Review

The Challenge of Unconventional Superconductivity

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Science  08 Apr 2011:
Vol. 332, Issue 6026, pp. 196-200
DOI: 10.1126/science.1200181

Abstract

During the past few decades, several new classes of superconductors have been discovered that do not appear to be related to traditional superconductors. The source of the superconductivity of these materials is likely different from the electron-ion interactions that are at the heart of conventional superconductivity. Developing a rigorous theory for any of these classes of materials has proven to be a difficult challenge and will remain one of the major problems in physics in the decades to come.

Superconductivity is an exotic state of matter that has intrigued scientists ever since its discovery in mercury in 1911 (1). It is sobering to realize that after 100 years, there are whole classes of superconducting materials that we still do not fully understand. Even for the first (“conventional”) superconductors, it took half a century to develop a theory, starting with the groundbreaking work of Cooper in 1956 (2) and Bardeen, Cooper, and Schrieffer (BCS) in 1957 (3). These ideas rapidly developed into a rigorous theory several years later (4) because for electron-ion interactions, a controlled many-body perturbation expansion is possible that greatly reduced the complexity of the problem (5). Unfortunately, no such luxury exists for the more recently discovered unconventional superconductors, and therefore the development of rigorous theories explaining these materials has proven to be a real challenge.

Conventional and Unconventional Superconductors

To appreciate these issues, we need to first understand what superconductors are all about, and how unconventional ones differ from their more conventional counterparts. Superconductors are not only perfect conductors (their electrical resistance drops precipitously to zero below a transition temperature Tc), but also exhibit the so-called Meissner effect (6), where they expel magnetic fields. As noted by Fritz London (7), this implies that electrons in superconductors behave in a collective manner. Bosons, which have integer values of a fundamental property known as spin, can behave in this fashion, whereas electrons, which are fermions that have half-integer spins, typically do not. In 1956, this apparent contradiction was resolved by Leon Cooper (2), who demonstrated that the presence of even an arbitrarily small attractive interaction between the electrons in a solid causes the electrons to form pairs. Because these “Cooper pairs” behave as effective bosons, they can form something analogous to a Bose-Einstein condensate. Rather than being real-space molecules, however, Cooper pairs consist of electrons in time-reversed momentum states and consequently have zero center-of-mass momentum. Because a pair of identical fermions is antisymmetric with respect to the exchange of one fermion with another, the spin and spatial components of the Cooper pair wave function must have opposite exchange symmetries. Thus, these pair states are either spin singlets with an even-parity spatial component, or spin triplets with odd parity.

The spin singlet pair state with an isotropic spatial component (s-wave) turns out to be the one realized in conventional superconductors (3). Even though electrons repel each other because of the Coulomb force, at low energies there can be an effective attraction resulting from the electron-ion interaction. This can be understood by considering that a metal is formed by mobile electrons detaching themselves from the atoms that form the crystalline lattice (these atoms then become positive ions). Such a mobile electron attracts the surrounding ions because of their opposite charge. When this electron moves, a positive ionic distortion is left in its wake. This attracts a second electron, leading to a net attraction between the electrons. This mechanism works because the ion dynamics is slow relative to the electrons—a consequence of the fact that the ions are much heavier than the electrons. However, the interaction at shorter times becomes repulsive because of the Coulomb interaction between the electrons; this retardation is what is responsible for limiting Tc (8). Until the discovery of cuprates, the highest known Tc was only 23 K.

The resulting wave function for the pairs turns out to be peaked at zero separation of the electrons—that is, an s-wave state. By an “unconventional” superconductor, we mean one that is not s-wave. Typically, this means that the pair wave function, also known as the order parameter, is not uniform in momentum space, although there can be exceptions, as may be the case for the new iron-based superconductors.

3He—The First Unconventional Superfluid

The first unconventional material was not a superconductor, however, but rather superfluid 3He (9). Because the atoms are neutral objects in this case, there is no analog of the electron-ion interaction mentioned above. In fact, the helium atoms behave as hard-core objects, which acts to suppress s-wave pairing. One of the first proposals to explain the pairing was based on van der Waals attraction between the atoms at larger separations (10), resulting in the formation of d-wave pairs. This pair state suppresses the influence of the hard core because it has a node for zero separation of the atoms, and the d-wave state optimizes the separation of the atoms to take advantage of the van der Waals attraction. A p-wave state, which has a node as well, was also proposed, but because of fermion antisymmetry, this odd-parity state is associated with spin triplet pairs. In this case, the attraction was speculated to be a result of exchange forces (11); 3He was thought to be nearly ferromagnetic, so an atom would prefer to align its spin with its pair partner. When superfluidity was seen in 3He several years later, it was soon realized that this was indeed a consequence of p-wave pairing (12). Two different superfluid phases, known as A and B, exist below a few millikelvin; these were also explained on the basis of exchange forces (13). Given the simplicity of the liquid state, the pair interaction was eventually quantified using known normal-state interaction (Landau) parameters. This analysis revealed that many factors contribute to the pair interaction, including density, spin, and transverse current interactions (12). Hence, it can be misleading to claim that one mechanism is the sole cause of pairing in unconventional superconductors.

Heavy-Fermion Superconductors

Of course, neutral atoms are not the same as charged electrons. Surprisingly, superconductivity with Tc = 0.5 K was discovered in 1979 in CeCu2Si2 (Fig. 1A) by Steglich’s group (14), and then in several uranium alloys such as UPt3 and UBe13 (15) a few years later. These materials contain magnetic 4f and 5f ions, which previous experience indicated would have been incompatible with superconductivity. After all, magnetic impurities are well-known sources of pair breaking in conventional superconductors (16). But these materials are more intriguing because the ions exhibit the Kondo effect. In such systems, the mobile conduction electrons have a tendency to form a bound resonance with the localized f electrons of the magnetic ions (17). When these magnetic ions form a regular lattice, this “Kondo” lattice is characterized by an electronic specific heat coefficient of order 1000 times that of conventional metals such as copper, with a correspondingly large spin susceptibility. This “heavy” Fermi liquid can form a number of ordered states, including magnetic order and, intriguingly, unconventional superconductivity (15).

Fig. 1

(A) Schematic phase diagram of the heavy-fermion material CeCu2Si2–xGex. Two superconducting domes are present, one (red) associated with a quantum critical point of an antiferromagnet (at pc1), the other (green) with a volume collapse transition (at pc2) (58). (B) Phase diagram of CeRhIn5 (temperature versus pressure). The superconducting (SC) phase with a critical temperature Tc abuts an antiferromagnetic (AF) phase with a Néel temperature TN, with a small coexistence region in between and a paramagnetic (PM) phase at higher temperatures (59).

Given some similarities with 3He, where the liquid phase is near a solid phase with magnetic order, these heavy-fermion metals were initially thought to be p-wave superconductors as well (18). But subsequent work has indicated that this is a complex problem (19). First, the concept of s-wave, p-wave, etc., must be taken with a grain of salt because of the presence of a crystalline lattice that breaks translational symmetry. Second, not only are multiple f orbitals involved, there are multiple conduction electron orbitals as well. Finally, spin-orbit effects are large for Ce and U ions, and they play a qualitatively different role than in light 3He atoms. In fact, although more than 30 years have elapsed since the discovery of heavy-fermion superconductors, the actual symmetry of the Cooper pairs of any of them has not yet been unambiguously determined.

Perhaps the closest we have come is UPt3 (20). This material, like 3He, has several superconducting phases (Fig. 2A). Because of the hexagonal symmetry of the lattice and the fact that the spin and orbital angular momentum of the pairs are linked because of spin-orbit coupling, this implies that either two different superconducting states have nearly identical transition temperatures, or that the order parameter is doubly degenerate. The latter is more likely, as it is thought that the small temperature range separating the two phases at zero magnetic field is a result of a weak lifting of the hexagonal symmetry of the lattice caused by the presence of small magnetic moments on the uranium ions. A variety of thermodynamic data indicate that the order parameter probably vanishes along lines that are perpendicular to the c axis of the crystal. This restricts the order parameter symmetry to be either E1g (“d-wave”) or E2u (“f-wave”). Very recently, measurements have been carried out that are sensitive to the phase of the order parameter (21). They indicate that the order parameter behaves as exp(2iϕ), where ϕ is the azimuthal angle within the hexagonal plane. If a line of nodes is indeed present, this rules out E1g in favor of E2u. Such an f-wave order parameter is indeed an exotic beast (Fig. 2B). But it is highly doubtful that the actual order parameter is so simple: UPt3 has a complex Fermi surface (which separates occupied from unoccupied states) formed from five different energy bands, and the spin-orbit coupling is so strong that even the single-particle states are best characterized by states of total angular momentum J.

Fig. 2

(A) Phase diagram of UPt3. Three different superconducting phases, A, B, and C, are present (60). (B) f-wave (E2u) Cooper pair wave function in three-dimensional momentum space proposed for phase A (top) and phase B (bottom) on the basis of phase-sensitive tunneling (61). In comparison, an s-wave wave function characteristic of conventional superconductors would be a simple sphere.

Less is known about other heavy-fermion superconductors. Much recent work has gone into the so-called 115 series, with a formula unit CeXIn5 where X is a transition metal ion (Fig. 1B). Available data are consistent with a nondegenerate order parameter that is singlet in nature, leading to the speculation of d-wave pairs. But to date, no phase-sensitive measurements have been performed. Of perhaps greater interest are the plutonium analogs, one of which has a Tc of 18 K, almost an order of magnitude higher than those previously known for heavy-fermion superconductors (22). Even less is known about this material given the challenges of working with plutonium, but its discovery indicates that perhaps even more striking examples await us in the future. Already there are heavy-fermion superconductors, such as UGe2 and URhGe, that are simultaneously ferromagnetic and superconducting. And perhaps related to these systems is Sr2RuO4 (23), a multiband layered transition metal oxide that appears to be a p-wave superconductor, although the exact nature of the order parameter is still being debated. Available phase-sensitive measurements are certainly consistent with a p-wave state (24).

A fundamental difference from 3He, though, is that most heavy-fermion superconductors are actually nearly antiferromagnetic, and in some cases (most notably in the Ce 115 series), superconductivity and antiferromagnetism coexist (Fig. 1). This was realized in 1985 when strong antiferromagnetic spin correlations were seen by neutron scattering in UPt3 (25), leading to the publication of three theoretical papers advocating that antiferromagnetic spin fluctuations were the source of d-wave superconductivity (2628). Superconductivity seems to be maximal at a point where magnetism disappears (Fig. 1); this was first appreciated in 1998 when superconductivity was discovered under pressure in CePd2Si2 and CeIn3 (29). A phase transition suppressed to zero temperature is known as a quantum phase transition (30), and it is thought that critical fluctuations associated with this quantum critical point could be the source of pairing (29).

Cuprates

In the same year, 1986, that the above-mentioned spin fluctuation papers were published, a small group from an IBM lab in Zürich made a startling discovery: superconductivity near 40 K in the layered cuprate La2–xBaxCuO4 (31). At first, this result did not attract much attention [in the past, there had been a number of claims of “unidentified superconducting objects” (USOs)]. But after its reproduction by several groups, a flurry of activity was unleashed, leading shortly to the discovery of superconductivity above 90 K in YBa2Cu3O7 (32). The technological implications were profound, given that liquid nitrogen, instead of helium, could be used for cooling. Surprisingly, this class of materials violated most if not all of the empirical search rules set down by Bernd Matthias; these rules were based on the previous record high Tc materials, which were cubic transition metal alloys, whereas cuprates are obtained by doping carriers into a parent material that is an insulating magnetic oxide. Not surprisingly, theorists speculated that the solution for the cuprate puzzle was a two-dimensional variant of the d-wave superconductivity mentioned above in the heavy-fermion context, Embedded Image symmetry (33).

But before this, a very different theory appeared that, for better or worse, would change the face of physics (34). Philip Anderson, the Nobel laureate, proposed instead that cuprates would exhibit a novel phase of matter where the spins formed a liquid of singlets—the so-called RVB (resonating valence bond) state—on the basis of previous work he had done in the 1970s on frustrated magnets. The name RVB was motivated by the classic work of Linus Pauling on benzene rings, where the carbon bonds fluctuate between single and double bonds. Anderson argued that such an RVB state was the consequence of several unique properties of cuprates: The materials are quasi–two-dimensional, the copper ions have spin ½, and the parent phase is a Mott insulator—that is, a state with an odd number of electrons per unit cell that is insulating because of many-body correlations. These effects, he speculated, would act to melt the expected antiferromagnetic (Néel) lattice into this spin-liquid phase. Upon carrier doping, these singlets would become charged, resulting in a superconducting state. Although the original proposal was for a uniform RVB (s-wave) state, subsequent work found that the free energy was actually minimized for a d-wave state (35). Although undoped cuprates were soon found to form a Néel lattice (but with a reduced moment), a few percent of doped holes was sufficient to destroy this state (Fig. 3A).

Fig. 3

(A) Schematic phase diagram of the cuprates (62). (B) Cooper pair wave function for YBa2Cu3O7 as a function of the planar azimuthal angle measured by phase sensitive tunneling (63). The red and blue lobes denote the opposite signs of the Embedded Image state. The size difference of the two lobes is due to the orthorhombicity of this material, which leads to mixing in of a small s-wave component.

What is unquestionable is that the exchange interaction J for cuprates is very large, on the order of 1400 K, and as such is an attractive source for pairing. This became very relevant in the mid-1990s, when it was shown by phase-sensitive tunneling (36) that the pairing state was indeed d-wave (Fig. 3B). But such a large J is also relevant for the more traditional spin fluctuation–based approaches, and there is currently much debate about which of these two approaches, RVB (37) or spin fluctuations (38), is the more appropriate. The lack of resolution of this debate is connected to the fact that for electronic-only models, we do not have a controlled perturbation expansion to work with, as we do for the electron-ion interactions underlying conventional superconductors.

Moreover, cuprates are complex systems with a variety of important interactions, including electron-ion. This has become increasingly obvious in attempts to explain their phase diagram (Fig. 3A). After the Néel order is destroyed by doping, there are four apparent regions of the phase diagram: (i) a pseudogap phase where an energy gap is present, (ii) a strange metal phase characterized by a resistivity linear in temperature, (iii) a Fermi liquid phase with largely normal transport properties, and (iv) a d-wave superconducting phase. In the RVB approach, the pseudogap is a spin gap phase resulting from spin singlet formation, whereas in the spin fluctuation approach, it is a fluctuating version of the Néel phase. But experiments now indicate an intriguing variety of phenomena associated with the pseudogap phase, including nematic (39) correlations (where the C4 rotational symmetry of the square lattice is spontaneously broken), and a novel form of magnetism, arising from either orbital currents or antiferromagnetism that is associated with the oxygen sites in the CuO2 unit cell of the cuprates (40). At lower temperatures, charge-density wave, spin-density wave, and superconducting correlations become apparent in a variety of measurements. From this very complicated soup, high-temperature superconductivity arises. Because superconductivity is created from the normal state, these phenomena must be understood before we will ever have a true understanding of the origin of high-temperature cuprate superconductivity.

Several authors have pointed out the similarities of the phase diagram of cuprates (Fig. 3A) with that of heavy fermions (Fig. 1). Accordingly, it has been proposed that superconductivity is mediated by quantum critical fluctuations, but again, the nature of the purported quantum critical point (which in the case of the cuprates is “hidden” under the superconducting dome) is being actively debated. Is it associated with antiferromagnetism, charge-density waves, spin-density waves, nematic correlations, orbital currents, or a combination thereof?

Organic Superconductors

If the physics of doped Mott insulators is indeed the key to cuprates (37), this will also have relevance to organic superconductors (41). These materials were discovered well before the cuprates and were an equal surprise to the community; as Bernd Matthias once quipped, “there aren’t any!” Well, they are indeed real, and the two-dimensional variety has a phase diagram intriguingly similar to their cuprate counterpart (Fig. 4A). In the BEDT-TTF salts, one has a lattice of molecular dimers, with one spin ½ degree of freedom per dimer. These are arranged in a triangular fashion, which has a tendency because of frustration to suppress magnetic order. It was indeed such a lattice that was the motivation for the original RVB idea. These materials at ambient pressure typically have a Mott-insulating ground state, becoming superconducting under pressure (42). Available evidence indicates a d-wave state, but as with other unconventional superconductors other than cuprates, the true pairing symmetry has yet to be unambiguously determined. Much of the recent attention regarding these materials has been devoted to those compounds that seem to exhibit a spin-liquid ground state in the Mott phase, as well as to the question of whether a Fermi surface of spin degrees of freedom (a so-called spinon Fermi surface) is indeed realized (43), as originally proposed by Anderson for the cuprates (34). As with the cuprates, there has been an interesting debate regarding RVB versus spin fluctuation approaches for the pairing mechanism.

Fig. 4

(A) Phase diagram of the organic superconductor κ-(ET)2Cu2(CN)3 (temperature T versus pressure) (63). No magnetic order has been detected in the Mott insulator phase. (B) Phase diagram (temperature T versus doping x) of the pnictide superconductor Ba(Fe1–xCox)2As2 (64). The antiferromagnetic (orthorhombic) phase occurs below TN (Ts) and is labeled by AFM (Ort). The normal (tetragonal) phase is denoted by Tet, and the superconducting phase, which occurs below Tc, by SC. Note the similarities of the phase diagrams presented in Figs. 1, 3, and 4.

Iron-Based Superconductors

This brings us to the newly discovered iron-based pnictide and related superconductors (44). Although involving iron rather than copper, and arsenic rather than oxygen, there are enough similarities to cuprates for these debates to again occur for this class of compounds (45). Unlike the cuprates, it appears that all five of the 3d orbitals of the iron are involved in the electronic structure near the Fermi energy. Although the undoped material is also antiferromagnetic, unlike in the cuprates, it is metallic. With doping, the magnetic state is suppressed and a high-temperature superconducting phase appears (Fig. 4B). It has been speculated that the order parameter is a so-called s± state, where the Fermi surfaces around the Γ point of the Brillouin zone have an order parameter with one sign, and those around the M point of the zone the opposite sign (46). Like the cuprates, and as has been speculated for the heavy fermions, this order parameter satisfies the condition that it changes sign under translation by the magnetic ordering vector of the parent phase. Such a state tends to remove the detrimental effects of the on-site Coulomb repulsion between the electrons, and naturally appears in spin fluctuation models. The jury, though, is still out on this question. Certainly this order parameter is consistent with what is known from angle-resolved photoemission, which shows rather isotropic energy gaps around each Fermi surface (47), but thermal conductivity (48) and other measurements indicate for certain dopings—gaps that appear to be d-wave–like in nature. And although the correlations appear to be weaker than in the cuprates because of the multi-orbital nature of the electronic structure (the electrons can avoid one another by occupying different d orbitals), there has been a rather heated debate on whether the magnetism is itinerant (as in the spin-density wave state in chromium) or more localized (as in the case of the cuprates). Interestingly, strong nematic effects similar to what have been observed in the cuprates have been seen in these materials as well (49).

Theory and Outlook

From a theoretical point of view, it has become increasingly obvious that unconventional superconductivity is a very tough problem. Even for the simple case of one d orbital with an on-site Coulomb repulsion, which has been considered by many to be the minimal description for cuprates, we do not know whether this model is indeed superconducting. Quantum Monte Carlo simulations of this model have given conflicting results, the issue being the infamous fermion sign problem that plagues such simulations. Even if the model turns out to be superconducting, there are many who feel that it is not sufficient—for instance, because of the rather large electron-ion effects that exist even for cuprates (50), or because a three-band model may be necessary to describe the physics (51). Indeed, multiband models are unavoidable in many cases, most prominently for heavy fermions, ruthenates, and pnictides. And, as opposed to conventional superconductors, in electronic-only mechanisms there is no really controlled perturbation theory to work with. Theorists have attempted to get around this by suggesting that an effective expansion parameter might exist, exploiting the fact that the collective spin fluctuations are “slow” relative to individual fermion degrees of freedom. On the other hand, it is well known that some higher-order terms are as large as the lowest-order term in such theories (52). Even in the RVB approach, most of the work has been done at an effective mean-field level reminiscent of BCS theory. Attempts to go beyond this by considering “gauge” fluctuations associated with constraints on a given copper site (such as no double occupancy) have met with limited success (37). In both the spin fluctuation and RVB “gauge” approaches, large N expansions (where N is the degeneracy of the electronic states) have been attempted, motivated by earlier work in heavy-fermion materials. But even in heavy fermions where the orbital degeneracy of the f electrons is large, N is typically 2 in the low-energy sector because of crystal field splittings, so the relevance of these approaches is controversial. Even the string theorists have gotten into the act, suggesting that the AdS/CFT (anti–de Sitter/conformal field theory) methods they have used to study certain strong coupling gauge theories may be relevant to condensed matter systems such as cuprates (53). Certainly, in the coming years, there will be increasing attention given to constructing rigorous nonperturbative strong coupling theories in the hope that we can “solve” the problem of unconventional superconductivity, at least to most people’s satisfaction.

Since the discovery of cuprates, several new families of superconductors have been discovered. MgB2 appears to be a conventional superconductor, and its properties were explained rather quickly by the standard strong-coupling approach for electron-ion interactions (54). It is revealing that this simple material was missed by both the experimentalists and theorists for such a long period of time. Alkali-doped buckyballs also have a substantial Tc near 40 K, and although at first sight their superconductivity appears to be conventional, there is increasing evidence that Mott physics may be involved here as well (55). And the discovery of pnictides again took the community by surprise. Because of this, there is no doubt that new classes of superconductors await our discovery. Whether current beliefs will aid us in this quest remains to be seen. The discovery of large values of the exchange constant J in iridium oxides (56) has led a few of us to propose that these materials might also be high-temperature superconductors. But attempts to dope these materials have so far not led to a superconducting phase. There are also a number of layered nitrides that have a comparatively high Tc near 26 K (57). Perhaps different variants are out there that will take us into the Tc range of cuprates. Regardless, what is apparent is that such discoveries will only occur with a proper investment in materials synthesis, guided by a good intuition of where to look. If this occurs, the future will indeed be bright.

References and Notes

  1. Supported by the U.S. Department of Energy (DOE) Office of Science under contract DE-AC02-06CH11357, and by the Center for Emergent Superconductivity, an Energy Frontier Research Center funded by the DOE Office of Science under award DE-AC02-98CH1088.
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