Abstract
Cuprates, ferropnictides and ferrochalcogenides are three classes of unconventional high temperature superconductors, who share similar phase diagrams in which superconductivity develops after a magnetic order is suppressed, suggesting a strong interplay between superconductivity and magnetism, although the exact picture of this interplay remains elusive. Here we show that there is a direct bridge connecting antiferromagnetic exchange interactions determined in the parent compounds of these materials to the superconducting gap functions observed in the corresponding superconducting materials: in all high temperature superconductors, the Fermi surface topology matches the form factor of the pairing symmetry favored by local magnetic exchange interactions. We suggest that this match offers a principle guide to search for new high temperature superconductors.
Introduction
In a conventional superconductor, superconductivity emerges from a normal metallic state below a critical transition temperature T_{c}, when pairs of electrons with opposite momenta near the Fermi surface (FS) are bound together to form a Cooper pair by an attractive force generated through absorption and emission of phonons. The pairing strength can be determined by measuring an energy gap, Δ, which equals 1.76k_{B}T_{c} in a standard BCS superconductor^{1}. Within this traditional picture of superconductivity, magnetism is considered to be an enemy of superconductivity because it breaks Cooper pairs. Furthermore, if the phases of Cooper pairs change signs in the reciprocal space, even nonmagnetic impurities are harmful to superconductivity^{2}. In contrast, the three known classes of highT_{c} superconductors (HTSCs) apparently violate many of these conventional wisdoms^{3,4}. First, the superconductivity in these HTSCs develops from a ‘bad metal’ state whose resistivity is several orders of magnitude higher than those of BCS superconductors. Second, strong magnetism is involved in the ‘bad metal’ parent state and the superconductivity occurs when longrange magnetic order is suppressed. Third, the ratio of Δ/k_{B}T_{c} is much larger than 1.76^{5,6,7,8,9,10,11}. Finally, the superconducting states are rather robust against impurities^{4,12}, contrary to conventional superconductors.
The studies of highT_{c} cuprates have led to many different proposals of superconducting mechanisms^{3,13,14,15}. In general, we can divide these proposals into three categories: (i) superconductivity without involving magnetism; (2) superconductivity mediated by spin fluctuations; (iii) superconductivity emerging from doped antiferromagnetic (AF) Mott insulators. The first category ignores the connection between magnetism and superconductivity. The difference between the second and the third categories is that the former suggests superconductivity mainly depends on the properties near Fermi surface in reciprocal space and Cooper pairs are formed by the retarded attractive force generated through emitting and absorbing “gluon”, in this case, magnons, while the latter emphasizes local interactions in real space and shortrange magnetic exchange interactions are responsible for superconducting pairing. The discoveries of ironbased superconductors demonstrate again strong interplay between magnetism and superconductivity, as their phase diagrams are very similar to those of cuprates. However, while ironbased superconductors can reach T_{c} comparable to single layer cuprates, they are much more itinerant than cuprates^{4}. Moreover, between two classes of ironbased superconductors, ferropnictides are also more itinerant than ferrochalcogenides even through T_{c} in both materials are also similar. Therefore, a unified mechanism of high T_{c} must reconcile the variation of the degree of itinerancy, a property in the reciprocal space, with magnetic interactions in real space. Here we show that, disregarding many microscopic electronic differences among the three classes of HTSCs, there exists a basic paradigm to unifiedly understand both cuprates and ironbased superconductors, including ferropnictides and ferrochalcogenides: the key ingredients in the determination of high T_{c} and pairing symmetries are local AF exchange interactions in real space and Fermi surface topology in reciprocal space that matches to the pairing form factor provided by the AF interactions. Such a paradigm will help to predict new highT_{c} superconductors and provide a guide to modify the properties of a material to increase T_{c}.
Results
Effective magnetic exchange interactions
First, we examine the magnetic exchange interactions of parent compounds of HTSCs. In all three classes of HTSCs, the transition metal atoms form a tetragonal square lattice. Their parent compounds exhibit distinct magnetically ordered states^{16,17,18,19}, as shown in Fig. 1.
In cuprates, the magnetic order is a checkerboard AF state with an ordered wavevector (π,π) as shown in Fig. 1a. This state can be naturally derived from a Heisenberg model where only the nearest neighbor (NN) AF interaction J_{1} is important and longer range magnetic exchange interactions can be ignored. Microscopically, J_{1} is generated by the superexchange mechanism mediated through oxygen atoms located in the middle of two NN copper atoms.
In ferropnictides, the magnetic order is a collinear AF (CAF) state with an ordered wavevector (π, 0)^{16} as shown in Fig. 1b. This magnetic state can be obtained in a J_{1} – J_{2} Heisenberg model with J_{1} < 2J_{2}^{20,21,22,23,24}, where J_{2} is the 2_{nd} NN magnetic exchange interaction. The measurement of spin wave excitations in the parent compounds of ferropnictides indicates that both J_{1} and J_{2} are AF^{25}.
In the 11ferrochalcogenide, FeTe, the magnetic order is a bicollinear AF (BCAF) state with an ordered wavevector ^{17,18} as shown in Fig. 1c. To obtain this magnetic state, a third NN ( 3_{rd} NN) AF exchange coupling J_{3} is needed^{26,27,28}. In fact, the analysis of spin wave excitations in FeTe shows that a ferromagnetic (FM) J_{1} and an AF J_{3} must be included while J_{2} does not differ significantly from ferropnictides^{29}. The magnetic exchange interactions are confirmed again in the 122ferrochalcogenide, K_{0.8}Fe_{1.6}Se_{2}, which exhibits a block AF state with an ordered wavevector ^{19}. Analyzing spin wave excitations in the blockAF state yields similar magnetic exchange interactions as FeTe^{30}.
Table 1 summarizes important AF exchange interactions in five different HTSCs. It is worth to note that in both statically ordered CAF and BCAF phases of ferropnictides and ferrochalcogenides, the spin wave excitations suggest that J_{1} must have different values on links with different spin configurations. This difference can be explained if a NN biquadratic effective spin interaction^{24,28} is included. However, since such a term does not play a role in providing superconducting pairing, we will not discuss it further.
Reciprocal form factors of pairing symmetries and determination of high T_{c}
Second, we examine the possible pairing symmetries and their reciprocal form factors determined from the corresponding magnetic exchange interactions. For an swave and dwave spin singlet pairing superconductor, only AF exchange interactions play a role in pairing electrons. The explicit pairing forms determined from the AF magnetic models discussed above for five different HTSCs are listed in Table 1 and their detailed derivation is explained in the supplementary information.
Finally, after knowing the form factors of possible pairing symmetries, one can determine the pairing symmetry and the transition temperature. Taking a oneband system with a single AF magnetic exchange interaction as an example^{31}, T_{c} is determined by the following selfconsistent meanfield equation as where and . is the band dispersion and f_{α}(k) is the corresponding pairing form factor determined by the AF exchange interaction J_{α}. The function g(x) is always positive and has its maximum value on FSs. In order to obtain nonvanishing T_{c} in Eq.1, the band dispersion has to be strongly renormalized so that J_{α} is comparable to the band width. Ironbased superconductors are multiband systems. Similar meanfield treatment has been studied in Refs.^{32,33}. These studies also show that the pairing symmetry and T_{c} are mainly determined by the weight of the form factors near FSs.
Here, rather than performing calculation within a theoretical model, we take band structures of HTSCs measured by angleresolved photoemission spectroscopy (ARPES) and calculate the overlap between the pairing form factors and the FSs, , which is the value of the quantity on the right side of Eq.1 at zero temperature that approximately determines T_{c}. The quantitative results of the overlap in five typical HTSCs are summarized in Table 1, and the detailed formula to evaluate the overlap is explained in the supplementary information. One can visualize this overlap by plotting FS and gap function in the same reciprocal space, as shown in Fig. 2.
To demonstrate the importance of this overlap in achieving high T_{c}, we illustrate the details of FS and superconducting gap of the three classes of HTSCs determined by ARPES: (i) In Fig. 3a, we show a typical FS of cuprates (optimally doped Bi_{2}Sr_{2}CaCu_{2}O_{8+x}^{5}). In this case, it is clear from Table 1 and Fig. 2 that the dwave form, cosk_{x} − cosk_{y}, has a much larger overlap with the FS than the swave form. Indeed, ARPES results strongly support this kind of dwave form, as shown in Figs. 3d, 3g^{5}. (ii) In Fig. 3b, we show the FSs of ferropnictides (optimally hole doped Ba_{0.6}K_{0.4}Fe_{2}As_{2}^{6,8}). There are two hole pockets at Γ, one hole pocket at Z and one electron pocket at M in the unfolded Brillouin zone. In this case, it is also clear from Table 1 and Fig. 2 that the swave form factor cosk_{x}cosk_{y} provided by the 2_{nd} NN AF J_{2} has the maximum overlap with the FSs. Consequently, in a doping region where electron and hole pockets are reasonably balanced, an swave with a symmetry form cosk_{x}cosk_{y} should dominate in the superconducting state, which has also been observed by ARPES, as shown in Figs. 3e, 3h^{6,8}. However, with a high percentage of hole or electron doping, which destroys the balance between electron and hole pockets, the AF NN J_{1} can start to take effect on the pairing symmetry. For example, in the case of heavily holedoped systems where the FSs are dominated by the hole FS pockets at Γ (Z), the dwave form cosk_{x} – cosk_{y} can strongly compete with the swave form cosk_{x}cosk_{y}. Indeed, there are strong experimental evidence for gap nodes in the heavily holedoped superconductor KFe_{2}As_{2} (T_{c} ∼ 3K)^{34,35}. Such a competition will weaken superconductivity as shown in refs.^{32,33}. (iii) In Fig. 3, we plot the FSs of ferrochalcogenides for FeTe_{0.55}Se_{0.45}^{36}, where one hole pocket at Z and one electron pocket at M are observed. In this case, the electron pocket dominates over hole pockets. The swave symmetry cosk_{x}cosk_{y} still has a good overlap with FSs. However, unlike the case of ferropnictides, here the NN interaction J_{1} is FM so that there is no competition from the dwave form cosk_{x} − cosk_{y}. Thus, we still expect a dominant swave pairing. The presence of a significant 3_{rd} AF J_{3} adds interesting effect on the gap function. For an swave, an AF J_{3} provides an additional pairing form, cos2k_{x} + cos2k_{y}, which takes large values at both hole and electron pockets as well. However, unlike cosk_{x}cosk_{y} which takes opposite sign between Γ(Z) and M, the form takes the same sign at Γ(Z) and M. Therefore one expects that the pairing form in these materials should be proportional to cosk_{x}cosk_{y} − δ(cos2k_{x} + cos2k_{y}) with δ being positive. This pairing form exactly describes what is observed in FeTe_{0.55}Se_{0.45} by ARPES, as shown in Figs. 3f, 3i^{36}. The same analysis can also be applied to the recently discovered HTSC KFe_{1.7}Se_{2}, which only has electron pockets at M^{9,10,11,37}. With both J_{2} and J_{3} being AF, the absence of hole pockets allows the gap function in the electron pockets to take large values to achieve high T_{c}.
Predictions of possible high temperature superconductors
The paradigm established here allows us to predict possible magnetic interactions and FSs in undiscovered HTSCs. It is clear that the presence of strong local AF interactions is necessary. Assuming that these interactions are known, we can discuss the possible matching FSs which can lead to highT_{c} superconductivity in several common lattice structures. In Figs. 4a, 4b, we draw two possible FSs that can lead to high T_{c} for a tetragonal lattice structure. The FS in Fig. 4a leads to an swave superconductor for a strong NN AF interaction while the one in Fig. 4b leads to a dwave superconductor for a strong 2_{nd} NN AF interaction. In Figs. 4c, 4d, we draw two FSs that can lead to swave pairing symmetry in a honeycomb lattice when the NN and 2_{nd} NN AF exchange interactions dominate respectively. The detailed reciprocal pairing forms are given in the supplementary information. The prediction for a triangle lattice with NN AF exchange interactions and swave pairing symmetry is similar to Fig. 4c with a rotation of 30 degrees of all FS around the center Γ point. We do not address dwave paring symmetry in a honeycomb lattice here because the dwave superconducting state will most likely break the timereversal symmetry.
Discussion
While the paradigm described here is a phenomenological, or at most, a semimicroscopic understanding of HTSCs, it is already a powerful guide to understand many unconventional properties in these materials.
The paradigm suggests that the effect of electronelectron correlations is very important to high T_{c}. It can strengthen local AF exchange interactions as well as cause strong renormalization of band structures. However, a strict Mottinsulating state is not a necessity of high T_{c}. The AF exchange interactions rely more sensibly on the electronic properties of the atoms which mediate superexchange interactions, such as oxygen in cuprates and As or Se(Te) in ironbased superconductors, rather than onsite interaction U.
The paradigm also suggests that the sign change of superconducting orders in reciprocal space is not due to the positive Josephson couplings between different FSs. Instead, it is determined together by local AF exchange interactions and FS topology, namely the sign change behavior is a derivative product, rather than an origin to cause high T_{c} in the first step that has been proposed in many weak coupling theories^{38,39,40}. Of course, Josephson couplings derived in weak coupling theories and superconducting pairing provided by local AF exchange interactions can collaborate with each other to drive higher T_{c}. Such as in ferropnictides, the collaboration can happen if there are positive Josephson couplings between the hole pockets at Γ(Z) and electron pockets at M. However, in KFe_{2}Se_{2}, due to the absence of hole pockets at Γ(Z), the positive Josephson couplings between two electron pockets at M will damage superconductivity if it is swave pairing. A verification of swave pairing symmetry in KFe_{2}Se_{2} will be an important support for the paradigm since the positive Josephson coupling between two electron pockets results in a dwave pairing symmetry^{41}.
The paradigm further provides qualitative explanations of strong pairing, short coherence length and impurity insensitivity in HTSCs. The strong pairing and the short coherence length result from instantaneous and shortrange attractive force generated by AF exchange interactions. The superconducting states in all HTSCs are rather robust against nonmagnetic impurities, since the pairing force is determined rather locally and the sign change is due to its form factor derived from local AF exchange. Therefore, if the local AF exchange interactions are not significantly altered by the impurities, the pairing force is stably maintained.
A hidden assumption of the paradigm is that the pairing force can be smoothly derived from the local AF exchange interactions existed in the magnetic parents, which suggests that the leading AF exchange interactions should not be drastically modified in doped materials. This assumption can be tested directly by measuring highenergy spin excitations or other spin properties in doped compounds. In cuprates, recent experiments using resonant inelastic xray scattering have reported that many superconductors, encompassing underdoped YBa_{2}Cu_{4}O_{8} and overdoped YBa_{2}Cu_{3}O_{7}, exhibits damped spin excitations (paramagnons) with dispersions and spectral weights similar to those of magnons in undoped cuprates^{42}. In ferropnictides, similar results have been obtained in the study of BaFe_{2−x}Ni_{x}As_{2} by neutron scattering experiments^{43}. Moreover, it has also been shown that the AF J_{2} in Li_{1−x}FeAs^{44}, which is already selfdoped, is similar to other parent compounds^{25,29,30}. In ferrochalcogenides, there is a rather robust incommensurate magnetic excitation in all superconducting FeTe_{1−x}Se_{x} samples^{45,46,47}, which suggests J_{3} is rather robust.
In summary, we have demonstrated that the superconducting gap symmetry and amplitude of cuprate, ferropnictide, and ferrochalcogenide HTSCs, as observed by ARPES, can be naturally determined by the local AF exchange interactions of their magnetic parent compounds collaborating with the FS topology in the superconducting offspring compounds. By identifying local AF exchange and collaborative FSs as key ingredients of HTSCs, we are able to predict magnetic configuration, FS topology and pairing symmetry of several undiscovered HTSCs. We believe that this phenomenological description can help to develop a microscopic theory of unconventional hightemperature superconductivity.
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Acknowledgements
We thank P. Richard for many valuable suggestions for the manuscript and H. Miao for assistance in constructing figures. We acknowledge B. A. Bernevig, P.C. Dai, D.L. Feng, S. Kivelson, D.H. Lee, H.H. Wen, X.G. Wen, Z.Y. Weng and T. Xiang for valuable discussions. This work is supported by National Basic Research (973) Program of China (grants No. 2010CB92300), National Science Foundation of China (NSFC11190024) and Chinese Academy of Sciences (grant No. 2010Y1JB6).
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Affiliations
Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
 Jiangping Hu
 & Hong Ding
Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA
 Jiangping Hu
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Contributions
H.D. performed experiments, J.P.H. did most of theory, J.P.H. and H.D. wrote the paper.
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The authors declare no competing financial interests.
Corresponding author
Correspondence to Hong Ding.
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