Colloquium: Unconventional fully gapped superconductivity in the heavy-fermion metal ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$

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.

Figure 1

(a) Contribution of the $4f$ electrons to the specific heat of ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ plotted as $C_{4f}/T$ vs $T$ on a logarithmic scale. The solid line is a guide for the eye. Inset: crystal structure of ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ (${\mathrm{ThCr}}_2{\mathrm{Si}}_2$-type structure, space group $I4/mmm$), where the green, red, and blue spheres correspond to Ce, Cu, and Si atoms (see labels), respectively. From [226]. (b) Temperature dependence of the resistivity of ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ on a logarithmic temperature scale. From [212].

Figure 2

(a) Resistivity $\rho (T)$, ac susceptibility ${\chi }_{\mathrm{ac}}(T)$, and (b) specific heat as $C/T$ vs $T$ for polycrystalline ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ indicating bulk superconductivity at $T_{\mathrm{c}}\approx 0.5\mathrm{K}$. The Pauli susceptibility ($T>T_{\mathrm{c}}$) shown in the inset amounts to ${\chi }_P=82\times {10}^{-9}{\mathrm{m}}^3/\mathrm{mol}$ ([1]). Note that the normal-state values of both $\rho (T)$ and $C(T)/T$ point to non-Fermi-liquid behavior. In (b) data of two samples are displayed that have the same nominal composition and were prepared in the same way. From [227].

Figure 3

Upper critical magnetic field $B_{\mathrm{c}2}$ vs $T$ of a ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ single crystal for fields applied within ($\parallel$) and perpendicular to ($\perp$) the Ce planes obtained from $\rho (T)$ measured parallel to the respective field. There is only a moderate anisotropy, but a large initial slope at $T_{\mathrm{c}}$ is found for $B_{\mathrm{c}2}(T)$. Note the shallow maximum of $B_{\mathrm{c}2}(T)$ near $T=0.15\mathrm{K}$ as reflected in the inset by the reentrant $\rho (T)$ behavior for $B\ge 2.4\mathrm{T}$. From [23].

Figure 4

(a) Temperature-pressure-magnetic-field phase diagram of a single crystal of $A/S$-type ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$. From [120]. (b) Magnetic-field–temperature diagram of single crystal ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$, where positions of the field-induced bicritical point (BCP) and QCP are also displayed. From [274].

Figure 5

Temperature-pressure phase diagram of ${\mathrm{CeCu}}_2({\mathrm{Si}}_{1-x}{\mathrm{Ge}}_2{)}_2$ that exhibits two superconducting domes (for $x=0.1$), one centered around a lower pressure $p_{\mathrm{c}1}$ associated with an antiferromagnetic QCP, while the dome at higher pressures is near a possible valence transition. The diamonds, circles, triangles, and squares correspond to compositions with $x=0.25$, 0.1, 0.05, and 0.01, respectively. The dashed line displays the anticipated line of first-order valence transitions, ending in a critical point somewhere between 10 and 20 K. The solid lines are a guide for the eye. From [285].

Figure 6

(a) Neutron-diffraction intensity map of the reciprocal $(hhl)$ plane around $\mathbf{Q}=(0.21,0.21,1.45)$ in A-phase ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ at $T=50\mathrm{mK}$ and 1 K. (b) Main heavy Fermi surface sheet in ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ indicating the columnar nesting with wave vector $\mathit{\tau }$. From [246].

Figure 7

Spin dynamics in ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$. (a) Low-energy spin excitations in $S$-type ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ at ${\mathbf{Q}}_{\mathrm{AFM}}$ and $T=0.07\mathrm{K}$ in the superconducting state ($B=0$) and the normal state ($B=2\mathrm{T}$). From [247]. (b) Scaling of the normal-state quasielastic response in $S$-type ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ at ${\mathbf{Q}}_{\mathrm{AFM}}$ and at $B=B_{\mathrm{c}2}=1.7\mathrm{T}$ indicating universal scaling of the dynamical susceptibility ${\chi }^{\prime \prime }{T}^{3/2}$ vs $\omega /T^{3/2}$. (c) Dispersion of the spin excitations in the normal and superconducting states of $S$-type ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$. (d) Relaxation rate $\mathrm{\Gamma }$ and inverse susceptibility $\chi (\mathbf{Q}{)}^{-1}$ of the normal-state magnetic response at $\mathbf{Q}={\mathbf{Q}}_{\mathrm{AFM}}$ in $S$-type ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ vs $T^{3/2}$. (b)–(d) From [22].

Figure 8

Dependence of the superconducting transition temperature of ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ polycrystals on substitutions for Ce and Cu. Adapted from [225].

Figure 9

Temperature dependence of the spin lattice relaxation rate $1/T_1(T)$ of ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$, obtained from Cu-NQR measurements for (a) various superconducting and nonsuperconducting polycrystals, and (b) single crystals of superconducting ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ under hydrostatic pressure, as well as ${\mathrm{LaCu}}_2{\mathrm{Si}}_2$. No Hebel-Slichter peak at $T_{\mathrm{c}}$ is observed and a $T^3$ dependence is found in the superconducting samples down to around 0.1 K. (a) From [91]. (b) From [64].

Figure 10

(a) Temperature dependence of the electronic contribution to the specific heat as $C_{\mathrm{e}}/T$ of ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ down to temperatures of 0.04 K. Lower inset: analysis of the data using a nodeless two-gap model. Upper panel: the data as a contour plot. (b) The field dependence of the electronic specific-heat coefficient at various temperatures for fields along the (left panel) $[100]$ and (right panel) $[001]$ directions. At 0.06 K, linear behavior is observed at low fields for both field orientations. Inset in the right panel: $C_{\mathrm{e}}/T$ as a function of the in-plane azimuthal field angle $\varphi$, which remains constant. From [105].

Figure 11

(a) Temperature dependence of the magnetic penetration depth shift of an $A/S$-type ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ single crystal, measured using the tunnel-diode-oscillator-based method. The solid line shows the fit to the low-temperature data with a model for a fully gapped superconductor, where the lack of nodes in the gap is corroborated by an exponent $n$ that is consistently larger than 2 (inset). From [171]. (b) Temperature dependence of the in-plane thermal conductivity ${\kappa }_{\mathrm{a}}/T$ of ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$, which in the superconducting state ($H=0$) extrapolates to zero at $T=0$, demonstrating fully gapped superconductivity. Also shown are the data in the normal state, which demonstrates the validity of the Wiedemann-Franz law. From [280].

Figure 12

Fermi surface and quasiparticle dispersion of $S$-type ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$ from ARPES measurements. (a) Three-dimensional bulk Brillouin zone (black solid lines) and the projected surface Brillouin zone (red dashed lines) of ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$. (b) Experimental $k_x\text{−}{k}_y$ map at 10 K taken with 135 eV photons. The red arrow indicates the in-plane component of the SDW ordering wave vector ${\mathbf{Q}}_{\mathrm{AFM}}$ observed using neutron diffraction (Sec. 3b). (c) Band dispersion along $\overline{\mathrm{\Gamma }}\text{−}\overline{M}$ at 10 K. (d) Enlargement of the heavy-electron band near $E_F$. From [277].

Figure 13

Projection of the renormalized heavy Fermi surface and ordering wave vector ${\mathbf{Q}}_{\mathrm{AFM}}=(0.215,0.215,0.53)$ onto the $k_z=0$ plane of the 3D Brillouin zone at zero temperature. The dashed lines indicate the nodes of the individual components of the $d+d$ pairing. The diagonal black and vertical and horizontal green dashed lines denote the nodes of ${\mathrm{\Delta }}_3(\mathbf{k})\propto d_{x^2-y^2}$ and ${\mathrm{\Delta }}_1(\mathbf{k})\propto d_{xy}$, respectively; see Eq. (1). The effective gap is determined by the addition in quadrature of the two components. Since the nodes of the ${\mathrm{\Delta }}_3$ and ${\mathrm{\Delta }}_1$ components do not overlap except at isolated points of the Brillouin zone, the $d+d$ pairing is always gapped on the Fermi surface. The wave vector for the peak of the observed antiferromagnetic fluctuations projected onto the $(k_x,k_y)$ plane ${\mathbf{Q}}_{\mathrm{AFM}}^{||}$ connects parts of the cylindrical Fermi surface near the edges (the red pill shapes), where ${\mathrm{\Delta }}_3\propto d_{x^2-y^2}$ has the opposite sign, leading to the emergence of a pronounced peak inside the superconducting gap in INS experiments. From [171].

Figure 14

Single Cu plane in the unit cell of ${\mathrm{CeCu}}_2{\mathrm{Si}}_2$. The four sites labeled (1)–(4) correspond to Cu $d_{x^2-y^2}$ orbitals in the plane. The dashed-line circles represent the Ce sites projected onto the Cu plane. From [154].

Figure 15

Temperature dependence of the superfluid density derived from penetration depth measurements using the tunnel-diode-oscillator-based method. (a),(b) Results fitted to the superfluid density of an $S$-type sample with an isotropic two-gap model and $d+d$ band-mixing pairing model, respectively. (c),(d) Corresponding results for the $A/S$-type sample. From [171].