Unconventional Superconductivity G. R. Stewart Department of Physics, University of Florida, Gainesville, FL USA 32611 Gregory Randall Stewart Department of Physics 2001 Museum Road University of Florida Gainesville, FL 32611 USA *Email: [email protected] 001 352 3929263 Acknowledgements: This work was supported by the United States Department of Energy, Office of Basic Energy Sciences, under contract number DE-FG02-86ER45268. Abstract: “Conventional” superconductivity, as used in this review, refers to electron-phonon coupled superconducting electron pairs described by BCS theory. Unconventional superconductivity refers to superconductors where the Cooper pairs are not bound together by phonon-exchange but instead by exchange of some other kind, e. g. spin fluctuations in a superconductor with magnetic order either coexistent or nearby in the phase diagram. Such unconventional superconductivity has been known experimentally since heavy fermion CeCu2Si2, with its strongly correlated 4f electrons, was discovered to superconduct below 0.6 K in 1979. Since the discovery of unconventional superconductivity in the layered cuprates in 1986, the study of these materials saw Tc jump to 164 K by 1994. Further progess in high temperature superconductivity would be aided by understanding the cause of such unconventional pairing. This review compares the fundamental properties of 9 unconventional superconducting classes of materials - from 4f-electron heavy fermions to organic superconductors to classes where only three known members exist to the cuprates with over 200 examples – with the hope that common features will emerge to help theory explain (and predict!) these phenomena. In addition, three new emerging classes of superconductors (topological, interfacial – e. g. FeSe on SrTiO3, and H2S under high pressure) are briefly covered, even though their “conventionality” is not yet fully determined. Keywords: superconductivity; unconventional; symmetry; heavy fermion; cuprate PACS index categories: 74.20.Mn Nonconventional mechanisms 74.25.-q Properties of superconductors 74.70.-b Superconducting materials other than cuprates 74.72.-h Cuprate superconductors 1. Introduction Superconductivity, the phenomenon where the resistivity falls to zero below a certain critical temperature Tc (discovered in Hg at 4.2 K in 1911) and the magnetic flux is expelled from the bulk of the superconductor (a phenomenon discovered in 1933), was explained as due to electron-phonon coupling by Bardeen, Cooper, and Schrieffer (BCS) in 1957. The BCS weak- coupled theory describes the condensation into the superconducting state as due to the exchange of phonons between electrons of opposite spins, an s=0 singlet ground state. Thus the average phonon frequency, <>, or equivalently the characteristic Debye temperature, D, (a phenomenological cutoff of the phonon frequencies) plays an important role in the BCS expression for Tc BCS Tc <>exp(-1/N(0)V) eq. 1 where N(0) is the electronic density of states at the Fermi energy and V is an average electron- phonon coupling strength parameter. The BCS weak coupling theory has N(0)V<1. Many superconductors (e. g. the over 25 superconducting elements [1] and various classes including A15 superconductors [2] - useful for high field magnets), are generally believed to be described by the BCS model, as extended by various improvements (called Eliashberg theory) that encompass stronger coupling. Defining “Unconventional Superconductor” (UcS) as a material where the Cooper pairing deviates from the BCS description, where the attraction between pairs and driving mechanism for condensation into the superconducting state might come from, e. g., exchange of spin fluctuations, is not a definition new to this review. This is consistent, e. g., with the definition of UcS in the phenomenological theory review of UcS by Sigrist and Ueda [3] from 1991. There are several other ways (as discussed thoroughly in section 2) to define an UcS. Based on a discussion of symmetry [4] by Tsuei and Kirtley, any long range ordering transition is accompanied by a lowering of symmetry. In a BCS superconductor the only symmetry broken is the one dimensional global gauge symmetry, U(1), caused by the macroscopic phase coherence that occurs below Tc. In an UcS, one or more additional symmetries (e. g. time reversal symmetry or reflection symmetry) are broken. p-, d-, and f-wave as well as s± (theorized for iron based superconductors [IBS]) pairing symmetries all break reflection symmetry. Thus, another definition of UcS could be a superconductor in which at least two symmetries are broken at Tc. As we will see, although a majority of the 9 superconducting UcS classes discussed herein exhibit such additional symmetry breaking (e. g. time reversal and reflection symmetry breaking in the hole-doped cuprates), others (including recently discovered classes or classes with only a few members) do not. Thus, we will remain with the “non-BCS” definition of UcS as being the best representation of the fundamental theme of this review. Here, in the Introduction, a short overview discussion will help the reader follow the discussion of the nine classes of UcS, and the three additional classes. As can be inferred from the short discussion above, it is not at present possible to state the cause of UcS, if indeed (as seems unlikely) there is only one such cause. The present review aims to summarize the properties of the various classes of UcS in a way that points to fundamental similarities. One of the questions important for understanding UcS is: what characteristics of materials are causes of, or at least consistent with, UcS? 1.) As we will see in this review, most UcS have strong electron correlations through their d-electrons (cuprates, IBS, Sr2RuO4, cobalt oxide hydrates, layered nitrides) or f-electrons (heavy Fermions and coexistent superconductivity and ferromagnetism). Only the organic UcS rely on pairing between p-electrons. 2.) The main instability found in UcS is antiferromagnetism (as will be seen herein in phase diagrams and discussions for the heavy Fermion 115 structure, CeCu2Si2, U(Pd,Ni)2Al3, for both electron- and hole-doped cuprates, for the majority (but not all) of the IBS, the non- centrosymmetric superconductors like CePt3Si that are strongly correlated, organics, and cobalt oxide hydrates.) In addition there are coexistent superconducting and ferromagnetic compounds like UGe2. Another instability found in some UcS is charge density waves (hole-doped cuprates, theorized in some heavy Fermion superconductors, and cobalt oxide hydrates.) 3.) Two dimensional behavior based on the crystalline structure plays an important role and is widespread in many of the UcS: cuprates (where the ratio of resistivities along the c-axis vs the ab-plane can be larger than 1000), IBS (c/ab ~ 100), Sr2RuO4, layered metal nitride halides, cobalt oxide hydrates, and the 115 structure heavy Fermion superconductors, while there is quasi-1d behavior in some of the organics. This d<3 behavior can play an important role in creating UcS since according to the Mermin Wagner theorem fluctuation effects for d<3 become more important. Such fluctuation effects are predicted (as discussed below) to be a possible superconducting pairing mechanism (superconducting ‘glue’) in these same aforementioned superconductors. 2d antiferromagnetic fluctuations are repulsive, so a Fermi surface with a sign change in the order parameter where the repulsive interaction is large can cause (see the s± model discussed below for the IBS) attraction and condensation into a superconducting state. Put another way, this sign change in the order parameter under translation by the magnetic ordering vector of the parent phase “tends to remove the detrimental effects of the on-site Coulomb repulsion between the electrons.” (Norman [5]). Thus, details of the Fermi surface topology are important and 2d crystalline structures are indeed widespread in UcS. Of course, there are obvious counterexamples, for instance the indisputably 3d nature of the cubic unconventional heavy Fermion superconductor UBe13. With all of these named properties being ‘consistent’ with UcS, there still remains no ability to a priori predict unconventional superconductivity. Put another way, there is no ‘microscopic’ theory of UcS, like the BCS theory for conventional superconductivity, since - due to the strong correlations - solving the pairing in UcS is a non-perturbative problem. As Norman has put it [5] “developing a rigorous theory for any of these classes of materials [in the UcS] has proven to be a difficult challenge, and will continue to be one of the major problems in physics in the decades to come.” It is the goal of this review to provide a comprehensive overview of UcS in one place to aid in this challenge. Fig. 1 (color online): Time line of the discovery of some unconventional superconductors The first superconductor identified as unconventional was CeCu2Si2, Tc=0.6 K, reported [6] by Steglich et al. in 1979. Rather than electron-phonon coupling, the pairing in CeCu2Si2 has been described as due to some sort of fluctuations (either quantum critical and/or antiferromagnetic.) Steglich et al.’s discovery led to the discovery of other unconventional superconductors of the same class as shown in Fig. 1 (UBe13 in 1983, UPt3 in 1984, . ), where this class is known as heavy fermion superconductors (HFS) due to the large (sometimes > 100 rest mass of the electron) effective mass m* of the conduction electrons, where these