Role of the orbital degree of freedom in iron-based superconductors

Almost a decade has passed since the serendipitous discovery of the iron-based high temperature superconductors (FeSCs) in 2008. The question of how much similarity the FeSCs have with the copper oxide high temperature superconductors emerged since the initial discovery of long-range antiferromagnetism in the FeSCs in proximity to superconductivity. Despite the great resemblance in their phase diagrams, there exist important disparities between FeSCs and cuprates that need to be considered in order to paint a full picture of these two families of high temperature superconductors. One of the key differences lies in the multi-orbital multi-band nature of FeSCs, in contrast to the effective single-band model for cuprates. Due to the complexity of multi-orbital band structures, the orbital degree of freedom is often neglected in formulating the theoretical models for FeSCs. On the experimental side, systematic studies of the orbital related phenomena in FeSCs have been largely lacking. In this review, we summarize angle-resolved photoemission spectroscopy (ARPES) measurements across various FeSC families in literature, focusing on the systematic trend of orbital dependent electron correlations and the role of different Fe 3d orbitals in driving the nematic transition, the spin-density-wave transition, and implications for superconductivity.


Introduction
Thirty years after the historic discovery of cuprate high temperature superconductors, the mechanism for high temperature superconductivity remains the biggest challenge in condensed matter physics despite tremendous amount of theoretical and experimental efforts. The discovery of iron-based superconductors 1 provides a great opportunity to identify the important ingredients that are common to both families of high Tc materials and to test the theoretical models that have been formulated for cuprates. Comparing FeSCs with cuprates, the most striking similarity is the common phase diagram, in which unconventional superconductivity appears in the vicinity of other competing phases, such as the pseudogap phase and the charge order in cuprates and the spin-density-wave (SDW) phase and nematic phase in FeSCs 2-3 . The emergence of superconductivity always takes place with the suppression of these competing phases. Such a remarkable resemblance has raised our hope for a unified theory of high temperature superconductivity and has motivated many theorists to take the strong coupling approach to describe FeSCs.
On the other hand, FeSCs also appear to distinguish themselves from cuprates in various aspects, including metallicity of the parent phase, crystal structure of the conduction layer, spin symmetry of the antiferromagnetic order, as well as the underlying electronic structure. Prior to establishing a unified understanding of the physics in cuprates and FeSCs, we first need to understand whether these differences are trivial nuances or critical ingredients that cannot be neglected. Among them, the most fundamental difference is the multi-orbital multi-band nature of the underlying electronic structure in FeSCs. In contrast to cuprates, for which essential physics seems to take place in a single effective band and Fermi surface, there are at least three out of five Fe 3d orbitals that are active near the Fermi level (E F ) in FeSCs, forming multiple Fermi surface sheets. The complexity of theoretical treatment for a multi-band system has led to various proposals for minimal models for FeSCs, in which the orbital degree of freedom is often ignored for simplicity. While these models capture some underlying physics, the question is whether they miss important orbital related physics.
The lack of systematic experimental studies on the role of different Fe 3d orbitals may be part of the reason that orbital physics in FeSCs has not garnered as much attention as they perhaps deserved. In this review, we summarize experimental evidence of various orbital dependent phenomena in the electronic structure from angle-resolved photoemission spectroscopy literature. transitions. In the end, we discuss the non-trivial implications of multi-orbital nature on the superconducting pairing mechanism of these materials.

The normal state
All FeSC compounds, regardless of their compositions, share in common planes containing iron and pnictogen (P/As) or chalcogen (S/Se/Te), which is located alternatingly above and below the Fe lattice (Fig. 1a). The difference among FeSC families is the composition and structure of interlayers between the Fe planes. In some cases, these interlayers form charge reservoirs that donate charge carriers to the Fe planes. The normal state is defined as the phase outside the boundaries of the magnetic and structural transitions (Fig. 1a). All FeSC compounds share in common the basic electronic structure consisting of Fe 3d bands near E F , with the d xz , d yz , and d xy orbitals most active near E F . The basic electronic structure of FeSC in the normal state is illustrated in Fig. 1c, where three hole-like bands reside near the Brillouin Zone (BZ) center, Γ, and two electron-like bands near the BZ corner, M (except in the case of the 122 structure, where this point is called the X point). For different doping levels and structural subtleties, the overall or relative positions of the hole and electrons bands may vary in energy, leading to different Fermi surface topologies with varying number of hole pockets at Γ and electron pockets at M (Fig. 1d). For undoped parent compounds, the hole pockets at Γ and electron pockets at M are of similar sizes. For hole-doped compounds, the hole pockets at Γ enlarge while the electron pockets shrink. For heavily electron-doped compounds such as the electron-doped chalcogenides, the hole pockets disappear while large electron pockets remain at M.
One of the fundamental questions after the discovery of the FeSCs was whether it is appropriate to model them as localized systems or itinerant systems. On one hand, the observed large spectral weight in the fluctuating magnetic spectrum 5 tends to suggest the former, while the high density of states found near E F compared to the cuprates 6 seems to suggest the latter. As we know now, neither picture is fully complete. We will show in the following that there is a large systematic spread of electron correlation strength among different FeSCs, and more importantly, this occurs in a strongly orbital-dependent way.
This trend can be qualitatively seen by comparing representative compounds from the iron phosphides to the iron arsenides and to the iron chalcogenides (Fig. 2). Electron correlation renormalizes the electronic bandwidth. From ARPES data, one way to quantify the strength of correlation is to extract the ratio of the non-interacting bandwidth calculated from local density approximation (LDA) and experimentally measured bandwidth, which is the renormalization factor 7 . Fig. 2a-c shows the band dispersions along the high symmetry direction Γ-M observed in the paramagnetic state of a phosphide (SrFe 2 P 2 ), an arsenide (NaFeAs), and a chalcogenide (FeSe 0.44 Te 0.56 ), plotted in the same energy window, where the d yz band is marked in green and d xy blue. There are two observations to make: i) the bandwidth of all bands systematically narrows from phosphide to the arsenide to the chalcogenide, which is also seen in the increase of the renormalization factors, and ii) the bandwidth of d xy narrows at a much faster rate than that of d yz from the phosphides to the chalcogenides. These two observations suggest that both the overall electron correlation strength and orbital-selectivity, which is defined here as the ratio in the band renormalization between the d xy band and d yz band, increase from the iron phosphides to the iron chalcogenides. In the following section, we systematically demonstrate these trends and discuss factors that correlate with these two parameters.
Considering the three active orbitals near E F , the d xz and d yz orbitals are bound by C 4 symmetry to be degenerate in the tetragonal state. The d xy orbital, however, does not necessarily need to behave in the same way as d xz /d yz . Hence we examine each orbital in turn. For the bands near E F , the d yz band is the band that can be observed most completely under common polarization setups, with its band top at Γ and band bottom between Γ and M (Fig. 1d). Here we extract its bandwidth from ARPES literature and compare with available LDA calculations (Table 1).  (Fig. 3a,c). As d yz is extended in the out-of-plane direction and d xy is mostly in plane, the lack of clear trend in both renormalizations against lattice a suggests that electron hopping is dominated by the indirect path of Fe-Pn/Ch-Fe rather than that of direct Fe-Fe bond. The d yz renormalization factor also has strong correlation with related structural parameters such as anion height and bond angle, which both directly relate to the Fe-Pn/Ch bond length 8 . Similar dependence of correlation strength on structural parameters is also seen in optical data within the BaFe 2 As 2 family 9 and photoemission data within the heavily electron-doped iron chalcogenides 10 . For other structural parameters such as the lattice constant c, the correlation with bandwidth renormalization is not obvious across families (Fig. 3b,d).
Structural parameters are not the only factors that correlate with the overall correlation strength.
A second factor is electron filling 7 . In Fig. 2e we plot the d yz bandwidth renormalization versus the electron filling for doped compounds of two series, the BaFe 2 As 2 series and the LiFeAs series. To put this plot in perspective, we also overlay the electron (Co) and hole (K) doped phase diagram of BaFe 2 As 2 on the horizontal axis. The data points for this series range from those taken from KFe 2 As 2 (n = 5.5) to BaNi 2 As 2 (n = 8). Here we note that the undoped parent compounds of FeSC has n = 6, which is not half-filling as the case of the parent compounds of the cuprates. True half-filling for the Fe 3d orbitals is n = 5. This explains the asymmetry of the overall correlation with respect to the undoped parent compounds of FeSC 7,9 . The electron correlation is weak far away from n = 5 for the heavily electron-doped compounds, and diverges towards n = 5 on the hole-doped side. However, it is interesting to note that, under this scenario, the known undoped iron pnictides are effectively on the electron-doped side of the true half filling. In analogy to the cuprates, there may be an equivalent regime of superconductivity on the hole-doped side, as has been recently theoretically suggested 11-12 . In Fig. 2f we plot all the compounds (including those with different electron fillings) with sufficient information from literature on a 2D plot with electron filling and the Fe-Pn/Ch bond length, and use color to indicate the strength of d yz bandwidth renormalization factor. Here we see as demonstrated before, both reducing electron filling towards n = 5 and lengthening the Fe-Pn/Ch bond length increase overall correlation.
Next, we discuss factors that affect the orbital-selectivity. Since the d xy orbital is the most correlated of the orbitals, quantitatively, we extract the ratio of the renormalization factors of the d xy band and the d yz band. In Fig. 2h we plot this for all compounds we found from literature against the Pn/Ch-Fe-Pn/Ch bond angle. A clear trend is seen where smaller bond angle, as in the case of the chalcogenides, leads to strong selectivity; whereas bigger bond angle, as in the case of the phosphides, leads to almost no selectivity. This is because smaller bond angle results in a vertically elongated tetragon, which reduces hopping more dramatically for the in-plane d xy orbital than d xz /d yz orbitals, considering hopping is dominantly mediated through the Pn/Ch, which reside out of the Fe plane. Besides the bond angle, Fig. 2g also shows another factor that correlates with orbital-selective correlation-the overall electron correlation represented by the d yz renormalization factor. It demonstrates that among all FeSCs, the stronger the overall correlation, the stronger the orbital-selectivity in the d xy orbital. This has been discussed in previous theoretical work as due to the tendency towards a Hund's metal phase, where Hund's coupling increases orbital differentiation and independence by suppressing inter-orbital correlations 8,13-14 . In Fig. 2i, we plot the orbital-selectivity as a function of both bond angle and overall correlation strength for all compounds. We see that these dependences are not only true within a family of FeSC, but systematically spread among all FeSC compounds.
As orbital-selectivity increases, an interesting phenomenon occurs in the strongly correlated members of FeSCs-a tendency towards an orbital-selective Mott phase (OSMP). In Fig. 4a, we plot the bond length dependence of the renormalizaton factors of the d xy orbital for undoped FeSCs (n = 6), which is the most correlated orbital. Comparing to the equivalent plot for d yz (Fig.   2d), we see that the dynamic range of the d xy orbital is five-fold of that of d yz . This again showcases the strong orbital differentiation among the FeSCs towards the strongly correlated members. When this differentiation is strong, as in the iron chalcogenides, the normal state of these materials is sufficiently close to an OSMP such that raising temperature has been observed to push them into the OSMP where the d xy orbital completely loses its spectral weight while other orbitals remain itinerant (Fig. 4b) 15 . This has in fact been observed universally for different iron chalcogenides including Fe(Te,Se), KFe 2 Se 2 , and monolayer FeSe film grown on SrTiO 3 [16][17][18][19] and Nb:BaTiO 3 /KTaO 3 heterostructures 20 . Even for the most correlated iron arsenide, KFe 2 As 2 , evidence for decoherence of the d xy orbital has been observed by transport measurements 21 . Here we see that the tendency towards the OSMP in the chalcogenides is not accidental, but grows naturally from an increasingly selective orbital correlation systematically in all FeSCs.
Both the spread of electron correlation strength and the increasing orbital selectivity are perhaps what makes it difficult to develop a unified theoretical model for all FeSCs. On one side of the spectrum the orbitals are mostly itinerant with almost no orbital selectivity while on the other side strong orbital decoupling due to Hund's coupling results in one of the orbitals approaching localization. Nonetheless, this orbital-selectivity is a behavior unique and essential to the multi-orbital physics of the FeSCs in contrast to the single-band physics of the cuprates. Its importance is especially evident for the compounds on the strongly correlated side such as the chalcogenides. These behaviors have led to interesting theoretical proposals where one redefines the phase diagram using an average orbital filling, showing a gradual approach from superconductivity towards a Mott insulating state via an OSMP 13 , while another proposes that in the strongly orbital-selective regime, the d xy orbital is near half-filling 14 . Both suggest parallels reminiscent of the cuprate problem.

The nematic state
In a typical phase diagram of iron-based superconductors, the material in the underdoped regime goes through two transitions and enters the nematic state and the SDW state at low temperatures 2-3 . The multi-orbital nature not only leads to orbital-dependent band renormalization in the normal state, but also plays an important role in driving the system into these symmetry-breaking states. The nematic phase is marked by a tetragonal to orthorhombic structural transition at T S , where C 4 rotational symmetry is broken down to C 2 symmetry without breaking the translational symmetry. With the orthorhombic distortion, material forms natural twin domains. Hence for macroscopic probe like ARPES where the beam spot is larger than the typical domain size (Fig. 5a) Next, we discuss the detailed band reconstruction around these two folded points using data from detwinned BaFe 2 As 2 , which can be revealed by using different polarizations. For the folded Γ-M x cut (Fig. 6b), along the AFM direction, the d xy electron band from M x folds unto the d xz and d yz hole bands around Γ, but little or no SDW gap appears along this high symmetry direction where folded bands cross. Along the orthogonal FM direction at this same point (Fig.   6c), the d yz electron band crosses the d xz /d yz bands, forming an SDW gap on the order of ~30meV between the d yz bands, which can be seen in both the d yz hole band and the d yz electron band. For the folded M y -Γ' point ( Fig. 6d), along the AFM direction, the d xz electron band crosses the d xy hole band, opening an SDW gap that saddles E F , with a gap size bigger than 50meV. Along the orthogonal FM direction at this point (Fig. 6e), the d xy hole and electron bands cross each other, opening an SDW gap that is larger than 50meV. Overall, we see that the SDW

20
The level of material-dependence reported in the past decade has been somewhat puzzling and perhaps disappointing for the ultimate goal of finding a simple unifying description of superconductivity. However, this may be well expected when we consider the multi-orbital nature of the FeSCs. As this is a new dimension which has been lacking from the machinery developed out of the cuprate problem, theoretical work taking into account the orbital degree of freedom has been very limited, but several work have already showed promise. From the strong coupling approach using a multiorbital t-J 1 -J 2 model, one study showed that the orbital-selectivity results in a gap anisotropy that is also orbital-dependent 79 . From a weak-coupling approach, a very recent theoretical study based on spin fluctuations taking into consideration the orbital-selective renormalization that modulates the coherent spectral weight of different orbitals was able to reproduce the observed momentum-dependent gap structure of monolayer FeSe and LiFeAs 80 . Other theoretical works have also proposed interesting mechanisms by which the FeSCs and cuprates could be united [81][82] . These work importantly demonstrate that behind the apparent gap variations amongst FeSCs there may be a common underlying pairing mechanism, and the source of the material-dependence may be the different degree of orbital-selective correlation effects, which tune the dominant orbitals that are manifested.

Discussion
We first summarize the key findings of the four major phases discussed: o The d xy orbital is also observed to participate by exhibiting a splitting in energy that is comparable to that of d xz /d yz orbitals, suggesting an anisotropic hopping origin rather than ferro-orbital order.
• The spin density wave order: o Band folding occurs, producing SDW gaps that are the largest in the d xy orbital, moderate in the d yz orbital, and smallest in the d xz orbital.
• The superconducting state: o Superconducting gaps are generally observed to be multi-gap, suggesting dominance of intra-orbital pairing.
o Gap functions cannot be described by single trigonometric gap functions, and also vary among families, suggesting the complex role of intra-orbital pairing and multi-orbital FS.
Having discussed the normal state, the nematic state, the magnetic state, and the superconducting state separately, we now discuss the relationship between these phases. From the normal state properties, we see that there is a systematic spread of electronic correlation over all the FeSCs, with a large dependence on certain structural parameters such as the bond length and bond angle.
As has been shown, the superconducting temperature, T C , is also highly dependent on the bond angle 83 . Hence superconductivity is expected to be optimized at intermediate electron correlation strength. The nematic phase and the collinear SDW phase are often discussed together. Here we see that the two orders can be strongly coupled, as in most iron arsenides, but not necessarily always the case, as in FeSe. Regardless of the strength of coupling of these two orders, we see that the spectral signature and magnitude for the nematic order is the same across different materials. The nature of these two phases to superconductivity is competitive. As has been reported, the spectral order parameters of these two orders both decrease at the onset of superconductivity 84 , similar to the macroscopic order parameters of the lattice orthorhombicity and the magnetic moment 85 .
For all three phases discussed, we also see a strong orbital-dependence. Hund's coupling suppresses orbital interaction, separating the d xy orbital from the largely degenerate d xz /d yz . This coupled with crystal field splitting effectively makes the d xy orbital the most strongly correlated, as seen in the normal state, and in some cases close to half-filling while the overall filling is      Table 1 Table 1 for references used in generating all panels.)  Table 1      50 Figure 6