tends to overscreen phonons as
well and render them unstable. This is of course a problem for other
“excitonic” mechanisms as well; one can achieve pairing and keep phonons
stable only through invoking vertex corrections that appear in electron-
electron but not in electron-ion vertices.

  *
Figure 10:  (a) Phase diagram for pairing in Hubbard-Holstein model [97].
$g(0)$ is the bare electron-electron interaction due to electron-phonon
coupling. (b) Interorbital susceptibility mixing $xz$ and $xy$ orbitals; (c)
Intraorbital susceptibility for $xz$ orbitals.

Of all alternatives to the spin fluctuation model, the one that has received
most attention in the context of FeBS is the orbital fluctuations model of
Kontani and co-workers [98, 97, 99]. The possible importance of orbital
fluctuations was pointed out by several authors early on due to the
possibility of orbital ordering in the Fe $d$ states at the orthorhombic
transition [100, 84, 101, 102]. The stripe magnetic order will drive an
orbital ordering to some degree even in itinerant models. Fluctuating order of
this type, taken in isolation, can in principle lead to an attractive
mechanism for pairing, although it is hard to disentangle such orbital
fluctuations from spin fluctuations of the same symmetry.

It is useful to note that orbital fluctuations of this type are present in the
standard fluctuation exchange approach (3.1), and of course driven by the
interorbital Coulomb matrix elements $\bar{U}^{\prime}$, $\bar{J}^{\prime}$.
It is normally assumed (and verified by ab initio calculations [103]) that
$\bar{U}>\bar{U}^{\prime}$, but one can ask, taking $\bar{U}^{\prime}$ as an
independent parameter, what happens to the effective electron pairing vertex?
One may show that for sufficiently large $\bar{U}^{\prime}$,$\bar{J}^{\prime}$
the instability in the charge/orbital channel dominates the spin channel
contribution to $\Gamma_{ij}(\bi{k},\bi{k}^{\prime})$, Equation (3.1), even
for purely electronic interactions. In the former channel the interorbital
pair vertex becomes peaked at $(0,0)$ (compare figure 10), such that the
leading instability occurs in the $A_{1g}$ channel but without sign change,
i.e. a $s_{++}$ state.

These strong orbital fluctuations are unphysical in the required limit
$\bar{U}^{\prime}>\bar{U},$ $\bar{J}^{\prime}>\bar{J}$, relying on electronic
interactions alone. However, as pointed out by Kontani and Onari [97], certain
in plane Fe phonons can in principle “bootstrap” the interorbital processes
such that they dominate the spin fluctuation part of the interaction (a
different phonon was considered by Yanagi [104]). The electron-phonon coupling
to these phonons is included in an RPA-type calculation [97], and it is found
indeed that effective interorbital couplings $\tilde{U}^{\prime}$,
$\tilde{J}^{\prime}$, renormalized by the effective electron-electron
interaction due to phonons $g(0)$, enhance $s_{++}$ pairing. The phase diagram
of this model for fixed $\tilde{U}^{\prime}/\tilde{U}$ is shown in figure 10.
Note that orbitals 2 and 4 in the Kontani-Onari scheme are $xz$ and $xy$.

  * (a)  (b)  (c)

Figure 11: Diagrams for the electron-phonon vertex renormalization. Straight
red lines are electron propagators, wavy green line is the phonon propagator,
and the curly magenta line is a high-energy electronic excitation (orbital
fluctuation).

While the concept of orbital fluctuation pairing is quite clear, there are
several open questions concerning this idea. Probably the most intriguing one
is that how model calculations of Kontani and Onari should be reconciled with
the density functional calculations. Indeed, electron-phonon coupling in the
linear harmonic adiabatic approximation is included in the standard linear
response calculations of the phonon frequencies and coupling strength. These
calculations find only very moderate coupling strength for representative FeBS
[105], and include diagrams of the type shown in figure 11(a), where the
vertex is computed adiabatically, as a derivative of the electronic Green’s
function with respect to ionic displacement. An enhancement found by Kotani
and Onari can only come from vertex corrections. However, vertex corrections
of the type shown in figure 11(b) are excluded by virtue of the Migdal
theorem, and vertex corrections of the type shown in figure 11(c), where the
springy line is a high-energy electronic excitation (like an orbital
fluctuation) are included in the adiabatic density functional calculations
(any orbital repopulation due to a static ionic displacement is accounted
for). Thus, the slow (compared to the electronic scale) orbital fluctuations
are excluded by the Migdal theorem, and fast ones (compared to the phonon
scale) are already included in the DFT calculations. This appears to leave
rather little room for strong renormalization of the electron-phonon coupling.

Similarly, one can ask how such a huge enhancement will affect the
corresponding phonon self-energy. Indeed pretty much the same vertex
corrections enter the equation on the phonon frequency and one would expect
that if the density functional theory fails so miserably in calculating the
electron-phonon coupling for a particular mode, it should also drastically
overestimate the frequency of this mode compared to the experiment. Yet this
is definitely not the case [106].

Thus, the question to which extent this interesting orbital-fluctuation model
is operative in real compounds remains open. It is possible that, due to
varying interactions strengths, orbital fluctuations dominate in some of the
FeBS and spin fluctuation in the others. The authors of these works believe
that orbital fluctuations, and thus $s_{++}$ pairing, are dominant in most of
these systems, pointing to weak impurity scattering [107], the broad neutron
scattering resonance observed in most FeBS [98], and the natural explanation
[99] of the Lee plot (maximum of $T_{c}$ within family at tetrahedral Fe-As
angle [108]) as evidence in favor of this scenario. We postpone the question
of whether or not $s_{++}$ pairing has in fact been observed to Section 5.1.

###  3.3 Multiband BCS theory.

The famous BCS formula is derived in the assumption that the pairing amplitude
(superconducting gap, order parameter) is the same at all points on the Fermi
surface. The variational character of the BCS theory makes one think that
giving the system an additional variational freedom of varying the order
parameter over the Fermi surface should always lead to a higher transition
temperature. For a case of two bands with uniform order parameters in each of
them this problem was solved first in 1959 by Matthias, Suhl, and Walker [109]
and by Moskalenko [110]. It can be easily generalized onto a general
$\bi{k}$-dependent order parameter. In the weak coupling limit it reads

$\Delta(\bi{k})=\int\Lambda(\bi{k},\bi{k}^{\prime})\Delta(\bi{k}^{\prime})F[\Delta(\bi{k}^{\prime}),T]d\bi{k}^{\prime},$
(19)

where summation over $\bi{k}$ implies also summation over all bands crossing
the Fermi level. A strong coupling generalization in the spirit of Eliashberg
theory is straightforward. Here the matrix $\Lambda$ characterizes the pairing
interaction, and
$F=\int_{0}^{\omega_{B}}dE\tanh(\frac{\sqrt{E^{2}+\Delta^{2}}}{2T})/\sqrt{E^{2}+\Delta^{2}}$
. The intermediate boson frequency sets the cut-off frequency. Assuming that
the order parameter $\Delta$ varies little within each sheet of the Fermi
surface, while differing between the different sheets, Equation (19) is
reduced to the original expression of [109, 110]:

$\Delta_{i}=\sum_{j}\Lambda_{ij}\Delta_{j}F(\Delta_{j},T),$ (20)

where $i,j$ are the band indices and $\Lambda$ is an asymmetric matrix related
to the symmetric matrix of the pairing interaction $V$,
$\Lambda_{ij}=V_{ij}N_{j}$, where $N_{i}$ is the contribution of the $i$-th
band to the total DOS. It can be shown that in the BCS weak coupling limit the
critical temperature is given by the standard BCS relation,
$kT_{c}=\hbar\omega_{D}\exp(-1/\lambda_{eff}),$ where $\lambda_{eff}$ is the
largest eigenvalue of the matrix $\Lambda.$ The ratios of the individual order
parameters are given by the corresponding eigenvector. Note that although the
matrix $\Lambda$ is not symmetric, its eigenvalues (but not eigenvectors!) are
the same as those of the symmetric matrix $\sqrt{N}V\sqrt{N}$.

Furthermore it is evident from Equation 20 that unless all $V_{ij}$ are the
same, the temperature dependence of individual gaps does not follow the
canonical BCS behavior. For instance, in a two-band superconductor where the
intraband coupling dominates, the smaller gap opens initially at a very small
value, and only at a temperature corresponding to its own superconducting
transition (not induced by the larger gap) it starts to grow. This effect is
gradually suppressed as the interband coupling approaches the geometrical
average of the intraband couplings, but as the interband coupling starts to
dominate the gaps again show non-BCS temperature dependence. In this limit,
however, it is the larger gap that deviates more from the BCS behavior.

It was realized in 1972 [55] that Equations (19) and (20) may have solutions
even when all elements of the interaction matrices $\Lambda$ are negative,
i.e., repulsive. The simplest example is an off-diagonal repulsion:
$V_{11}=V_{22}=0$, $V_{12}=V_{21}=-V<0$. In this case the solution reads:
$\lambda_{eff}=\sqrt{\Lambda_{12}\Lambda_{21}}=|V_{12}|\sqrt{N_{1}N_{2}},$
$\Delta_{1}(T_{c})/\Delta_{2}(T_{c})=-\sqrt{N_{2}/N_{1}}$. At lower
temperatures the gap ratio becomes somewhat closer to 1.

It is important to bear in mind that actual FeBS materials have more than two
bands — rather four or five. These may all have different gap magnitudes, and
possibly angular dependences, as discussed in Section 5.

###  3.4 Disorder in multiband superconductors.

  *
Figure 12:  Schematic representation of two Fermi surface pockets with
superconducting gaps indicated with their signs. Top: interband scattering by
impurities mixes $\Delta_{1}$ and $\Delta_{2}$. Bottom: intraband scattering
mixes states on each pocket.

Experiments on impurity substitution [111, 112, 113] and proton irradiation
[114] have given the impression that $T_{c}$ is suppressed generally more
slowly than the maximum rate obtained for pure interband scattering in the
symmetric model, which is identical to the AG universal $T_{c}$ suppression
curve. It is worth advising the reader to interpret $T_{c}$ suppression
experimental results with caution, for several reasons. First, in some cases
not all the nominal concentration of impurity substitutes in the crystal.
Second, “slow” and “fast” $T_{c}$ suppression cannot be determined by plotting
$T_{c}$ vs. impurity concentration, but only vs. a scattering rate directly
comparable to a theoretical scattering rate (see below), which is generally
difficult to determine. The alternative is to plot $T_{c}$ vs. residual
resistivity change $\Delta\rho$, but a) this is only possible if the $\rho(T)$
curve shifts rigidly with disorder, and b) if comparisons with theory include
a proper treatment of the transport rather than the quasiparticle lifetime
[115]. Finally, the effect of a chemical substitution in a Fe-based
superconductor is quite clearly not describable solely in terms of a potential
scatterer, but the impurity may dope the system or cause other electronic
structure changes which influence the pairing interaction.

Given the many uncertainties present in the basic modeling of a single
impurity, as well as the multiband nature of the Fe-based materials, it is
reasonable to assume that systematic disorder experiments may not play a
decisive role in identifying order parameter symmetry as they did, e.g. in the
cuprates. Nevertheless, one can perhaps draw useful qualitative conclusions
about the effects of impurities on the various types of superconducting states
under discussion by attempting to study models of pairbreaking by impurities
which are generalizations of the conventional Abrikosov-Gor’kov (AG) approach
[116].

####  3.4.1 Intra- vs. interband scattering.

In conventional 2-band superconductors with two different isotropic gaps,
nonmagnetic impurities can either scatter quasiparticles between bands or
within the same band. Interband processes (see figure 12) will average the
gaps and can thus lead to some initial $T_{c}$ suppression, after which
$T_{c}$ will saturate until localization effects become important. Interband
scattering is much more profound in a sign-changing 2-band system [117, 118,
119], where nonmagnetic impurities with interband component of the scattering
potential are pairbreaking even if the gaps and densities of states are equal
on both bands (symmetric model). In such a situation, $T_{c}$ will eventually
be suppressed to zero at a finite critical concentration as in the theory of
scattering by magnetic impurities in a 1-band $s$-wave system [116]. In the
context of the FeBS, these general considerations were pointed out early on by
several groups [29, 120, 121, 122, 123]. A given type of chemical impurity in
a given host will be characterized crudely by an effective interband potential
$u$ and an intraband potential $v$, and results for various quantities in the
superconducting state will depend crucially on the size and relative strengths
of these two quantities.

Most calculations are performed in the framework of the $T$-matrix
approximation to calculate the average impurity self-energy for pointlike
scatterers $\hat{\Sigma}^{imp}(\omega_{n})$,

$\hat{\Sigma}^{imp}=n_{imp}\hat{\mathbf{U}}+\hat{\mathbf{U}}\hat{{G}}(\omega_{n})\hat{\Sigma}^{imp}({\mathrm{i}}\omega_{n}),$
(21)

where $\hat{\mathbf{U}}=\mathbf{U}\otimes\hat{\tau}_{3}$, $n_{imp}$ is the
impurity concentration, and the $\tau_{i}$ are Pauli matrices in particle-hole
space. Here ${\mathbf{U}}$ is a matrix in band space, frequently taken for
simplicity to be represented by constant intra- and inter-band potentials $v$
and $u$, respectively, such that
$(\mathbf{U})_{\alpha\beta}=(v-u)\delta_{\alpha\beta}+u$. This completes the
specification of the equations which determine the Green’s functions

$\hat{G}(\bi{k},\omega)^{-1}=\hat{G^{0}}(\bi{k},\omega)^{-1}-\hat{\Sigma}^{imp}(\bi{k},\omega),$
(22)

where $\hat{G}^{0}$ is the Green’s function for the pure system. Note that the
self-consistent $T$-matrix approximation includes only diagrams corresponding
to multiple scattering from a single impurity, and is well-known to have some
pathologies in two dimensions [124]. In the context of impurities in an
$s_{\pm}$ state, it has been claimed to produce inaccurate results in the
statistics of subgap states [19]. Nevertheless, for qualitative purposes–and
we will argue below that one cannot go beyond a qualitative analysis here
anyway–it seems quite adequate.

####  3.4.2 Effect on $T_{c}$.

  *
Figure 13:  Critical temperature for various $\tilde{\sigma}$ and $\eta$ as a
function of (a) the impurity scattering rate $\Gamma_{1}$ and (b) the
effective interband scattering rate $\tilde{\Gamma}_{12}$. The parameters are:
$N_{2}/N_{1}=2$, coupling constants are
for $\langle\lambda\rangle>0$: $\lambda_{11}=3$, $\lambda_{12}=-0.2$,
$\lambda_{21}=-0.1$, $\lambda_{22}=0.5$,
for $\langle\lambda\rangle=0$: $\lambda_{11}=2$, $\lambda_{12}=-2$,
$\lambda_{21}=-1$, $\lambda_{22}=1$,
for $\langle\lambda\rangle<0$: $\lambda_{11}=1$, $\lambda_{12}=-2$,
$\lambda_{21}=-1$, $\lambda_{22}=1$. From [125].

Properties in the presence of disorder can depend sensitively on the coupling
constants $\lambda_{ij}$ of the two-band superconductor (see Section 3.3) as
well, since these enter the BCS gap equations including impurities. In general
even the two-band problem can seem to be quite dependent on many parameters
which are difficult to determine as a practical matter. Recently a
simplification was pointed out in [125] whereby the suppression of $T_{c}$ can
be expressed solely in terms of a universal pairbreaking parameter

$\tilde{\Gamma}_{12}=\Gamma_{1(2)}\frac{(1-\tilde{\sigma})}{\tilde{\sigma}(1-\tilde{\sigma})\eta\frac{(N_{1}+N_{2})^{2}}{N_{1}N_{2}}+(\tilde{\sigma}\eta-1)^{2}},$
(23)

where $\tilde{\sigma}=(\pi^{2}N_{1}N_{2}u^{2})/(1+\pi^{2}N_{1}N_{2}u^{2})$ and
$\Gamma_{1(2)}=n_{imp}\pi N_{2(1)}u^{2}(1-\tilde{\sigma})$ are cross-section
and normal state scattering rate parameters, respectively. The parameter
$\eta=v/u$ is the ratio of intra-band to inter-band scattering. In the weak
scattering (Born) limit, $\tilde{\sigma}\to 0$, while for $\tilde{\sigma}\to
1$ the unitary limit (strong scattering) is reached. Note the strange result
that $\tilde{\Gamma}_{12}\rightarrow 0$ in the unitary limit, i.e. nonmagnetic
impurities do not affect $T_{c}$ in an $s_{\pm}$ state [119]. While this may
be an artifact of the 2-band model used, it is an indication that a rather
robust set of parameters may produce significantly weaker effects of $T_{c}$
suppression than expected.

Expressed in terms of (23), all $T_{c}$ suppression curves collapse onto one
of three “universal” curves, depending on whether the average pairing strength
parameter $\langle\lambda\rangle$ is positive, negative, or zero, as shown in
figure 13. It is clear that in the case where intraband scattering dominates,
even in the “standard” $s_{\pm}$ scenario with $\langle\lambda\rangle<0$, the
$T_{c}$ suppression will be much slower than the AG result; thus experimental
results need not be taken as evidence against the $s_{\pm}$ state. In the
interesting and relatively unexplored $\langle\lambda\rangle>0$ case, a
transition below $T_{c}$ from $s_{\pm}$ to $s_{++}$ with increasing disorder
is possible [125].

####  3.4.3 Anisotropic states.

  *
Figure 14: (Color online) Normalized spectral gap $\Omega_{G}(\phi)/T_{c0}$
vs. angle $\phi$ on the Fermi surface for an extended $s$-wave state
$\Delta(\phi)=\Delta_{0}(1+r\cos 2\phi)$, with $\Delta_{0}/T_{c0}=1$, $r=1.3$
and Born limit scattering rate $\Gamma/T_{c0}=0$ (dashed), 0.3 (dotted), 1.0
(solid), and 3.1 (dashed-dotted). From [126].

There is substantial evidence from low-energy thermodynamics and transport
experiments (Section 5) that low-energy quasiparticle excitations are present
in many Fe-based materials, indicating either very small minimum gaps or true
nodes of the order parameter on one or more Fermi surface sheets. The effect
of nonmagnetic impurities on $s$-wave states of this type also depends on the
character of the scattering, in particular whether or not inter- or intraband
processes dominate. If intraband scattering processes are considered by
themselves, they simply average the angular structure of the order parameter
on each Fermi surface sheet, as in the conventional $s$-wave case [127].
$T_{c}$ will fall initially and then saturate. If gap nodes are present, they
will be lifted by the averaging process at a critical value of disorder
(figure 14), and give rise to a crossover at the lowest temperature from power
laws in $T$ to exponential behavior [126] with increasing disorder. Thus if
intraband scattering dominates, such nodal or near-nodal systems will display,
in the clean limit, the low-energy excitations characteristics of nodes, while
dirty systems will be gapped with reduced $T_{c}$.

In the early literature on disorder in Fe-based system it was frequently
assumed, to the contrary,that the order parameter was isotropic $s_{\pm}$, and
that interband scattering dominates. In this case, under special circumstances
(see Section 3.4.4), bound states at the Fermi level may be induced and mimic
the effect of nodes in some experiments [120, 121]. In such a situation, the
opposite behavior with disorder is to be expected: clean systems will display
exponential $T$-dependence and dirty systems the power laws expected from
impurity-induced residual densities of states.

####  3.4.4 Single impurity problem.

In the most general and presumably realistic situation, anisotropic multiband
order parameters are present with both intra- and interband scattering.
Intraband scattering probably dominates in most situations (see below) and for
intermediate to strong impurity potentials, interband scattering effects are
largely irrelevant. To understand why this is the case, we consider the single
impurity problem in a symmetric $s_{\pm}$ state. Equation (21) for
$\hat{\Sigma}^{imp}$ is essentially identical as the $T$-matrix for a single
impurity, whose poles at $\Omega_{0}$ indicate the existence of impurity bound
states [128]. In general, energies nested near the gap edge correspond to weak
pairbreaking, while energies near the Fermi level correspond to strong
pairbreaking. A plot of the single impurity resonance energy position in a
symmetric $s_{\pm}$ state is given in figure 15, and shows that in order to
influence the states near the Fermi level a very specific fine tuning of
interband and intraband scattering is required. For the symmetric model, this
corresponds to $\eta=u/v=1$ in the intermediate to strong potential range, but
this criterion will be different in the asymmetric model $N_{1}\neq N_{2}$,
$\Delta_{1}\neq\Delta_{2}$. It thus seems a priori unlikely that the impurity
band in an isotropic $s_{\pm}$ state explanation for experiments indicating
low-lying quasiparticle states is correct. For further discussion, see Section
5.

  *
Figure 15: Energy of single impurity bound state $\Omega/\Delta_{0}$ in
symmetric $s_{\pm}$ superconductor with gaps $\pm\Delta_{0}$ as a function of
interband ($u$) and intraband ($v$) scattering (A.F. Kemper, private
communication).

The single impurity problem has been considered in more detail in a 2-band
model in [123, 129, 130, 131, 132, 133, 134, 135] and in a 5-band model in
[136]. These calculations show a much richer structure of the local density of
states around a single impurity than is found, e.g. in the one band $d$-wave
problem [128], as might be expected, and a complicated dependence on
interaction and impurity parameters. It seems unlikely, given this complexity,
that the combination of STM imaging of impurity states [128] and theories of
this type will be able to provide definitive information on order parameter
symmetry or structure in these systems.

One approach to reducing the number of parameters and making theories of this
type more predictive has been to try to calculate impurity potentials from
first principles methods. For example, Kemper et al [137] calculated the
nonmagnetic and magnetic impurity potentials for a Co substituting for an Fe
in BaFe2As2 within density functional theory. Nakamura et al [138] later
performed similar calculations for several impurity types. In principle, such
calculations can provide important input into phenomenological treatments of
disorder by specifying $u$ and $v$ in band space, but band-resolved results of
this type have not yet been given. Kemper et al found in their calculations
the nonmagnetic potential was significantly larger than the magnetic one, and
that the interband scattering was perhaps a factor of three smaller than
intraband. In orbital space, results from [137] and [138] disagree
substantively for the Co potential, but it is not clear whether this arises
from the fact that the former calculations were performed in the spin-
polarized phase, or due to a different treatments of the nonlocal LDA
potential [138].

####  3.4.5 Magnetic impurities.

A qualitative rule of thumb when considering the effect of magnetic and
nonmagnetic impurities on multiband, anisotropic superconductors is as
follows: When a nonmagnetic impurity scatters a pair from one point on the FS
into another point, such that the order parameter does not change sign,
scattering is not pair breaking; if the order parameter flips its sign, it is
pair breaking. For a magnetic impurity, the opposite is true: scattering with
an order parameter sign change is not pairbreaking, otherwise it is. However,
quantifying this rule of thumb may be complicated and results are sometimes
counterintuitive.

In terms of concrete calculations for magnetic impurities in $s_{\pm}$ states,
Golubov and Mazin [118] showed that for the symmetric model considered above,
$T_{c}$ is suppressed by magnetic and nonmagnetic impurities at the same rate
in the disorder averaged theory. The analogous single impurity problem was
treated by Akbari et al [139] within an Anderson model approach to a rare
earth impurity in such a system in the $s_{\pm}$ state. In both situations
only quantitative differences in the scattering from magnetic impurities
relative to the usual $s$-wave case were found. In special symmetric
situations, interband magnetic scattering in an $s_{\pm}$ state can give rise
to arbitrarily weak pairbreaking, whereas intraband scattering is strongly
pairbreaking as expected from AG theory [132].

####  3.4.6 Orbital effects.

In Section 3.4.4 above it was argued that fine tuning of intra-and interband
pairing amplitudes was required in order to create substantial pairbreaking in
an $s_{\pm}$ state, e.g. create bound states near the Fermi level. In a more
realistic approach, however, these parameters ( $u$ and $v$) should not be
considered arbitrary, For example, if one starts from a local atomic picture
where an impurity is assumed to modify the orbital occupation energies of the
Fe-derived $d$ states, and then transforms to a band basis, it is easy to see
that intraband scattering terms of the same order as interband ones are
automatically generated by the unitary transformation from orbitals to bands.
This was the basis of the argument made in [107] that interband scattering
generically leads to $u\simeq v$ and therefore to large $T_{c}$ suppression in
the symmetric $s_{\pm}$ model; this was taken as evidence against $s_{\pm}$
pairing. However as seen in Section 3.4.4, a fine tuning is required to
produce significant pairbreaking, so we consider it generically unlikely that
for a given set of four or five Fermi surface pockets, with differing density
of states and order parameter magnitudes, that the condition for maximal
(Abrikosov-Gor’kov like) $T_{c}$ suppression will be achieved accidentally.
This point of view is also borne out by the argument in [125] and exhibited in
figure 13.

###  3.5 Dimensionality.

In this section we will concentrate on the qualitative effects due to 3D
dispersion of the electronic bands. It should be kept in mind that
dimensionality also plays a role in magnetism, in particular, it may affect
the extent of the magnetic part of the phase diagram and magnetic-orthorhombic
splitting [140], and also it can manifest itself through anisotropy of elastic
properties, phonon dispersion and electron-phonon coupling (which does not
seem to be the case in FeBS). Again, we will not discuss these effects, but
only the effects of dimensionality on the band structure, and, via the latter,
on superconductivity.

Such effects can be, roughly speaking, divided into three groups. First, there
is a generic issue of the correspondence between the number of carrier and
their density of states. Let us compare conventional superconductors, MgB2 and
B-doped diamond. Both exhibit hole-doped covalent bonds, and electron-phonon
matrix elements are practically identical. Note that it is rather difficult to
dope such bonds, so the number of carriers is small in both cases. Yet in the
quasi-2D $\sigma$-bands of MgB2 this small amount creates a sizeable DOS
(recall that in ideal 2D parabolic bands DOS does not depend on the carrier
concentration at all), and a critical temperature of 40 K, while in Boron-
doped diamond the DOS remains small, according to the small number of holes,
and so does $T_{c}$.

The second effect is the geometry of the Fermi surface. As we know, in the
spin-fluctuation model the structure of the order parameter is defined by the
interplay between the $q$-dependence of the spin fluctuations and the shape of
the Fermi surface. This is an interesting possibility that has been explored
mostly by that part of the community which traces the spin fluctuations to
nearest- and second nearest neighbor superexchange (see Section 3.1). In that
case the nodal lines are fixed in reciprocal space at $k_{x}=\pm\pi/2$, and at
$k_{y}=\pm\pi/2$. If at some particular $k_{z}$ a Fermi surface expands as to
cross these lines, actual gap nodes develop.

Finally, the last group of effects is related not so much to possible changes
with $k_{z}$ of the Fermi surface shape but to the orbital composition of the
states forming the Fermi surface. These may come from two sources. First, near
the $\Gamma$ point, besides the ubiquitous $xz/yz$ band, occasionally other
bands may cross the Fermi level, including the $z^{2}$ band that is very
dispersive in the direction. This band hybridizes with the $xz/yz$ band
everywhere except the high-symmetry planes, leading to Fermi surface pockets
that rapidly change their character as $k_{z}$ changes (see figure 16).

  *
Figure 16: $d_{z^{2}}$ orbital character on the Fermi surface of 5% electron
doped Ba-122 according to DFT. The two outer hole Fermi surfaces are clipped
at $\pm\pi/2c$ to show the parts carrying most of the $d_{z^{2}}$ character.

Another, more subtle source of 3D effect related to the orbital character of
the bands comes from the fact that the electron bands, as discussed in Section
2, are never pure $xz/yz$, but always have an admixture of the $xy$ symmetry,
in the outer barrel. This fact was first noted and explained by Lee and Wen
[39], and elaborated in a review paper by Andersen and Boeri [141]. The
relevant physics also controls the warping and the twisting of the electron
FSs.

  *
Figure 17: Band dispersion in the unfolded BZ corresponding to a 1-Fe unit
cell. Major orbital contributions are labeled.

To understand this, we will start as usual with the unfolded band structure
(figure 17), corresponding to a single Fe unit cell (figure 16(b)). The
unfolded Fermi surface geometry of the electron pockets is, essentially,
defined by their ellipticity and its variation with $k_{z}$. The ellipticity
at a given $k_{z}$ in the unfolded zone is determined by the relative position
of the $xy$ and $xz/yz$ levels of Fe, and the relative dispersion of the bands
derived from them. Indeed, the point on the Fermi surface located between
$\Gamma$ and $\mathrm{X}$ has a purely $xy$ character, while that between
$\mathrm{X}$ and $\mathrm{M}$ a pure $yz$ character. At the $\mathrm{X}$ point
the $xy$ state is slightly below the $yz$ state, but has a stronger
dispersion, therefore depending on the system parameters and the Fermi level
the corresponding point of the Fermi surface may be more removed from
$\mathrm{X}$, or less. In the 1111 compounds, for instance, the dispersion of
the $xy$ band is not high enough to reverse the natural trend, so the Fermi
surface remains elongated in the $\Gamma\mathrm{M}$ (1,0) direction.

Both $xy$ and $xz/yz$ orbitals point away from the Fe-Fe bond (as opposed to
the $x^{2}-y^{2}$) orbital, therefore their hopping mainly proceeds via As
(Se) $p-$orbitals. The $xy$ states near the $\mathrm{X}$ point mainly hop
through the $p_{z}$ orbital (see [141] for more detailed discussions), and
$xz$ ($yz$) via $p_{y}$ ($p_{x}$) orbitals (not because of the orbitals’
shape, but because of their phases at $\mathrm{Y}$[$\mathrm{X}$]). If there is
a considerable interlayer hopping between the $p$ orbitals, whether direct (11
family) or assisted (122 family), the ellipticity becomes $k_{z}$-dependent.
For instance, in FeSe there is noticeable overlap between the Se $p_{z}$
orbitals, so that they form a dispersive band with the maximum at $k_{z}=0$
and the minimum at $k_{z}=\pi/c$. Obviously, hybridization is stronger when
the $p_{z}$ states are higher, therefore the Fermi surface ellipticity is
practically absent in the $k_{z}$=0 plane, while it is rather strong in the
$k_{z}=\pi/c$ plane, which leads to formation of the characteristic “bellies”
in the Fermi surface of FeSe. On the other hand, $p_{x,y}$ orbitals in FeSe
hardly overlap in the neighboring layers, so the $xz$ and $yz$ bands have very
little $k_{z}$ dispersion, so that the inner barrels of the electronic pockets
in this compound are practically 2D.

In 122, the interlayer hopping proceeds mainly via the Ba (K) sites, and thus
the $k_{z}$ dispersion is comparable, but opposite in sign (for instance, at
the $\Gamma$ point the hopping amplitudes from Ba $s$ to As $p_{z}$ orbitals
above and below have opposite signs, while those for the As $p_{x,y}$ orbitals
have the same signs) for the $xy$ and $xz/yz$ bands. Note this difference
between the 122 and 1111 materials is related simply to the structural
difference between “in-phase” and “out of phase” FeAs layers in the unit cell.
As a result, when going from the $k_{z}=0$ plane to the $k_{z}=\pi/c$ plane
the longer axis of the Fermi pocket shrinks, and the shorter expands. In
BaFe2As2 the average ellipticity is very small, while the As-Ba overlap is
large, so that the actual ellipticity changes sign when going from $k_{z}=0$
to $k_{z}=\pi/c$. On the other hand, in Se based 122 systems the Se-K hopping
is quite small, so ellipticity is small for all $k_{z}$.

Importantly, the symmetry operation that folds down the single-Fe Brillouin
zone when the unit cell is doubled according to the As (Se) site symmetry is
different in the 11 and 1111 structures, as compared to the 122 structure. In
the former case, the operation in question is the translation by
$({\pi},{\pi},0)$,without any shift in the $k_{z}$ direction, in the latter by
$({\pi},{\pi},{\pi})$. Thus the folded Fermi surface in 11 and in 1111 has
full fourfold symmetry, while that in the 122 has such symmetry only for one
particular $k_{z}$, namely $k_{z}=\pi/2c$. Furthermore, in 122 the folded
bands are not degenerate along $\tilde{M}\tilde{X}$ as they were in 11/1111.
Finally, there is a considerable (at least on the scale of the superconducting
gap) hybridization when the folded bands cross (except for $k_{z}=0$). As a
result, although the band structure calculations for 122 materials actually
produce two detached (except for one plane) cylinders for the electronic FSs,
one cannot “unfold” these two cylinders as if one of them was folded down into
the other. The actual folded FSs intersect, yet we do not observe these
intersections because of hybridization induced by the As (Se) potential.

All these effects of 3-dimensionality of the electronic structure manifest
themselves in the 3D gap structure, tending to complicate the simple 2D
theoretical pictures. We comment below on some of the most significant ways in
which this occurs.

##  4 Gap symmetry.

The first question to be asked regarding the pairing state in a novel
superconductor is, what is the symmetry of the order parameter? Of course, the
symmetry itself does not fully describe the structure of the gap. For
instance, even a full-symmetry ($s$-wave, or, synonymously, $A_{1g},$
symmetry) order parameter may have “accidental”, that is, not required by
symmetry, nodes, as long as these nodes transform into each other by all point
group operations. Yet, establishing the right symmetry is arguably the most
important step towards uncovering the full gap structure.

###  4.1 Triplet or singlet?

In materials with an inversion center, so-called centrosymmetric, any Cooper
pair can be characterized by its parity, in the sense that the spatial part of
its wave function (the order parameter) can either change sign or remain the
same under the inversion operation. Electrons being fermions, the full wave
function is always antisymmetric. Since inversion corresponds to swapping the
pair components, if they have total spin $S=1$ (triplet), the spatial part of
the wave function should be odd, and if they have $S=0$ (singlet) it should be
even.

A triplet Cooper pair with $S_{z}=\pm 1$ can screen an external magnetic
field, just as individual electrons can. The spin-orbit interaction can
prevent this for some directions, but not for others. On the other hand, a
singlet pair has no net spin and does not contribute to magnetic
susceptibility as $T\rightarrow 0$. Thus for singlet superconductivity one can
expect the uniform spin susceptibility to diminish below $T_{c}$. The easiest
and the most accurate way to probe the latter is via the Knight shift. This
experiment has been performed on several FeBS including Ba(Fe1-xCox)2As2
[142], LaFeAsO1-xFx [143], PrFeAsO0.89F0.11 [144], Ba1-xKxFe2As2 [145, 146],
LiFeAs [147, 148], and BaFe2(As0.67P0.33)2 [149], and it was found that the
Knight shift decreases in all crystallographic directions. This effectively
excluded triplet symmetries such as $p$-wave or $f$-wave.

###  4.2 Chiral or not?

Another way to classify the superconducting state is according to whether it
breaks time-reversal symmetry or not, and according to whether it is chiral
(finite expectation value of the magnetic moment operator in the ground state)
or not. A singlet chiral superconducting state is allowed in tetragonal
symmetry [27], and is transformed under the symmetry operations as
$xz+{\mathrm{i}}yz$ (of course the complex conjugate state,
$xz-{\mathrm{i}}yz$, is also allowed). There are several ways to detect
$T$-breaking and chirality, but the simplest is probably the $\mu SR$
spectroscopy. This technique is sensitive to small local magnetic fields, as
long as they are static. In a chiral superconductor, as long as the balance
between the two conjugate states is broken, which should be happening near
crystallographic defects, spontaneous orbital current appear, and should be
visible by the $\mu$SR technique. Such an effect was observed, for instance,
in Sr2RuO4 [150], but not in FeBS [151]. In principle, a chiral state can also
be generated by mixing two 1D representations, e.g. a so-called
$d+id^{\prime}$ (note a $s+id$ state, also possible in principle, is
$T$-breaking but is not chiral). However, this requires separate transition
temperatures corresponding to the two representations; since this has not been
observed, we do not discuss this possibility further.

###  4.3 $d$ or $s$?

Having excluded triplet ($p$ and $f$) symmetries, as well as the chiral state,
we are left with the following singlet pairing possibilities (in a 3D system
with tetragonal symmetry): $s$-wave [$A_{1g}$]; $d(xy)$ [$B_{2g}]$;
$d(x^{2}-y^{2})$ [$B_{1g}$]; $g$-wave $(xy(x^{2}-y^{2}))$ [$A_{2g}]$; and
$d(xz\pm yz)$ [$E_{g}$]. It is important to note that, while $d$-wave does not
necessarily imply the existence of gap nodes, in combination with a quasi-2D
Fermi surface centered around the $\Gamma Z$ line such nodes are unavoidable,
either vertical for the $A$ and $B$ symmetries, or horizontal, for the $E$
symmetry. As will be discussed later in this review, the surface probes, such
as ARPES and tunneling show full gaps with no nodes, and, at least for some
compounds bulk probes show exponential low-temperature behavior.

There are also direct experiments that provide evidence against $d$-wave. The
$d$-wave representations, $B_{1g}$ and $B_{2g}$, should not exhibit any
Josephson current when weakly coupled to a known $s$-wave superconductor, by
symmetry, if the current flow precisely along the $z$ axis. However, such
current was observed in the 122 single crystals [152]. This observation is
difficult to explain away by deviation from the correct geometry, because the
observed current was strong and showed a well-defined Fraunhofer diffraction.

Another piece of evidence comes from the so-called anomalous Meissner, or
Wohlleben effect. This effect was predicted in the beginning of the cuprates
era [153] and since then has been routinely observed in $d$-wave
superconductors. In a nutshell, this effect appears in polycrystalline samples
with random orientation of grains. For any $d$-wave superconductor one expects
roughly 50% of weak links to have a zero phase shift, and 50% a $\pi$ phase
shift. One can show that in this case the response to a weak external magnetic
field is paramagnetic, i.e., opposite to the standard diamagnetic
superconducting response. This effect has been searched for in FeBS [154], but
not found.

These separate pieces of evidence strongly suggest that the pairing symmetry
is $s$, and not $d$. However, we want to stress that direct testing similar to
that performed in cuprates, namely a single-crystal experiment with a 90∘
Josephson junction forming a closed loop, is still missing, and it is highly
desirable for experimentalists to perform this ultimate test. In addition, it
should be borne in mind that no law of nature forbids different FeBS materials
from having different order parameter symmetries, although our previous
experience with other novel superconductors tends to argue against this.
Indeed, there are several proposals that, while most FeBS are $s$-wave, those
with unusual Fermi surfaces with only one type of pocket can be $d$ wave, see
Section 5.3.

##  5 Gap structure.

###  5.1 Does the gap in FeBS change sign?

Even though there is convincing evidence that the point symmetry of the order
parameter is for most, if not for all compounds, $s$-wave, it does not tell us
much about the actual structure of the order parameter and the excitation gap.
As opposed to the $d$-wave case, where nodes are mandated on the hole pockets
by symmetry, in an extended $s$-wave scheme they may appear on either type of
pocket if higher harmonics in the angular expansion of the order parameter are
sufficiently large. As discussed in Section 3.1, there are microscopic reasons
why this may be the case. Moreover, since nodeless $s_{\pm}$, nodeless
$s_{++}$, and an extended $s$ with accidental nodes all belong to the same
symmetry class, the difference between them is only quantitative (but
important). In this regard, several experiments appear relevant.

####  5.1.1 Spin-resonance peak.

One obvious effect that was mentioned even in the very first paper proposing
the $s_{\pm}$ scenario [29] and later elaborated in detail [36, 155, 156], is
the neutron spin resonance. Neutron scattering is a powerful tool to measure
the dynamical spin susceptibility $\chi_{s}(\bi{q},\omega)$. For the local
interactions (Hubbard and Hund’s exchange, see Equation (3.1)), $\chi_{s}$ can
be obtained in the RPA from the bare electron-hole bubble
$\chi_{0}(\bi{q},\omega)$ by summing up a series of ladder diagrams to give

$\displaystyle\chi_{s}(\bi{q},\omega)=\left[I-U_{s}\chi_{0}(\bi{q},\omega)\right]^{-1}\chi_{0}(\bi{q},\omega),$
(24)

where $I$ is a unit matrix in orbital space and all other quantities are
matrices as well.

The fact that $\chi_{0}(\bi{q},\omega)$ describes particle-hole excitations
has interesting consequences in the case of an unconventional superconducting
state. Excitations are gapped below approximately $2\Delta_{0}$; (at $T=0$)
only above this threshold does $\mathrm{Im}\chi_{0}(\bi{q},\omega)$ become
non-zero. The term arising from the anomalous Green functions is proportional
to

$\sum{k}\left[1-\frac{\Delta{k}\Delta_{\bi{k}+\bi{q}}}{E_{\bi{k}}E_{\bi{k}+\bi{q}}}\right]...$
(25)

where $...$ represents the kernel of the BCS susceptibility (see e.g. [157]).
At the Fermi level,
$E_{\bi{k}}\equiv\sqrt{\varepsilon{k}^{2}+\Delta{k}^{2}}=|\Delta{k}|$. If
$\Delta{k}$ and $\Delta_{\bi{k}+\bi{q}}$ have the same sign, the coherence
factor in square brackets in (25) vanishes, leading to a smooth increase of
the magnetic response with frequency above the $T=0$ threshold of
$\Omega_{c}=\min\left(|\Delta{k}|+|\Delta_{\bi{k}+\bi{q}}|\right)$. In case of
unconventional superconductors [158], when for a given $\bi{q}$,
$\mathrm{sgn}\Delta{k}\neq\mathrm{sgn}\Delta_{\bi{k}+\bi{q}}$, the coherence
factor is non-zero and the imaginary part of $\chi_{0}$ possesses a