Impurities in multiband superconductors

High-T_c cuprates are known for their high critical temperature, unconventional superconducting state, and unusual normal state properties. Fe-based superconductors, with T_c up to 58 K in bulk materials [24], and probably up to 110 K in FeSe monolayer at the SrTiO₃ substrate [25–29], stand in second place after cuprates. When superconductivity in iron-based materials was discovered, the question immediately arose — how similar are they to cuprates? Let us compare some of their properties.

At first glance, the phase diagrams of cuprates and many Fe-based superconductors are similar. In both cases, the undoped materials exhibit antiferromagnetism, which vanishes with doping; superconductivity occurs 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 iron-based materials the competition between these orders can take several forms. In LaFeAsO, for example, there appears to be a transition between the magnetic and superconducting states at a critical doping value, whereas in the 122 systems (BaFe₂As₂ and similar ones) 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 lies in the fact that undoped cuprates are Mott insulators, whereas iron-based materials 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, rule out effects of correlations in iron-based materials, but they may be moderate or small. In any case, density functional theory-based approaches describe the observed Fermi surface and band structure reasonably well in 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 reveal pseudogap behavior in both one-particle and two-particle charge and/or spin excitations, while a similar robust behavior is absent in iron-based materials. Generally speaking, the term 'pseudogap' implies a dip in the density of states near the Fermi level. There are, however, a wide variety of unusual features of the pseudogap state in cuprates. For example, a strange metal phase near optimal doping level in hole-doped cuprates is characterized by linear-T resistivity over a wide range of temperatures. In iron-based materials, 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 [30]. There are, however, indications of a pseudogap formation in densities of states of some pnictide bands; see, e.g., Refs [31, 32].

The mechanism of doping deserves additional discussion. Doping in cuprates is accomplished by replacing one of the spacer ions with another ion having a different valence, as in La_2−xSr_xCuO₂ and Nd_2−xCe_xCuO₂, or adding extra out-of-plane oxygen, as in YBa₂Cu₃O_6+δ. The additional electron or hole is then assumed to dope the plane in an itinerant state. In iron-based materials, the nature of doping is not completely understood — similar phase diagrams are obtained by replacing the spacer ion or by in-plane substitution of Fe with Co or Ni. For example, LaFeAsO_1−xF_x, Ba_1−xK_xFe₂As₂, and Sr_1−xK_xFe₂As₂ belong to the first case, while Ba(Fe_1−xCo_x)₂As₂ and Ba(Fe_1−xNi_x)₂As₂ belong to the second one. Whether these heterovalent substitutions dope the FeAs or FeP plane as in the cuprates was not initially clear [33], but now it is well established that they affect the Fermi surface, consistent with formal electron count doping [34, 35]. Another mechanism to vary electronic and magnetic properties takes advantage of the possibility of isovalent doping with phosphorous in BaFe₂(As_1−xP_x)₂ or ruthenium in BaFe₂(As_1−xRu_x)₂. 'Dopants' can act as potential scatterers and change the electronic structure because of the difference in ionic sizes or simply by diluting the magnetic ions with nonmagnetic ones. In iron-based materials, therefore, some of the doping mechanisms are connected with the changes in the transition metal layer. But the phase diagrams of all Fe-based materials in the first approximation 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 chalcogen above the Fe plane has frequently been noted as playing some role in the overall doping dependence [36–38].

It is well established that the superconducting state in cuprates is universally d-wave. By contrast, the gap symmetry and structure of the iron-based materials 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 iron-based materials, depend on the Fermi surface geometry, orbital character of bands, and degree of correlations [15, 39].

Conventional superconductors behave differently depending on the type of introduced impurities. So, nonmagnetic impurities do not suppress superconducting critical temperature T_c according to Anderson's theorem [40], while, on the contrary, magnetic impurities cause T_c suppression with the rate following the Abrikosov–Gor'kov theory [41].

Cuprate superconductors reveal a more complicated picture. Phase diagram asymmetry for hole- and electron-doped cuprates is closely related to the impact of nonmagnetic and magnetic impurities replacing copper sites on superconducting properties. In electron-doped systems (n-type), the situation is analogous to the conventional superconductors — nonmagnetic impurities weakly suppress T_c, while magnetic ones cause a collapse of superconductivity for an impurity concentration of about one percent, which is in close agreement with the Abrikosov–Gor'kov theory. These results follow from studies of both polycrystalline samples [42, 43] and Pr_2−xCe_xCu_1−yM_yO_4+z single crystals with M = Ni, Co [44].

Contrary to the n-type cuprates, hole-doped counterparts show a different behavior. Early studies on YBa₂Cu₃O₇ (Y-123) [45] revealed the suppression of superconductivity via replacement of copper not only by magnetic (Fe, Co, Ni), but also by nonmagnetic (Zn, Al, Ga) ions. Notice, however, that to compare effects of different types of impurities on T_c, it is more convenient to study the lanthanum-based system La_2−xSr_xCu_1−yM_yO₄ (with y being the amount of M = Fe, Co, Ni, Zn, Al, Ga), where all impurities are located in the CuO₂ layer, as opposed to the Y-123 system. In the latter system, copper in-plane sites are replaced by divalent ions, while trivalent ions generally occupy Cu–O chains, which reduces their effect on T_c and thus complicates interpretation of the results.

The problem described is absent in the La_1.85Sr_0.15Cu_1−yM_yO₄ compound, for which the following results were obtained in Ref. [46]: both magnetic impurity, Co, and nonmagnetic impurities, Zn, Al, Ga, result in almost the same T_c(y) dependence. At the same time, Fe causes the most rapid suppression of T_c, while Ni gives a slowest decrease in it, though both should be magnetic due to their atomic structure. To clarify the relation between superconductivity and the magnetic nature of impurities, static susceptibility measurements were made [46, 47]. They revealed the presence of an effective magnetic moment at the impurity site in all systems studied. Moreover, it become clear that the rate of T_c suppression has a weak correlation with the impurity valence. Furthermore, the magnitude of the moment strongly correlates with the critical impurity concentration at which T_c vanishes. This argues in favor of the magnetic mechanism of pairbreaking and against pairbreaking originating from the change in the hole doping level.

The authors of Ref. [46] suggested a qualitative explanation for the concentration dependence of T_c(y) and for the magnetic properties of impurities. It is based on indications that all impurities with zero spin (nonmagnetic Zn, Al, Ga, as well as Co³⁺ in the low-spin state) induce an effective magnetic moment in close proximity to the Cu²⁺ moment. That is, one has to consider the copper ion spin removed by the impurity. For an impurity with an open d-shell (Fe, Co, Ni), it is necessary to consider not only the removed copper spin but also its own impurity moment. The experimental value of the Fe³⁺ ion effective moment suggests that it resides in a high-spin state with S = 5/2 in the lanthanum system, and it generates an effective moment significantly larger than the Cu²⁺ moment. On the other hand, the anomalously small experimental value of the Ni²⁺ effective moment suggests that the spin should be no more than 0.32, instead of the expected S = 1. The authors of Ref. [46] explain this by the significant delocalization of the Ni spin state, in contrast to the strong localization of the Fe state.

Besides a qualitative explanation of the anomalous result of Cu with Ni replacement, proper treatment of the multielectron effects in the correlated band structure leads to a quantitative description of the T_c(y) dependence [48]. With the diamagnetic replacement of copper with zinc, the fraction of ions in configuration d¹⁰ (Zn²⁺) is equal to y. The model for such systems is the antiferromagnetic lattice of S = 1/2 spins with one empty site that behaves as one paramagnetic center due to the uncompensation of sublattices. As for the copper replacement with nickel, the nickel ion Ni²⁺, which formally should be in the d⁸ state with the spin S = 1, due to strong intraatomic Coulomb repulsion acquires an intermediate valence. The probability of it being in the nonmagnetic d¹⁰ state with the spin S = 0 is equal to , and the probability of the d⁹ state with S = 1/2 is . As follows from the summary of optical, photoemission, and magnetic data on La₂CuO₄, the weights of these states, and , are expressed via such parameters of the multiband p–d model of copper oxides [49] as energies of p and d holes in the crystal field and matrix elements of Coulomb interaction. This way, the nickel ion should have the effective spin instead of a nominal Ni²⁺ state with S = 1. Calculated values of and S = 0.36 [48] are in good agreement with the experimental data.

Therefore, with the substitution of nickel for copper, the amount of impurity ions in the d¹⁰ state (the same state as for zinc) is equal to . The probability of ions being in the d⁹ state is , and since their magnetic and charge characteristics are almost the same as those of copper, such ions should not suppress superconductivity. The resulting ratio of T_c(y) slopes for nickel and zinc impurities is , which is close to the experimental value of 0.38 for La_1.85Sr_0.15Cu_1−yM_yO₄ [46].

The change in the impurity effect with doping can be summarized as follows. Suppression of T_c by impurities in overdoped systems does not depend on the doping level p, while in underdoped samples it is strongly doping-dependent [50]. Given that the pseudogap state occurs exactly at low doping, the observed p-dependence emphasizes the importance of the ground state in the response of the system to the disorder.

It should be noted that an alternative to the chemical introduction of impurities is the creation of defects via fast neutron irradiation. Such a method benefits from avoiding some of the difficulties related to the replacement of some atoms with others. Suppression of T_c in this case, as well as other physical characteristics of cuprates being irradiated by neutrons, are extensively described in Refs [51–53].

Summarizing, strong electronic correlations causing the formation of local moments due to the presence of formally nonmagnetic impurities complicates significantly the interpretation of disorder effects on T_c suppression. Among other factors preventing the formulation of a consistent theory for the role of defects in superconductivity of cuprates are the absence of the theory for the correlated ground state, difficulties with controling the disorder parameters, and the presence of anisotropy in impurity scattering. Since a detailed discussion of cuprate physics with the important role of strong correlations is not the goal of the present review, here we mentioned only a few important points of impurity scattering. We direct the curious reader to other reviews, like Refs [54–60].

Nor are we going further into the details of d-wave superconductivity and related problems in cuprates. This topic is extensively reviewed in many papers concerning both the theories of impurity scattering in a d-wave superconductor [61–70] and the effect of impurities on observable properties of cuprates [71–79]. Let us just mention that the single-band d-wave superconductor can be approximately treated as a two-band superconductor with opposite signs of the gaps in different bands — the analogy of the s_± state [80]. In other words, parts of the Fermi surface with different signs of the order parameter are considered to originate from different bands. Though this is a rough approximation, it can give some qualitative results.

Under normal conditions, iron is ferromagnetic. Under pressure, however, once the Fe atoms form an hcp lattice, iron becomes nonmagnetic and even superconducting at T < 2 K [81], most probably due to electron–phonon interaction [82]. On the other hand, iron-based superconductors are quasi-two-dimensional materials with the conducting square lattice of Fe ions. The Fermi level is occupied by the 3d⁶ states of Fe²⁺. This was established in early DFT (density functional theory) calculations [16, 83, 84] which are in quite good agreement with the results of quantum oscillations and ARPES (angle-resolved photoemission spectroscopy). All five orbitals, d_x²−y², d_3z²−r², d_xy, d_xz, and d_yz, are at or near the Fermi level. This results in a significantly 'multiorbital' and multiband low-energy electronic structure, which could not be described within the single-band model. For example, within the five-orbital model [17] correctly reproducing the DFT band structure [85], the Fermi surface comprises four sheets: two hole pockets around the point (0, 0), and two electron pockets around the points (π, 0) and (0, π). Such k-space geometry results in the possibility of spin-density wave (SDW) instability due to the nesting between hole and electron Fermi surface sheets at the wave vector Q = (π, 0) or (0, π). Upon doping x, the long-range SDW order is destroyed. If electrons are doped, then for the large x hole pockets disappear, leaving only electron Fermi surface sheets, as observed in K_xFe_2−ySe₂ and in FeSe monolayers [26]. Upon increasing hole doping, first, a new hole pocket appears around the point (π, π) and then electron sheets vanish. KFe₂As₂ just corresponds to the latter case. ARPES confirms that the maximal contribution to the bands at the Fermi level comes from the d_{xz, yz} and d_xy orbitals [86, 87]. At the same time, as will be pointed out later, the presence of a few pockets and the multiorbital band character significantly affect superconducting pairing.

Soon after high-quality samples of cuprates were prepared, the d_x²−y² symmetry of the gap, with the cos k_x − cos k_y structure, was empirically established by penetration depth, ARPES, NMR, and phase sensitive Josephson tunneling experiments. No similar consensus on any universal gap structure has been reached even after several years of intensive research on high-quality single crystals of iron-based superconductors. There is strong evidence that small differences in electronic structure can lead to a strong diversity in superconducting gap structures, including nodal states and states with a full gap on the Fermi surface. The actual symmetry class of most of the materials may be generalized A_1g (extended s-wave symmetry), probably involving a sign change of the order parameter between Fermi surface sheets or its parts [15]. Understanding the symmetry character of the superconducting ground states, as well as a detailed structure of the order parameter should provide clues to the microscopic pairing mechanism in iron-based materials and thereby lead to a deeper insight into the phenomenon of high-temperature superconductivity.

The group-theoretical classification of gap structures in unconventional superconductors is rather complicated and has been reviewed in, e.g., Ref. [9]. 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. The easiest and most accurate way to probe whether the pair is spin-triplet is via the Knight shift measurements. These experiments have been performed on several iron-based materials, including Ba(Fe_1−xCo_x)₂As₂ [88], LaFeAsO_1−xF_x [89], PrFeAsO_0.89F_0.11 [90], Ba_1−xK_xFe₂As₂ [91, 92], LiFeAs [93, 94], and BaFe₂(As_0.67P_0.33)₂ [95]. It was found that the Knight shift decreases in all crystallographic directions. This effectively ruled out triplet symmetries, such as p-wave or f-wave.

Having ruled out the spin-triplet states, we focus first on simple tetragonal point group symmetry. In a three-dimensional tetragonal system, group theory allows only five one-dimensional irreducible representations according to how the order parameter transforms under rotations by 90° and other operations of the tetragonal group: A_1g ('s-wave'), B_1g ('d-wave', x² − y²), B_2g ('d-wave', xy), A_2g ('g-wave', xy(x² − y²)), and E_g ('d-wave', xz; yz). Notice that the s₊₊ and s_± states all have the same symmetry, i.e., neither changes sign if the crystal axes are rotated through 90°. By contrast, the d-wave state changes sign under such a rotation. Notice further that the mere existence of the hole and electron pockets leads 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, we can have individual rotations on single pockets and still preserve symmetry. For example, if for the d-wave case we rotate the gap in the hole pocket, 90°, but keep the electron pocket signs fixed, it still represents a B_1g state. B_2g states are also possible by symmetry and would have nodes on the electron pockets. Further, more complicated, gap functions with differing relative phases become possible when more pockets are present and when three-dimensional effects are included.

It is important to note that, while the d-wave does not necessarily imply the existence of gap nodes, in combination with a quasi-two-dimensional Fermi surface at the center of the Brillouin zone such nodes are unavoidable: either vertical for the B_1g, B_2g, and A_2g symmetries, or horizontal for the E_g symmetry. Since such a Fermi surface exists in pnictides, the experimentally proven absence of nodes on it would evidence against the d-wave symmetry. As for experiments, surface probes such as ARPES show full gaps on the central Fermi surface sheet. Moreover, the full gap on the whole Fermi surface is observed in tunneling and bulk probes in hole-doped systems, as well as in materials with a small electron doping.

There are also direct experiments that provide evidence against the d-wave. The Josephson current in the c-direction, when the studied superconductor is coupled to a known s-wave superconductor, would confirm the s-type of the former. Exactly such current was observed in 122 single crystals [96].

Another piece of evidence comes from the absence of the so-called anomalous Meissner effect (or Wohlleben effect) [97]. This effect manifests itself in polycrystalline samples with random orientation of grains. It was predicted at the beginning of the cuprates era [98] and since then it has been routinely observed only in d-wave superconductors. The Wohlleben effect appears due to the fact that the response to a weak external magnetic field is paramagnetic, i.e., opposite to the standard diamagnetic response of an s-wave superconductor. This happens because half of the weak links have a zero phase shift, and the other half have a π phase shift.

The described separate pieces of evidence strongly suggest that the pairing symmetry is s-wave, not d-wave. 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 lacking, and it is highly desirable to make an ultimate conclusion.

It should also be borne in mind that nothing forbids different iron-based materials from having different order parameter symmetries, although our previous experience with other superconductors tends to argue against this. Indeed, there are several theories claiming that, while most iron-based systems have the s-wave gap symmetry, those with unusual Fermi surfaces with either electron or hole pockets can have the d-wave symmetry of the order parameter [20, 99–103].

Notice that the term symmetry should be distinguished from the term structure of the gap. The latter is used 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. Let us illustrate this for the s-wave symmetry (Fig. 1). Fully gapped s-states without nodes on the Fermi surface differ only by a relative gap sign between the hole and electron pockets, which is positive in the s₊₊ state and negative in the s_± state. On the other hand, the gap vanishes in nodal s-states at certain points on the electron pockets. These states are called 'nodal s_±' ('nodal s₊₊') and are characterized by the opposite (same) averaged signs of the order parameter on the hole and electron pockets. Nodes of this type are sometimes described as 'accidental', since their existence is not dictated by symmetry, in contrast to the symmetry nodes of the d-wave gap. Therefore, they can be removed continuously, resulting in either an s_± or an s₊₊ state [104, 105].

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Superconducting states with different symmetries and structures of order parameters differently respond when subject to the disorder. As we have mentioned earlier, nonmagnetic impurities in the single-band s-wave superconductors do not suppress T_c, while magnetic impurities do it in accordance with the Abrikosov–Gor'kov theory [41]. In unconventional superconductors, suppression of the critical temperature as a function of parameter Γ characterizing impurity scattering may follow a complicated, yet very explicit, law. That is why many authors desire to attribute the observed T_c(Γ) dependence as pointing towards a particular gap structure is not surprising.

Several experiments on iron-based systems show that T_c suppression is much weaker than expected in the framework of the Abrikosov–Gor'kov theory for both nonmagnetic [106–112] and magnetic disorder [107, 113–116]. It is worth advising the reader to interpret T_c suppression 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 the theoretical scattering rate, which is generally difficult to determine from experiments. The alternative is to plot T_c vs residual resistivity change Δρ, but this is only possible, first, if the ρ(T) curve shifts rigidly with disorder and, second, if comparisons with theory include a proper treatment of the transport rather than the quasiparticle lifetime. Finally, the effect of a chemical substitution in an iron-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. The most promising alternatives are irradiation experiments, since the disorder is introduced without altering the chemical composition of the studied material. Experiments of this kind include irradiation by protons [109, 117, 118], neutrons [106], electrons [105, 112, 119, 120], and heavy ions [121–123]. There are some specific complications though. For example, consider the work on neutron irradiation by Karkin et al. [106]. As seen from other work by the same group [124], the structure of the studied material changed after neutron irradiation. The doping is also accompanied by changes to the structural parameters that correlate with the T_c changes [36–38, 125]. And it might seem that the problem of separating the role of defects and changes to structural parameters with neutron irradiation is not a simple one. Given the many uncertainties present in the basic modeling of a single impurity, as well as the multiband nature of iron-based materials, it is reasonable to assume that systematic disorder experiments may not play a decisive role in determining the order parameter symmetry and structure. Nevertheless, one can extract useful information from the qualitative effects appearing on the level of simple multiband models of disorder [126, 127].

In this review, we demonstrate the basics of impurity effects on multiband superconductivity using the simple model for iron-based materials as an example. In particular, within the -matrix approximation, we discuss the role of scattering on nonmagnetic and magnetic impurities for the s_± and s₊₊ states in a two-band model. We show that for the finite nonmagnetic impurity scattering rate, the transition from s_± to s₊₊ occurs, i.e., one of two gaps changes sign when going through zero. The transition happens for the positive sign of the superconducting coupling constant averaged over the bands. At the same time, T_c stays finite and almost independent of the impurity scattering rate, which is proportional to the impurity concentration and magnitude of the scattering potential. There are two cases for scattering on magnetic impurities when the transition temperature T_c is not fully suppressed, in contrast to the Abrikosov–Gor'kov theory, but the saturation of it appears in the regime of a large scattering rate. The first case is characterized by purely interband impurity scattering. At the same time, the s_± gap is preserved, while the s₊₊ state transforms into the s_± state with increasing magnetic disorder. The second case corresponds to the unitary limit of scattering with the gap structure remaining intact. The reason for the s_± ↔ s₊₊ transitions is the following: if one of the two competing superconducting interactions leads to a state robust against impurity scattering, then, although it was not dominant in the clean limit, it should become dominant while the other state is destroyed by impurity scattering. Since the transitions between the s_± and s₊₊ states go through the gapless regime, they should reveal themselves in thermodynamic and transport properties of the system and thus be observable in optical and tunneling experiments, as well as in photoemission spectroscopy. Because one of the gaps vanishes near the transition, ARPES should reveal the gapless spectra and, in the optical conductivity, the transition should result in the 'restoring' of the Drude frequency dependence of Re σ(ω). We ignored the complicated question of nonmagnetic and magnetic scattering channels coexisting due to its poor development in the multiband case at the time of writing.

The layout of the review is as follows. In Section 2 we present the Eliashberg formalism for a multiband superconductor and the -matrix approximation for the impurity self-energy. The approximation is then applied to the simple two-band model, in which either the s_± or s₊₊ state occurs, depending on the parameters. Section 3 contains a discussion of qualitative impurity scattering effects in the Born limit. In Sections 4 and 5, the roles of nonmagnetic and magnetic impurities are described, respectively. Section 6 is devoted to a short review of the experimental findings on the impact of impurities on the superconducting state of pnictides and chalcogenides. In Section 7, we discuss the effect of disorder on such experimentally observable dynamical characteristics as the density of states, the spectral function, optical conductivity, and the magnetic field penetration depth. Conclusions are contained in the final Section 8.

Nonmagnetic impurities in a conventional two-band superconductor with two isotropic gaps lead to scattering of quasiparticles between bands or within each band. Interband processes shown in Fig. 4 result in averaging the gaps and, therefore, to the initial suppression of T_c, after which T_c saturates and stays constant until localization effects become important [143, 144]. Interband scattering in a two-band system with the sign-changing order parameter leads to a much more complicated behavior [68, 80, 135, 145]. In this case, nonmagnetic impurities with the interband component of the scattering potential destroy the superconductivity even for equal magnitudes of gaps and densities of states (the so-called symmetric model). The reason is quite simple — interband scattering results in averaging the gaps in two different bands and, since Δ_a and Δ_b have opposite signs in the s_± state, their average goes to zero. Critical temperature T_c would then vanish for the finite critical impurity concentration, similar to the theory of magnetic impurity scattering in a single-band s-wave superconductor [41]. Such a characteristic feature of iron-based superconductors was quickly noticed by different groups of researchers [16, 132, 146–148].

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As for the effect of nonmagnetic and magnetic disorder on a multiband anisotropic superconductor, the simple (and naive!) qualitative rule of thumb is as follows: when a nonmagnetic impurity scatters a pair from one point on the Fermi surface to 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. As follows from calculations for some particular cases, however, such a naive qualitative rule collapses and quite unexpected results appear, which we discuss in Sections 4–7. But before going further, we apply the simplest Born limit in Section 3.2 to demonstrate the general results in single- and two-band models.

Here, we consider a weak coupling example, i.e., the case of λ_αβ ⪡ 1, and describe the effect of disorder on a macroscopic characteristic — the critical temperature of the superconducting transition T_c. Scattering on static impurities would results only in a decrease of T_c from its clean limit value T_c0.

Order parameter Δ_αn in a multiband superconductor in the clean limit is a solution of the equation that follows from expressions (13) and (15). Neglecting the frequency dependence of the coupling functions (which is the essence of weak coupling), we have

Summation over the Matsubara frequencies ω_n goes until a cut-off frequency ω_c. In the limit T → T_c0, we have Δ_α → 0 and g_2αn → −πN_αΔ_α/|ω_n|. Then, expression (25) becomes the equation for the critical temperature in the clean limit, i.e., for T_c0:

The single-band case is realized at λ_αβ = λδ_αβ:

where we have taken into account that ω_n = (2n + 1) πT_c0, (2N_c + 1) πT_c0 = ω_c, Ψ(1/2) = −γ − 2 ln 2 ≡ −C, and Ψ(z + 1) → ln z as z → ∞. Here, Ψ is the digamma function, and γ is the Euler constant. Since exp C/(2π) ≈ 1.13, the solution of the above equation gives the well-known expression for the superconducting critical temperature T_c0 = 1.13ω_c exp (−1/λ). Notice again that the coupling constant includes the density of states, λ ∝ N_a.

Expression (26) in a two-band case is a system of two equations in Δ_a and Δ_b gaps. From the consistency condition of the system (determinant of the corresponding matrix should be equal to zero), one can derive the expression for T_c0:

Let us consider the simplest case of treating the impurity scattering by replacing the 'bare' Green's function g_2βn in equation (25) with the full one (11) containing the renormalized order parameter and frequency :

where . As T → T_c, we have and .

The equation for the critical temperature becomes

As follows from a comparison with equation (26), T_c does not depend on impurity scattering if the following condition is satisfied:

In the Born approximation, only the contribution from double scattering at the same impurity is allowed, . We can now derive expressions for the frequency and order parameters:

where is the scattering rate parameter; the sign + (−) corresponds to nonmagnetic (magnetic) impurities. The different signs for magnetic disorder occur due to the spin operators accompanying the impurity potential in in H_imp.

Let us first consider the single-band case. Then, γ_ab = 0 and . Equations for nonmagnetic impurities are as follows: and , which immediately lead to relation (30). Therefore, T_c is independent of impurity concentration. This is the essence of Anderson's theorem.

For magnetic impurities, the appropriate equations are and . Thus, condition (30) is violated. Instead of it, we have

since for the T_c equation we are interested in the case of ω_n ⩾ 0. In the equation for the critical temperature, an additional factor 2γ_aa appears in the denominator:

In the ω_c → ∞ limit, the last equation takes the form

Combining this equation with the corresponding expression in the clean limit, 1 = λ [ln ω_c/(2πT_c0) − Ψ(1/2)], we finally have

which represents the formula for T_c suppression according to the Abrikosov–Gor'kov theory [41].

Let us now consider the two-band case. For impurity scattering within only one band ('intraband impurities'), γ_ab = 0, equations (31) and (32) for different bands are not coupled. Therefore, all conclusions made for the single-band case above are also true here for each band in the presence of both nonmagnetic and magnetic impurities.

When both intra- and interband nonmagnetic impurity scattering channels are present, from equations (31) and (32) we have

Evidently, if Δ_a = Δ_b, then condition (30) holds and, thus, T_c is independent of the disorder. Therefore, there is no impurity effect on the multiband isotropic s-wave superconducting state. If, however, Δ_a ≠ Δ_b, then condition (30) is violated and T_c will be suppressed by scattering on impurities.

For a magnetic impurity, equations (31) and (32) lead to

Obviously, condition (30) holds true only for Δ_b = −Δ_α and γ_aa = γ_bb = 0. That is, the s_± state with the equal absolute values of gaps is not susceptible to the scattering by magnetic impurities having an interband scattering channel only ('interband impurities'). In all other cases, T_c would decrease with increasing concentration and potential of magnetic impurities.

In the case of a nonmagnetic disorder, we can simplify the problem by reducing the dimension of matrices due to spin degeneracy. Thus, instead of expressions (9) and (10), we have 4 × 4 quasiclassical matrix Green's function in Nambu and band spaces:

where are Pauli matrices corresponding to the Nambu space.

The impurity potential matrix entering the -matrix equation (16) is , where . Without loss of generality, we set R_i = 0 for the single-impurity problem studied here. For simplicity, intraband and interband parts of the impurity potential are set equal to v and u, respectively, such that (U)_αβ = (v − u) δ_αβ + u.

From equations (16) and (34), we then have

Renormalizations of frequencies and gaps come from and respectively. Equations for and are derived through the replacement a ↔ b in the equations above. Considering the relation , we derive the following solution for and :

where

In what follows, apart from the general case, we will also consider two important limits: the Born, weak scattering, limit with πuN_{a, b} ⪡ 1, and the opposite limit of very strong scattering with πuN_{a, b} ⪢ 1, called the unitary limit.

It is convenient to introduce the generalized cross-section parameter

and the impurity scattering rate

Parameter η controls the ratio between intra- and interband scattering potentials:

With the introduced notations, we rewrite equations for frequency (12) and order parameter (13) taking the impurity self-energy (37), (38) into account:

where

Let us examine important limiting cases. In the Born limit, we have σ → 0 (weak scattering, πuN_{a, b} ⪡ 1); thus, D = 1, Γ_a = 2n_impπN_bu², and

where γ_aa = 2πn_impN_au²η² and γ_ab = 2πn_impN_bu². Evidently, for the finite interband scattering γ_ab, i.e., finite η, different bands are mixed in equations. This leads to the suppression of T_c, similar to the one following from the Abrikosov–Gor'kov expression (33).

In the unitary limit, with σ → 1 (strong scattering, πuN_{a, b} ⪢ 1), we have Γ_a = 2n_imp/(πN_a), and one must distinguish two cases:

(1) Uniform impurity potential with η = 1. We the have

where

Obviously, different bands are mixed in equations for the renormalized frequency and order parameter, so this leads to a suppression of T_c.

(2) All other cases with η ≠ 1. We then have

We get the same result as for the intraband impurities, since the other band (b) does not contribute to the equations. Surprisingly, Anderson's theorem works here independently of the gap signs in different bands. Thus, T_c should be finite for any impurity concentration.

In this way, there is a special case of T_c suppression in the unitary limit for the uniform impurity potential η = 1. Such a situation arises due to the structure of the denominator D in equations (42), (43). It vanishes at η = σ = 1 and we have to accurately pass first to the limit η → 1, and only then put σ → 1. This is a η = 1 case, which was considered in Ref. [148]. For all other values of η (even for a slight difference between intra- and interband potentials), impurities are not going to affect the critical temperature. Of course, from the physical point of view, the former situation is improbable, since it is hard to imagine an impurity in a multiorbital system where intra- and interband scatterings have equal intensities.

As T → T_c, the equations become significantly simplified, because the order parameter vanishes and . Thus, the linearized Eliashberg equations (12), (13) for the renormalization factors and gap functions [131] can be rewritten, in view of expressions (42), (43), as follows:

where we have introduced renormalized impurity scattering rates [126]:

After substituting Z_αn from Eqn (50) to (51), we obtain the equation for the critical temperature T_c:

The last term is finite only for α ≠ β. Therefore, intraband terms and are cancelled and do not contribute to T_c, in agreement with Anderson's theorem. From the expression for scattering rates (52), we recover explicitly the well-known but counterintuitive result that in the unitary limit —that is, nonmagnetic impurities do not affect T_c in the s_± state [68, 135].

Since T_c depends only on parameter , we call it the effective impurity scattering rate.

To determine T_c, we must solve numerically either equation (54) or Eliashberg equations (50), (51) and vary T to find the highest temperature at which nontrivial solution exists [126]. For definiteness, we choose N_b/N_a = 2. The resulting T_c and gap Δ_{α, n = 1} as functions of Γ_a in the s₊₊ state are depicted in Fig. 5. Generally, the superconductivity is not suppressed completely, though there is an initial drop of T_c due to the scattering between bands with initially unequal gaps. Notice that the system in the unitary limit seems not to care about disorder — neither critical temperature nor gaps depend on Γ_a. As seen from equations (46), (47), there is, however, an isolated point, η = 1, corresponding to the vanishing of determinant D. That is, superconductivity is suppressed for the uniform impurity potential, v = u (see Fig 5).

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Figure 6 demonstrates T_c as a function of Γ_a for the s_± state. As follows from calculations, T_c behavior is qualitatively different for different signs of the coupling constant averaged over the Fermi surface [126]:

where N = N_a + N_b is the total density of states in the normal phase. We choose the following coupling constants for illustrative purpose: (λ_aa, λ_ab, λ_ba, λ_bb) = (3, −0.2, −0.1, 0.5) for 〈λ〉 > 0 [133, 134], (1, −2, −1, 1) for 〈λ〉 < 0, and (2, −2, −1, 1) at 〈λ〉 = 0. For the first set in the clean limit, a critical temperature is T_c0 = 30 cm⁻¹, for the second set it is T_c0 = 27.96 cm⁻¹, and for the third set it is T_c0 = 31.47 cm⁻¹, which correspond to 43.1 K, 40.2 K, and 45.2 K. It should be noted that the strongest T_c suppression occurs in the Born limit for the pure interband potential, i.e., η = 0. In the opposite limit of pure intraband scattering with u = 0 (η → ∞), pairbreaking is absent because . Such a situation appears in the unitary limit. As for the dependence of T_c on (52), shown in Fig. 7, all cases with different sets of σ and η fall onto one of the universal T_c curves, depending on the sign of the coupling constant averaged over the Fermi surface, 〈λ〉. It is clearly seen from Fig. 7 that, depending on the sign of 〈λ〉, one gets two types of T_c behavior for the s_± state: (1) the critical temperature vanishes at a finite impurity scattering rate for 〈λ〉 < 0, and (2) for 〈λ〉 > 0, the critical temperature remains finite as . In the marginal case of 〈λ〉 = 0, we find that but with exponentially small T_c. Therefore, we have found the universal behavior of T_c controlled by a single parameter 〈λ〉.

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While the behavior of type (1) systems is in agreement with the qualitative statement that s_± superconductivity is destroyed by nonmagnetic interband impurities due to the 'mixing' of gaps with different signs [145, 149], the behavior of type (2) systems with 〈λ〉 > 0 is surprising. To understand what happens in this case, we calculated the gap Δ_αn for the first Matsubara frequency n = 1 at T = 0.016T_c0. Results are shown in Figs 8 and 9 for zero and finite intraband potential v, respectively. Coupling constants λ_αβ are chosen to have T_c0 ≈ 40 K. It is seen that gaps on both bands, Δ_{a(b) n}, converge to the same value Δ_Γa(b)→∞, while T_c quickly saturates. The initially negative order parameter Δ_bn (corresponding to the smaller gap) increases and at some point in time passes through zero and becomes positive. After that, since the gap signs for both bands are equal, we have the s₊₊ state. Due to Anderson's theorem, this state is robust against impurity scattering, thus having a finite T_c up to Γ_a → ∞. Therefore, T_c stays finite in type (2) systems due to the s_± → s₊₊ transition.

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The transition is also seen in gap functions Re Δ_α(ω) analytically continued to real frequencies, which are shown in Fig. 10.

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Similar to the s₊₊ state, there is no disorder effect on the critical temperature or gaps in the unitary limit except for the case of η = 1, where the s_± → s₊₊ transition occurs (see Fig. 9), which again makes the case of uniform scattering somehow unique [68].

There is a simple physical reason for the transition: with increasing interband scattering, order parameters on different Fermi surfaces 'mix' due to the scattering processes and converge to the same value. At the same time, the larger gap 'attracts' the smaller one that passes through zero and changes its sign. Similar effects were discussed in Refs [145, 149, 152, 153] for the two-band s₊₊ superconductor, and in Ref. [104], where node lifting in the extended s_± state on the electron pocket was investigated. The discovered s_± → s₊₊ transition allows us to explain the much slower suppression of critical temperature than that following from the well-known Abrikosov–Gor'kov equation. Qualitatively, this result was confirmed by agreement with the numerical solution of the Bogoliubov–de Gennes equations [154, 155].

In the case of magnetic impurities, we have to consider a Green's function matrix entering into equation (9) with the dimension 8 × 8. This considerably complicates the problem in comparison with the study of nonmagnetic disorder. The impurity potential for the noncorrelated impurities can be written out as , where

is the 4 × 4 matrix, with (...)^T being the matrix transpose, and S = (S_x, S_y, S_z) being the classic spin vector [156]. The vector is composed of Pauli τ-matrices, . The potential strength is determined by . For simplicity, the intraband and interband parts of the potential are set equal to and , respectively, such that (V)_αβ = ( − ) δ_αβ + . Then, V is given by

Components of the impurity potential matrix Û are then , and the matrix itself takes the form

Coupled -matrix equations (16) for the aa and ba components of the self-energy in the introduced notations become

The solution of the system of equations in the matrix form is given by

where . Renormalizations of frequencies and gaps come from

Equations for and are derived from above equations via replacement a ↔ b.

We assume that spins are not polarized and s² = 〈S²〉 = S(S + 1). Since s enters everywhere together with the components and of the impurity potential (see expression (58) for Û), without loss of generality we later set s = 1, assuming that and are renormalized to include s in themselves.

As follows from the calculations, similar to the results presented in Section 4, expressions for and are proportional to the impurity scattering rate Γ_{a, b} and contain the generalized cross-section parameter σ that helps to control the approximation for the 'strength' of impurity scattering. This ranges from the Born limit (weak scattering, π N_a, b ⪡ 1) to the unitary limit (strong scattering, π N_{a, b} ⪢ 1):

We also introduce the parameter η to control the ratio between intra- and interband scattering potentials, = η.

Expressions for at arbitrary temperature and η are too complicated and noninformative to write them down here. It is much more convenient to consider limiting cases. We also consider the three special forms of the impurity potential: the uniform potential with η = 1 ( = ), the interband-only potential with η = 0 ( = 0, ≠ 0), and the intraband-only potential with = 0, ≠ 0 (formally, η = ∞).

The Born limit corresponds to σ = 0. Eliashberg equations (12), (13) are then written out as follows:

One of the significant differences between this expression and the analogous one for nonmagnetic impurities [see formula (44) and (45)] is the minus sign before the term originating from impurity scattering in equation (68). In the presence of interband-only scattering (η = 0), we derive here the remarkable result

Indeed, for the s₊₊ state we have and equations written above correspond to the generalization of the Abrikosov–Gor'kov theory to the two-band case; therefore, impurities should suppress superconductivity. This, however, as we will see later, is not always true due to the complicated structure of equations, and their self-consistent solution may lead to unexpected results. On the other hand, for the s_± state we have and the sign of the last term in formula (70), originating from impurity scattering, changes and the equations become similar to expressions for a two-band superconductor with nonmagnetic impurities. That is, T_c is not suppressed by disorder, except when η = 1.

For the uniform impurity potential, we have η = 1, then and . Here, contributions from both the a and b bands are mixed, so we expect a suppression of T_c by the disorder [151].

When the interband component is absent (η = ∞), equations for different bands are decoupled:

and we have the suppression of superconductivity in each band, following the Abrikosov-Gor'kov theory.

It is remarkable that equations in the unitary limit are exactly the same as in the unitary limit for nonmagnetic impurities (46)–(49). Therefore, all conclusions about suppression of superconductivity for η ≠ 1 and η = 1 are the same.

Now, we write down the Eliashberg equations for special shapes of the impurity potential. For the intraband-only impurity potential ( = 0), terms in equations corresponding to bands a and b are separated, namely

where

For the impurity potential scattering solely between different bands ( = 0), equations for different bands decouple:

where

The results that follow were obtained by solving self-consistently frequency and gap equations (12) and (13) with the impurity self-energy from the solution of equations (59), (60) for both arbitrary finite temperatures below T_c and at T_c [127]. Hereinafter, we consider for illustrative purposes the case of N_b/N_a = 2 and choose coupling constants as (λ_aa, λ_ab, λ_ba, λ_bb) = (3, −0.2, −0.1, 0.5) for the s_± state with 〈λ〉 > 0 [133, 134], and as (3, 0.2, 0.1, 0.5) for the s₊₊ state. Critical temperature in the clean limit for both sets is T_c0 = 30 cm⁻¹, which corresponds to 43.1 K.

In Figs 11–13, we plot T_c and Matsubara gaps Δ_αn for the first Matsubara frequency ω_{n = 1} = 3πT as functions of Γ_a for both s_± and s₊₊ superconductors and for various values of σ. The real part of the analytical continuation of Δ_αn to real frequencies, the gap function Re Δ_α(ω), is shown in Fig. 14.

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First, we discuss the s_± state. Critical temperature T_c becomes insensitive to impurities for the pure interband scattering, = 0. This partially confirms qualitative arguments that the s_± state with magnetic impurities behaves like the s₊₊ state with nonmagnetic disorder [145, 149], and agrees with theoretical calculations in the Born limit [157]. For the initially unequal gaps, |Δ_α| ≠ |Δ_b|, there is an initial decrease in T_c for small Γ_a until the renormalized gaps become equal and then T_c saturates, since the analog of Anderson's theorem is achieved. For the finite , intraband scattering on the magnetic disorder averages gaps up to zero and, thus, suppresses T_c. On the other hand, in the unitary limit (σ = 1) a T → T_c we obtains

for arbitrary values of η, including the case of intraband-only impurities, 1/η = 0. This form of equation is the same as for nonmagnetic impurities and, thus, there is no impurity contribution to the T_c equation, in analogy to Anderson's theorem. The only exception here is the special case of uniform impurities, η = 1, when

Both gaps are mixed in the equation for ; thus, they tend to zero with an increasing amount of disorder. That is also true away from the unitary limit (see Fig. 13) and is the source for the claim that the uniform potential with = is a special case with the strongest T_c suppression.

In general, the multiband s₊₊ state should always be fragile against paramagnetic disorder, since magnetic scattering between bands having gaps of the same sign is equivalent to pairbreaking scattering within the single (quasi)isotropic band. Surprisingly, we found a regime with the saturation of T_c for the finite amount of disorder right after the initial downfall (similar to the one following from the Abrikosov–Gor'kov theory) (Fig. 12b). The saturation of T_c is observed for interband-only impurities, while the presence of the intraband magnetic disorder ultimately suppresses T_c to zero. However, depending on the 'strength' of scattering σ, a decrease in T_c may proceed quite slow compared to the one predicted by the Abrikosov–Gor'kov theory.

To understand the origin of the T_c saturation, we analyzed the gap function dependence on the scattering rate Γ_a (see Fig. 12). For the s₊₊ state after a certain value of the scattering rate, the smaller gap, Δ_b, becomes negative. What we see is the s₊₊ → s_± transition. As soon as the system becomes effectively s_±, scattering on magnetic impurities cancels out in the T_c equation, similar to Anderson's theorem, and T_c saturates. Before the saturation, the initial downfall akin to the one following from the Abrikosov–Gor'kov theory occurs. The transition is also seen in the frequency dependence of the gap function on a real frequency axis (see Fig. 14).

Similar to the s_± → s₊₊ transition for nonmagnetic disorder, there is a simple physical argument behind the s₊₊ → s_± transition here. Namely, with increasing interband magnetic disorder, the gap functions on different Fermi surfaces tend to the same value and, if one of the gaps is smaller than another, it crosses zero and changes sign. A similar effect has been mentioned in Refs [145, 149, 152] for a two-band systems with s₊₊ symmetry in the Born limit.

Notice that here we do not consider a time-reversal symmetry broken s_± + is₊₊ state. It may appear for T≲T_c in cases when translational symmetry is violated [158].

First, we discuss the transition from the s_± to the s₊₊ state induced by nonmagnetic impurities. A total density of states N(ω) calculated using expression (17) for systems with 〈λ〉 > 0 is plotted in Fig. 15a. Upon increasing the impurity scattering rate, the smaller gap closes, resulting in a finite residual density of states at the zero frequency, N(ω = 0), and then reopens. Such behavior is reflected in the temperature dependence of the London penetration depth (19), shown in Fig. 15b. Here, we present results correspondingly normalized to the plasma frequency ω_pα. Evidently, in a clean limit has an activation temperature dependence determined by the smaller gap, then transforms into the T_c behavior in the gapless state, and finally shows a new activation regime in the s₊₊ state. In other words, at Γ_a = 0 we have a typical two-gap dependence [168]. For larger values of the scattering rate, when the two gaps are almost equal, the temperature dependence of the penetration depth becomes like that in a single-band superconductor.

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The density of states N(ω) and the inverse square of the penetration depth in the case of magnetic impurities with = /2 and σ = 0.5 are shown in Fig. 16 for the s_± and s₊₊ superconductors. In the former case, we see the expected behavior with gradually decreasing gaps. Gapless superconductivity with the residual N(ω = 0) occurs for Γ_a > 10T_c0 when Re Δ_α(ω = 0) vanishes, which is seen in Fig. 14b. As for the s₊₊ state, the smaller gap vanishes upon increasing the impurity scattering rate Γ_a, leading to a finite residual density of states N(ω = 0). Then, the gap reopens and Δ_bn ≠ 0 until T_c reaches zero at Γ_a ∼ 20T_c0. Still, the superconductivity stays gapless with the finite N(0), because Re Δ_α(ω = 0) → 0, as seen in Fig. 14d. Penetration depth in the clean limit shows the activation behavior determined by the smaller gap. In the case of the s₊₊ state, the penetration depth becomes proportional to T² in the gapless regime, causing its significant reduction near Γ_a ∼ 4T_c0 (Fig. 16d), and then the penetration depth shows an activation temperature dependence in the s_± state after the transition.

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The presence of the gapless state should definitely manifest itself in ARPES spectra. The total measured photoemission current intensity I(k, ω) in the sudden approximation is given by

where M(k, ω) is the matrix element of one-electron dipole interaction, depending on the initial and final states of the photoelectron, photon energy, and its polarization, f (ω) is the Fermi function, and A_α(k, ω) is the spectral function. The last can be expressed through the analytical continuation of Green's function (6) to the real frequencies as

Notice that here we have to deal with a 'bare' dispersion ξ_kα, because the self-energy in our approximation does not depend on momentum and makes corresponding contributions neither to the dispersion nor to the chemical potential shift.

The contribution from the electron–boson interaction to the self-energy Σ_0α(k, ω) vanishes in the weak coupling approximation [169]. Therefore, we have Σ_0α(ω) → 0 and Σ_2α(ω) → Δ_α(ω) in the model with the isotropic self-energy. Then, the spectral function takes the following form [169]:

where

To be more specific, let Δ_b be the smaller gap. Two cases of superconductivity should be distinguished — one with the full total gap, and the other gapless. In the first case, A_α(k, ω) vanishes for energies below Δ_α. On the other hand, the spectral function of the same band in the gapless regime with Δ_b → 0 will behave in the same fashion as it would in the normal state:

The fermionic spectral function A_b(k, ω) for the band b calculated via expression (76) is plotted in Fig. 17. For the sake of argument, we show the calculations for |Δ_b| < |Δ_α| and the scattering on nonmagnetic impurities with η = 0.5 and σ = 0.5, although these results are retained for the case of magnetic impurities. In the clean limit (Fig. 17a), the behavior of A_b(k, ω) at small ω and ξ_kb is determined by the presence of the superconducting gap in the spectrum of excitations. On the other hand, at the s_±-to-s₊₊ disorder-induced transition, A_b(k, ω) shows an absence of the gap (Fig. 17b). With a further increase in the scattering rate Γ_a, when the transition already happened, the gap in the spectrum of the b-band reappears. Therefore, ARPES measurements in the superconducting state at different impurity concentrations would help to detect the disorder-induced transition.

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Considering nonmagnetic impurities as an example, here we demonstrate how the optical conductivity changes its behavior with increasing impurity scattering rate and, in particular, near the transition between the s_± and s₊₊ states. Figure 18 shows the optical conductivity Re σ(ω) = ∑_α Re σ_α(ω) calculated as the solution of equations (20) and (23) at different rates of disorder with η = 0.5 and σ = 0.5. Due to the presence of the superconducting gap, in the clean limit we have Re σ_α(ω) = 0 at small frequencies ω < 2Δ_α. With an increase in the impurity scattering rate in the s_± state, as opposed to the s₊₊ superconductor, the range of the zero value of Re σ_b(ω) for the band b diminishes and the peak above 2Δ_b becomes narrower. This is surely due to a decrease in the gap Δ_b upon approaching the s_± → s₊₊ transition (Fig. 10b). It is clearly seen in Fig. 18b that the Drude peak appears near the s_± → s₊₊ transition at Γ_a ∼ 1.2T_c0. This peak is typical for a normal metal because of vanishing the gap in the b-band, i.e., the gapless superconductivity regime in the course of transition. With a further increase in Γ_a, the optical conductivity regains the form of the full gap superconductor, though with a smaller width of the gap compared with its initial value.

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The behavior described differs significantly from the behavior of the s₊₊ superconductor, shown in Figs 18c and d, where the gaps converge in the limit of the infinite impurity scattering rate.

The temperature dependence of the optical conductivity Re σ(ω) at a fixed frequency for the s_± superconductor is plotted in Fig. 19. Evidently, the low-temperature contribution to the optical conductivity of the band b increases with increasing scattering rate Γ_a before the transition to the s₊₊ state at Γ_a ∼ 1.1T_c0, and then the contribution decreases.

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The imaginary part of the optical conductivity, Im σ(ω), is proportional to the real part of the polarization operator Π(ω), as seen from its definition (20). The frequency dependence of Π(ω) for the s_± and s₊₊ states in the presence of impurity scattering is illustrated in Fig. 20. There is a dip at frequency ω = 2Δ_α(ω) in the case of the s₊₊ superconductor. This agrees with the results for single-band superconductors [170]. In the s_± state, interesting features are observed for the band b: first, the location of the dip is a nonmonotonic function of the scattering rate, and, second, the dip disappears in the gapless regime near the s_± → s₊₊ transition.

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A comparison of theoretical and experimental temperature dependences of the optical conductivity and the London penetration depth on the dose of proton irradiation at THz frequencies [117] is drawn in Fig. 21. It is interesting to trace the behavior of the coherence peak in the real part of the optical conductivity σ₁(T, ω → 0). The peak is analogous to the Hebel–Slichter peak discussed in Section 7.4. With an increase in the irradiation dose, the peak disappears and then reappears (Fig. 21c). Such a behavior is a signature of the gradual closing of the smaller gap and its later reopening. It is exactly the process taking place at the s_± → s₊₊ transition. The general trend of the penetration depth behavior is the same in the theory and in the experiment, as evident from a comparison of Figs 21b and d. In the experiment, however, it was not possible to 'catch' the region of the s_± → s₊₊ transition itself. The latter is marked in Fig. 21b by the red arrow.

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In addition to the Knight shift, which allows one to distinguish between singlet and triplet pairing, NMR can probe the spin-lattice relaxation rate 1/T₁. Since we are going to discuss Fe-based materials, later we imply NMR in iron nuclei. The effect of the nuclei form factors is not very important here compared to cuprates, for example. It is confirmed by a good agreement between 1/T₁T data on different nuclei (⁵⁷Fe, ⁷⁵As, ⁵⁹Co, and ¹³⁹La) in 122 and 1111 systems [92, 95, 171, 172]. It was also experimentally examined [91] that the hyperfine coupling A_hf(q) most probably does not depend on the wave vector q.

The spin-lattice relaxation rate is determined by the spin susceptibility integrated over the Brillouin zone, viz.

As in the case with the spin resonance [15, 173–175], 1/T₁ carries information about the underlying gap symmetry and structure. For example, an isotropic s-wave state is characterized by a Hebel–Slichter peak just below T_c and an exponential low-T temperature dependence. It is well known that d-wave superconductors exhibit a weak or no peak and demonstrate behavior for T ⪡ T_c.

In the case of iron-based materials, the situation is somewhat more complicated. Typical data are given in Fig. 22a. Apparently, there is no peak below T_c and the temperature dependence does not follow the same simple power or exponential law. However, simple arguments can enable us to understand the main features revealed in experiments.

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In the case of a weakly coupled clean two-band superconductor below T_c, assuming that the main contribution to Im χ(q, ω) comes from interband interactions, we derive the following expression for the inverse NMR spin-lattice relaxation rate:

where E_k is the quasiparticle energy in the superconducting state, while k and k' = k + q lie on hole and electron Fermi sheets, respectively. Thus, q is the vector connecting hole and electron pockets. The above equation follows from the expression for the 'bare' susceptibility χ₀(q, ω) at zero temperature and for a vanishing frequency. It is written out in a special way to emphasize the role of coherence factors for the dominating interband processes. The coherence factor in square brackets in formula (81) gives rise to an important distinction between different symmetries of the gap. They play a similar role in the formation of the spin resonance peak in inelastic neutron scattering [173]. In the NMR 1/T₁ coherence factors, the internal sign is different from that in coherence factors entering the spin susceptibility related to neutron scattering. For the isotropic s₊₊-state with Δ_k = Δ_k' = Δ, we obtain

The denominator gives rise to a peak for temperatures T≲T_c near T_c, which is just the Hebel–Slichter peak. As pointed out earlier [16], it must be suppressed for the s_±-state. Indeed, if

which is a monotonically decreasing function with a decrease in temperature for T < T_c. The same can be demonstrated for a more general s_±-state with |Δ_k| ≠ |Δ_k'| [132].

It is well known that pair-breaking impurity scattering dramatically increases the subgap density of states just below T_c, and even a weak magnetic scattering can broaden and eliminate the Hebel–Slichter peak in conventional superconductors. In the case of a sign-changing gap, the similar effect manifests itself due to nonmagnetic interband scattering [145]. Since the Hebel–Slichter peak is not present in iron-based materials, even in the clean case [see equation (83)], the pair-breaking effect is more subtle: it changes an exponential behavior for T < T_c to a more power-law one. If the impurity-induced bound state lies at the Fermi level, the relaxation rate acquires a low-temperature Korringa-like term linear in temperature over a range of temperatures corresponding to the impurity bandwidth [176].

Qualitative arguments suggest that neither pure Born nor pure unitary limits with a simple isotropic s_±-state are well suited for explaining the observed 1/T₁ behavior: the former leads to an exponential behavior at low temperatures in a relatively clean system, while the latter to Korringa behavior. Various data on 1111 systems appeared to be between these two limits [89, 90, 171] (see Fig. 22b). Results of the 1/T₁ calculation for the simple s_± state are also shown there [132]. We observe that the s_±-state result exhibits no coherence peak and, as opposed to the Born and unitary limits, the intermediate case with σ not equal to 0 or 1 is capable of reproducing the experimental behavior of 1/T₁ [132, 146–148, 177]. This result, taken alone, should not be seen as evidence for an isotropic s_± state, since a strong gap anisotropy is probably present in some of these systems, and will also lead to a higher density of states for quasiparticles contributing at intermediate temperatures.

Regarding other systems, data obtained on BaFe₂(As_1−xP_x)₂ shows a term linear in temperature in 1/T₁ for an optimally doped sample, crossing over to something roughly approximating ∼ T³ above T≳0.1T_c [95, 178], consistent with reports of line nodes in this material from other probes. In Ba_0.68K_0.32Fe₂As₂, 1/T₁ shows an exponential decrease below T ≈ 0.45T_c, consistent with a full s_± gap [179]. Finally, NMR in the LiFeAs system also shows a full gap, which is consistent with other measurements [94].