Unconventional Superconductivity – Forty years of exciting developments

By Manfred Sigrist, ETHZ

This year in May the 25th anniversary of the discovery of Sr2RuO4 was celebrated with a dedicated workshop at ETH Zurich. Sr2RuO4 has been the second superconductor found in the class of perovskite transition metal oxides after the cuprates and has ever since played an important role in the field of unconventional superconductivity. One might expect that after 25 years such a workshop would be a mere wrap up combined with a nice party. But it turned out completely di fferent. Just a few months prior to the event new experimental data turned things upside down. Many ideas and concepts which had been developed over the years were suddenly put into question, making it clear to everybody that the job is not finished yet and we may actually be again at a starting point where (almost) everything is possible. And somehow a feeling of pioneering was ignited similar to the early days of unconventional superconductivity about 40 years ago.

After the great success of the BCS theory which explained all essential features of superconductivity so successfully, in the 1970s the field was considered essentially harvested, with little left to explore. Then in the 1980s the discoveries of the superconducting heavy Fermion compounds, the organic superconductors and most strikingly the cuprate high-temperature superconductors opened a new era for superconductivity in materials with strong electron correlations, usually rather a habitat of magnetism. The heavy Fermion superconductors CeCu2Si2, UBe13 and UPt3 triggered the first proposals of unconventional Cooper pairing of electrons much in analogy to superfluid 3He. At this time it was also found that both Th-doped UBe13 and UPt3 possess phase diagrams with multiple superconducting phases – a clear divergence from conventional superconductivity.

Let me explain briefly the terminology: conventional Cooper pairs bind electrons in the most symmetric spin-singlet pairing state – often called s-wave pairing – which originates from electron-phonon mediated attractive interaction. The term unconventional denotes any other pairing channel of lower symmetry, allowing also for the spin-triplet con figurations. While the s-wave pairing state is unique, there are unlimited many unconventional ones. Electron-phonon interaction is essentially a contact interaction and perfectly suitable for s-wave pairing. For unconventional pairing channels the pair wave functions involve higher relative angular momenta such that two electrons stay apart from each other and longer-ranged interactions are required, as, for example, provided by fluctuations of electronic degrees of freedom, most prominently spin fluctuations. Indeed magnetism despite being a competitor seems an almost ubiquitous companion of unconventional superconductivity in strongly correlated electron systems.

Ever since the early days of heavy Fermion and cuprate high-temperature superconductivity the pairing symmetry has been among the most urgent questions addressed, since it determines much of the phenomenology of a superconductor and had been believed to be the key to understand microscopic pairing mechanisms. But it is a rather non-trivial task to probe the Cooper pair wave function of a superconductor. Still nowadays the pairing states of the early heavy Fermion superconductors, CeCu2Si2, UBe13 and UPt3, have not been unambiguously identifi ed as well as for many others. In this respect the cuprates high-temperature superconductors are rather the exception. They emerge through carrier doping into a highly correlated insulator with antiferromagnetic groundstate. While the normal state, emerging from a carrier-doped highly correlated antiferromagnet, is extremely complex and remain mysterious in many respects to the present day, the superconducting phase has a robust and uncontested dx2-y2-wave symmetry. The pair wave function changes sign upon a 90° rotation around z-axis and has nodes along [110], features which have been reliably established by numerous experiments. Surely this symmetry had been hotly debated in the early years, but nowadays it seems rather natural that antiferromagnetic spin fluctuations would yield a dx2-y2-wave pairing state.

Figure 1: Schematic phase diagram of a superconducting “dome” (SC) around a quantum critical point of an antiferromagnetically ordered phase (AFM), as frequently observed in many Ce-based heavy Fermion superconductors. In the overlap region the two phases (AFM+SC) can also coexist.

The belief that magnetic fluctuations would be benefi cial for Cooper pairing is, probably, most beautifully supported through a series of Ce-based heavy fermion compounds where superconductivity appears in a limited range around a quantum critical point between a paramagnetic and magnetically order phase, where magnetic fluctuations are strongly enhanced (Figure 1). After the first observation of such a correlation in CeIn3 in 1998, the search for superconductivity at quantum phase transitions, not only magnetic ones, has become a guiding principle for many subsequent discoveries of new superconductors which are believed to be unconventional.

Heavy fermion superconductors hold further surprises connected with the so-called key symmetries for Cooper pairing – inversion and time-reversal symmetry – which guarantee the existence of degenerate partner electrons that can form a pairs of defi nite parity and vanishing total momentum.The series of UGe2, URhGe and UCoGe are ferromagnetic superconductors where superconductivity appears within a ferromagnetic phase, i.e. Cooper pairs have to cope with a Fermi surface that splits into a majority and minority spin part, thus time reversal symmetry being absent. This is likely enforcing polarized spin Cooper pairing, in a so-called non-unitary phase analogous to the A1-phase of super fluid 3He in a magnetic field. The discovery of CePt3Si, CeRhSi3 and CeIrSi3 has initiated the investigation of unconventional non-centrosymmetric superconductors. Their crystal lattice does not have an inversion center giving rise to antisymmetric spin-orbit coupling, which leads to rather complex momentum-dependent spin polarization of the Fermi surface and a ffects again Cooper pairing profoundly. In particular, the labelling by parity for the Cooper pair symmetry is not valid anymore and we encounter mixed-parity pairing where also spin singlet and triplet con figurations are not distinguishable. Rashba-type spin-orbit coupling, as realized in the above mentioned heavy Fermion superconductors, represents the probably best-known case of antisymmetric spin-orbit coupling. Theory suggests a complex phenomenology for unconventional pairing, but little has been explored experimentally so far. The most striking experimental fi nding, however, is the extremely high upper critical field for CeRhSi3 and CeIrSi3 of up to 40 T at T = 0 K, although their critical temperatures are only around 1 K, which originates from the structure of the antisymmetric spin-orbit coupling and strong quantum critical spin fluctuations.

The theme of topology as a central concept for electronic properties is omnipresent nowadays, in particular, in the context of topological insulators and many related systems. Historically the exploration of topologically non-trivial phases for superconductors is considerably older and even predates in some aspects the analysis of the Quantum Hall E ffect in topological terms. The two superfluid phases of 3He, the A- and B-phase, represent the two most basic topological Cooper pairing states: the A-phase is chiral and the B-phase is helical. For both the bulk-edge correspondence applies yielding dispersing quasiparticle modes at the surface (for the A-phase this may admittedly be more complex due to a continuous degeneracy in three-dimensional 3He and point nodes in the gap). The A-phase is then characterized by a chiral edge current and for the B-phase a (helical) spin edge current would flow. Both phases have spin-triplet character. While there are candidates of bulk topological superconductors, such as URu2Si2, SrPtAs and the initially mentioned Sr2RuO4, there are also routes to design topologically non-trivial superconducting phases on purpose.

A promising path uses the reduction of symmetries through arti ficial structuring combined with magnetism, in this way a ffecting again the key symmetries of Cooper pairing. On everyones lips are the Majorana Fermions which are supposed to appear in certain spin triplet superconductors as topologically protected edge modes. Conditions for their appearance have probably been achieved in quantum wires with superconductivity induced by contact with a conventional superconductor (proximity e ffect). Strong spin-orbit coupling at the junction and a magnetic fi eld applied in the right direction could eventually stabilize a spin triplet pairing state in the wire with zero-energy Majorana states. This type of device is advertised as one of the promising building blocks for topological quantum computers.

Figure 2 : Superlattice of superconducting CeCoIn5 and normal metal YbCoIn5. Locally non-centrosymmetric systems showing distinct features in high magnetic fields (adapted Yuji Matsuda group, Kyoto University).

Less noticed but by no means less fascinating are superconducting superlattices. An example is the superlattice of multilayers of CeCoIn5 and YbCoIn5 grown by MBE techniques. Here the heavy Fermion component CeCoIn5 contributes the superconducting part, while the standard metal YbCoIn5 acts as a spacer (Figure 2). CeCoIn5 as bulk superconductor has most likely a spin singlet d-wave symmetry and shows paramagnetic limiting, the destruction of Cooper pairs through spin polarization which might provide the condition for the long-sought Fulde-Ferrel-Larkin-Ovchinikov (FFLO) phase, marked by a spatial modulation of the superconducting order parameter at high-magnetic field and low-temperature. Actually a phase distinct from the usual mixed state (vortex phase in a type-II superconductor) has been detected in the range of the phase diagram where the FFLO state is expected. However, unexpectedly it also involves a peculiar incommensurate spin density wave state, known nowadays at the “Q-phase”. In the superlattice where several layers of CeCoIn5 are sandwiched between several YbCoIn5 layers in a repeated structure the top and bottom layer of each CeCoIn5 slab host superconductivity in a non-centrosymmetric environment. It is believed that this makes it rather robust against paramagnetic limiting, as observed in experiment in a consistent and systematic way for many diff erent CeCoIn5/YbCoIn5 superlattices. For such a system theory predicts that intriguing novel features of superconductivity could be tailored through structuring and the application of magnetic fields, which opens the door towards topological as well as spatially modulated superconducting phases (such as pair density waves) again by removing the key symmetries.

While the superconductivity appearing at the metallic interface of the LaAlO3 / SrTiO3 heterostructure connection two band insulators is likely conventional, the lack of mirror symmetry and the apparent presence of intrinsic magnetism set a stage for interesting features of superconductivity in environment of reduced symmetry. In particular, the possibility of influence the antisymmetric (Rashba) spin-orbit coupling together with the superconducting phase by means of electric gates provides tuneable devices.

The quickly rising ability to fabricate genuinely low-dimensional materials look very promising for the discovery of new unconventional superconducting phases through reduced symmetries imprinted in the electronic states through the crystal structure. The families of monolayer transition metal dichalcogenides, such as MoS2, MoSe2 or NbSe2, and also some iron-based superconductors like FeSe belong to this category, as they lack at least inversion symmetry, if not even time reversal symmetry through magnetic order. Under such conditions novel Cooper pair symmetries can arise, for example, the so-called Ising Cooper pairing due to the spin-orbit coupling in diff erent Fermi pockets. Much can be learned in this direction to guide us towards possible manipulation of Cooper pairing in a controlled way.

Going to mesoscopic length scales sample shaping using focussed ion beams (FIB) technology has become a further increasingly powerful tool of designing and probing superconductors in well defi ned geometries (Figure 3). Unconventional superconductors react strongly on con finement as their order parameters are susceptible to scattering at surfaces. Information on the order parameter structure may be obtained by symmetry lowering through shape and sample deformation. Tiny rings cut by FIB have led to the observation of half-flux quantization in Sr2RuO4, which is considered as a trademark of spin triplet pairing. This technique will also shape future experiments on and may be technology based on unconventional superconductivity.

Figure 3: Single crystla of the heavy Fermion superconductor CeRhIn5 structured by focussed ion beam for high-precision transport measurements. (Philip Moll group, EPFL / MPI CPFS, //www.cpfs.mpg.de/en/moll).

Among the unconventional superconductors Sr2RuO4 plays indeed a special role. When it was discovered in 1994 it was early on realized that the electrons with their well-de fined strongly renormalized Fermi liquid property would rather likely form spin triplet Cooper pairs analogous to superfluid 3He. This would make it an ideal test case of our understanding of unconventional superconductivity. Actually much of the experimental evidence points towards a chiral p-wave phase closely related to the 3He A-phase, which is topologically non-trivial making it highly attractive even to a wider community in condensed matter research. Over the past 25 years Sr2RuO4 became one of the best investigated unconventional superconductors. However, this most detailed examination also revealed that things are more complex than anticipated, teaching us the amazing intricacy of unconventional superconductors. The fact that there are several bands, similar to most of the iron-based high-temperature superconductors, and the rather complex magnetic fluctuations hamper a decisive conclusion on the microscopic mechanism superconductivity. No less complicated is the phenomenology, in particular, concerning the supposed chiral nature of the Cooper pairing, which left contradictory signatures in various experiments and triggered growing controversy on the pairing symmetry. The most recent twist to the story, however, is the NMR Knight shift indicating that the conclusion on the spin triplet pairing may have been premature. Within a few months of the first news a number of alternative proposals have appeared for the pairing symmetry as well as novel symmetry classi cation schemes taking the multi-orbital aspect more carefully into account. Sr2RuO4 is at a crossroad leaving again open where its voyage eventually will go. This and many other exciting developments show that the chapter of unconventional superconductivity is surely not closed, even after 40 years.

Switzerland and, in particular, MaNEP has played a key role in the exploration of unconventional superconductivity from the very beginning of heavy Fermions to the large variety of modern systems, in experiment as well as in theory. Just to mention the discovery of the cuprates high-temperature superconductors by Alex Müller and Georg Bednorz at IBM Ruschlikon in 1986 which was one of the most impactful events in the field and de fined much of the focus of our research for many years. To this day Switzerland holds also the record of the highest Tc at ambient pressure, with 134 K in HgBa2Ca2Cu3O8 found by Andreas Schilling, Hans-Ruedi Ott and collaborators in 1993. A strong tradition of research on unconventional superconductivity has been built up and carried early on by colleagues like Maurice Rice, Øystein Fischer, Hans-Ruedi Ott and Piero Martinoli who had also been the organizers of the very fi rst M2S conference held in Interlaken in 1988. Ever since then many research groups in the Swiss community succeed to be among the world-leading players in this fascinating field to the present day.

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