Oliver Stockert

The heavy-fermion metal CeCu$_2$Si$_2$ was the first discovered unconventional, non-phonon-mediated superconductor, and for a long time was believed to exhibit single-band $d$-wave superconductivity, as inferred from various measurements hinting at a nodal gap structure. More recently however, measurements using a range of techniques at very low temperatures ($T \lessapprox 0.1$ K) provided evidence for a fully-gapped superconducting order parameter. In this Colloquium, after a brief historical overview we survey the apparently conflicting results of numerous experimental studies on this compound. We then address the different theoretical scenarios which have been applied to understand the particular gap structure, including both isotropic (sign-preserving) and anisotropic two-band $s$-wave superconductivity, as well as an effective two-band $d$-wave model, where the latter can explain the currently available experimental data on CeCu$_2$Si$_2$. The lessons from CeCu$_2$Si$_2$ are expected to help uncover the Cooper-pair states in other unconventional, fully-gapped superconductors with strongly correlated carriers, and in particular highlight the rich variety of such states enabled by orbital degrees of freedom.

The heavy-fermion metal ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ was the first discovered unconventional, non-phonon-mediated superconductor and, for a long time, was believed to exhibit single-band $d$-wave superconductivity, as inferred from various measurements hinting at a nodal gap structure. More recently, however, measurements using a range of techniques at low temperatures ($T⪅0.1\mathrm{K}$) provided evidence for a fully gapped superconducting order parameter. In this Colloquium, after a historical overview the apparently conflicting results of numerous experimental studies on this compound are surveyed. The different theoretical scenarios that have been applied to understanding the particular gap structure are then addressed, including both isotropic (sign-preserving) and anisotropic two-band $s$-wave superconductivity, as well as an effective two-band $d$-wave model, where the latter can explain the currently available experimental data on ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$. The lessons from ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ are expected to help uncover the Cooper-pair states in other unconventional, fully gapped superconductors with strongly correlated carriers, and, in particular, highlight the rich variety of such states enabled by orbital degrees of freedom.

The disappearance of a charge-density-wave (CDW) transition in $\mathrm{Lu}{\left({\mathrm{Pt}}_{1-x}{\mathrm{Pd}}_x\right)}_2\mathrm{In}$ is thought to be closely linked to a maximum of the superconducting transition temperature in the system. We studied the superstructure and the phonon softening in $\mathrm{Lu}{\left({\mathrm{Pt}}_{0.5}{\mathrm{Pd}}_{0.5}\right)}_2\mathrm{In}$ by means of neutron and high-resolution inelastic x-ray scattering. A full phonon softening at ${\mathit{q}}_{\mathrm{CDW}}=\left(0.50.50\right)$ with the appearance of superstructure peaks could be clearly identified. This supports the CDW transition to be a continuous transition. The large tail of the superstructure intensity above the CDW transition indicates the importance of critical fluctuations. In general, the observed phonon dispersion relations are quite similar to those predicted by theoretical calculations. However, the softening occurs in a much narrower momentum range than predicted.

CePdAl is a prototypical frustrated Kondo lattice with partial long-range order (LRO) at $T_{\mathrm{N}}=2.7$ K. Previous bulk experiments under hydrostatic pressure found signatures for a quantum critical regime that extends from $p_{\mathrm{c}}\approx 0.9$ GPa, where LRO disappears, up to $\sim 1.7$ GPa. We employed extensive muon spin relaxation and rotation experiments under pressure. The continuous and complete suppression of LRO at $p_{\mathrm{c}}$ is confirmed. Above $T_{\mathrm{N}}\left(p\right)$ and beyond $p_{\mathrm{c}}$, an additional crossover scale $T^{*}\left(p\right)$ characterizes the change from pure to stretched exponential relaxation in zero field. Remarkably, $T^{*}\left(p\right)$ agrees with previously determined signatures of entropy accumulation above LRO. This coincidence microscopically evidences fluctuating frustrated spins at $T\le T^{*}$ with spin-liquid behavior. Power-law divergences of the temperature and longitudinal field dependences of the relaxation rate, with time-field scaling, at pressures between $p_c$ and 1.7 GPa characterize this regime as quantum critical.

Magnetic frustration in metals is scarce and hard to pinpoint, but exciting due to the possibility of the emergence of fascinating novel phases. The cubic intermetallic compound ${\mathrm{HoInCu}}_4$ with all holmium atoms on an fcc lattice exhibits partial magnetic frustration, yielding a ground state where half of the Ho moments remain without long-range order, as evidenced by our neutron scattering experiments. The substitution of In with Cd results in ${\mathrm{HoCdCu}}_4$ in a full breakdown of magnetic frustration. Consequently we found a fully ordered magnetic structure in our neutron diffraction experiments. These findings are in agreement with the local energy scales and crystal electric field excitations, which we determined from specific heat and inelastic neutron scattering data. The electronic density of states for the itinerant bands acts as a tuning parameter for the ratio between nearest-neighbor and next-nearest-neighbor interactions and thus for magnetic frustration.

The heavy-fermion compound ${\mathrm{CeCu}}_{6-x}{\mathrm{Au}}_x$ has become a model system for unconventional magnetic quantum criticality. For small Au concentrations $0\le x<0.16$, the compound undergoes a structural transition from orthorhombic to monoclinic crystal symmetry at a temperature $T_s$ with $T_s\to 0$ for $x\approx 0.15$. Antiferromagnetic order sets in close to $x\approx 0.1$. To shed light on the interplay between quantum-critical magnetic and structural fluctuations we performed neutron-scattering and thermodynamic measurements on samples with $0\le x\le 0.3$. The resulting phase diagram shows that the antiferromagnetic and monoclinic phase coexist in a tiny Au concentration range between $x\approx 0.1$ and 0.15. The application of hydrostatic and chemical pressure allows us to clearly separate the transitions from each other and to explore a possible effect of the structural transition on the magnetic quantum-critical behavior. Our measurements demonstrate that at low temperatures the unconventional quantum criticality exclusively arises from magnetic fluctuations and is not affected by the monoclinic distortion.

We report on the results of powder and single-crystal neutron diffraction to investigate the magnetic order in ${\mathrm{YbCo}}_2{\mathrm{Si}}_2$ below the Néel temperature $T_{\mathrm{N}}=1.7$ K in detail. Two different magnetically ordered phases can clearly be distinguished. At lowest temperatures a commensurate magnetic structure with a propagation vector ${\mathbf{k}}_1=\left(0.250.251\right)$ is found, while the intermediate phase ($T>0.9$ K) is characterized by an incommensurate magnetic structure with ${\mathbf{k}}_2=\left(0.250.0861\right)$. The magnetic structure in ${\mathrm{YbCo}}_2{\mathrm{Si}}_2$ is in marked contrast to all other known $R{\mathrm{Co}}_2{\mathrm{Si}}_2$ compounds ($R$ = rare earth element) likely due to some itineracy of the Yb $4f$ states being responsible for the magnetism.