Superconductivity: how the unconventional became the new norm

Superconductivity, discovered by Kamerlingh Onnes in Leiden in 1911, arguably still is the most intriguing solid state phenomenon [1]. The microscopic origin of superconductivity remained a big mystery for almost half a century until Bardeen, Cooper and Schrieffer introduced their 'BCS theory' in 1957 [2]. BCS theory, describes the superconducting state as a condensate of electron pairs or 'Cooper pairs' that are formed due to a phonon-mediated attractive interaction. The theory was (and still is) an amazing intellectual feat that almost closed the book. That is, until the 1970's when new 'unconventional' forms of superconductivity [3, 4] were discovered, culminating in the discovery of high-temperature superconductivity in layered cuprate compounds in 1986 [5]. While the record-high superconducting transition temperatures of the cuprates have not been surpassed since then, tremendous progress has been made in our understanding of how strong electron correlation effects and unusual Fermi surface topologies can lead to alternative pairing mechanisms, and how those pairing symmetries can be both engineered and exploited in a wide variety of low-dimensional materials systems. The field of superconductivity has become multi-facetted and has brought together researchers from various materials disciplines. This special issue of 'Superconductivity in the 2D Limit' provides a sampling of recent advances in superconductivity research in a diverse set of low-dimensional materials systems, and contains both original research contributions as well as broad overviews that tie together state-of-the-art advances made in various subfields, many of them being quite spectacular. The purpose of this preface is to provide an easy accessible introduction to the key issues and a proper context for the contributions to this special issue.

A superconductor represents a thermodynamic phase of matter that is characterized by dissipationless electrical conductivity in conjunction with the expulsion of magnetic flux from its interior, known as the Meissner effect [1]. Microscopically, the supercurrent is carried by Cooper pairs, whose pair wave functions become phase locked as they condense, like bosons, into a coherent macroscopic quantum state, i.e., all particles are described by the same wave function. Cooper pair formation is made possible, for instance, via the mutual exchange of virtual phonons [2]. It implies the existence of a superconducting gap because the energy of a bound pair is slightly less than that of two 'free' electrons. Phase locking is the result of spontaneous electromagnetic gauge symmetry breaking, which gives the condensate a form of rigidity from which the Meissner effect and other superconductivity phenomena such as persistent currents, flux quantization, and the Josephson effect can be inferred [1].

The transition from the low-temperature superconducting state to the high temperature 'normal' state, proceeds via a second order phase transition at a critical temperature T_C, and can be analyzed within the phenomenological Ginzburg–Landau (GL) framework [1, 6]. Here, the GL order parameter Ψ(r) can be identified with the pair wave function:∣Ψ(r)∣² measures the superfluid density, which in turn is related to the gap parameter Δ_k. In conventional superconductors, the pair wave function is isotropic, meaning that the Cooper pairs are in the l = 0 (s-wave) angular momentum state. Accordingly, the two spins form a singlet state. Higher angular momentum states are possible, depending on the details of the pairing potentials and Fermi surface topology [7]. In particular, if the electron repulsion is very strong and cannot be overcome by the phonon mechanism, electrons can minimize their Coulomb repulsion by pairing in higher angular momentum states (l > 0) where the superconducting gap ∣Δ_k∣ is suppressed along nodal planes in momentum space [7]. For example, the pairing symmetry in strongly correlated cuprate high-temperature superconductors is predominantly d-wave (l = 2) where the Cooper pairing is most likely mediated by spin-fluctuations.

For years, conventional wisdom suggested that the superconducting state in a low-dimensional system should be fragile. Reduced screening would suppress Cooper pairing while, according to the Mermin Wagner theorem, thermal fluctuations would inhibit long-range phase coherence [1]. These factors are furthermore aggravated by the detrimental role of defects in reduced dimensionality. Fluctuations in a two-dimensional (2D) superfluid or superconductor primarily involve the excitations of topological defects or magnetic vortices [8]. These defects are expected to be present as bound vortex-antivortex pairs, up to the so-called Berezinskii–Kosterlitz–Thouless (BKT) temperature T_BKT < T_C [8], above which the vortices unbind and the system becomes resistive [9]. Conclusive evidence of the BKT transition in, e.g., thin metal films is very hard to come by, as pointed out by Uchihashi [10] and by Brun et al [11] in this issue. In particular, the interplay between disorder and Coulomb interactions in 2D has been shown to preempt the BKT transition [12], thereby suppressing the formation of the condensate. Instead, to observe the BKT transition, most likely one would need ultraclean and highly crystalline thin films with an absolute minimum of vortex pinning sites, which is almost impossible to achieve in thin metal films. (Strong pinning is required for achieving high supercurrent densities in practical applications.) Interestingly, the BKT transition has reportedly been observed at the 2D superconducting LaAlO₃/SrTiO₃ interface, albeit at the very low temperature of 188 mK [13].

In recent years, numerous experimental studies on highly crystalline 2D superconductors, such as those acquired through epitaxial synthesis or mechanical exfoliation, have shown that superconductivity in these materials can be amazingly robust down to a few atomic layers [10, 11, 14–19]. Even a single atom layer of lead or indium on a Si(111) substrate retains a fairly high transition temperature (as compared to the bulk) of the order of 1.5–3.2 K [10, 11, 14]. Few atom layer thick Pb films can carry very large dissipationless currents, up to several MA cm⁻², which is a very sizeable fraction of the depairing current density [15]. Likewise, ionic-liquid-gated 2D materials such as MoS₂ and NbSe₂ can support giant in-plane magnetic fields of over 50 T [16, 20]. Arguably, an even more spectacular discovery is the observation of high-temperature superconductivity in a single unit-cell thick layer of FeSe grown epitaxially on SrTiO₃ and BaTiO₃ substrates with a superconducting transition temperature far greater than that of bulk FeSe or that of any other iron-based superconductor [14, 17]. As discussed by L Wang et al [14], Y Wang et al [18], and Linscheid [21], the substrate plays a key role in the T_C enhancement, possibly through charge transfer, epitaxial strain, or cross-interface electron–phonon coupling. The latter turns out to be strongly momentum dependent and may be very effective in mediating the pairing when the s-wave channel blocked by Coulomb interactions [18]. Finally, Bollinger and Bozovic have shown that the superconductivity in single-phase films, bilayers and superlattices of cuprate materials, made with an atomic-layer-by-layer epitaxy, is fully consistent with the notion that superconductivity in the cuprates is mostly 2D, but also that the transition temperature in a single CuO₂ plane can be as high as that of the bulk [19].

High-temperature superconductivity seems mostly associated with both unconventional pairing symmetry in the presence of strong Coulomb interactions and low (quasi 2D) dimensionality. Where do we go from here? First of all, we would like to point out that the number of materials systems addressed in this issue, and consequently the amount of new physics learned, is still rather limited. Baskaran [22] shows that when it comes to defining future directions in 2D superconductivity, we should not only stick to well-known 2D materials but also think outside the box. He argued that unconventional pairing symmetries are not limited to the well-known transition metal compounds. Coulomb repulsions are also known to be very strong in doped low-dimensional band insulators with low carrier density, such as LaO_1−xF_xBiS₂. These sp-bonded materials systems may be driven into an insulating 2D Wigner crystal state at x = 0.5. Then, introducing additional dopants and allowing for quantum fluctuations, the Wigner crystal state could 'melt' to form a superconductor. These ideas, and more generally the possibility of strong correlations in doped sp bonded materials systems, clearly merit further investigation.

Furthermore, the FeSe case strongly suggests the possibility of enhancing superconductivity through hetero-structure engineering. Practical devices can now be obtained by combining superconductors with 2D materials such as graphene [23] or layered transition metal dichalcogenides, topological insulators, and even ferromagnets, that can all be made superconducting through e.g. the proximity effect [23] or ionic-liquid gating [16]. In all of these endeavors, detailed structural knowledge of the interface is essential, as highlighted by Linscheid [21]. Another interesting question would be how an unconventional order parameter would be affected by non-magnetic lattice imperfections or line-defects such as surface steps, as was done in the works by Brun et al [10] and Kim et al [24] in the specific case of Pb on Si(111) and Ge(111), respectively, or when a key symmetry element such as time-reversal or structural inversion symmetry is broken [11, 16]. The latter would occur at a surface or interface and constitutes the basis of the Rashba spin–orbit coupling effect that produces a spin-split Fermi surface with opposite locking of the spin and momentum directions [25] (or a valley-dependent spin-momentum locking as in the case of MoS₂ [16]). This suggests the possibility, starting for instance with a conventional s-wave superconductor, of creating a novel superconductor with a hybrid singlet-triplet order parameter [11, 26]. Additional breaking of the time reversal symmetry, for instance by introducing an exchange field through the addition of magnetic impurities, could even lead to the existence of a topological p-wave superconductor with exotic Majorana edge states, which might be a promising testbed for Majorana physics and topological quantum computing [26].

While many attempts have already been made along those lines, this still is the beginning. Recent advances in 2D materials fabrication, heterostructure engineering, as well as the availability of very powerful surface analytical probes such as ultralow-temperature and high-magnetic field scanning tunneling microscopes and novel photoemission spectrometers that allow for rapid and 'easy' measurements of the Fermi surface contour with a superior energy and momentum resolution, have contributed immensely to the most recent advances in the understanding of unconventional superconductivity. Many more exotic superconductors and superconducting hybrid devices are likely to be discovered at an accelerating pace.