Superconducting phases of $f$-electron compounds

Intermetallic compounds containing $f$-electron elements display a wealth of superconducting phases, which are prime candidates for unconventional pairing with complex order parameter symmetries. For instance, superconductivity has been found at the border of magnetic order as well as deep within ferromagnetically and antiferromagnetically ordered states, suggesting that magnetism may promote rather than destroy superconductivity. Superconducting phases near valence transitions or in the vicinity of magnetopolar order are candidates for new superconductive pairing interactions such as fluctuations of the conduction electron density or the crystal electric field, respectively. The experimental status of the study of the superconducting phases of $f$-electron compounds is reviewed.

Figure 1

(Color online) Evolution of the total number of $f$-electron heavy-fermion superconductors. Systems included in this plot and covered in this review: 1979, $\mathrm{Ce}{\mathrm{Cu}}_2{\mathrm{Si}}_2$; 1984, $\mathrm{U}{\mathrm{Be}}_{13}$ and $\mathrm{U}{\mathrm{Pt}}_3$; 1986, $\mathrm{U}{\mathrm{Ru}}_2{\mathrm{Si}}_2$; 1991, $\mathrm{U}{\mathrm{Pd}}_2{\mathrm{Al}}_3$ and $\mathrm{U}{\mathrm{Ni}}_2{\mathrm{Al}}_3$; 1993, $\mathrm{Ce}{\mathrm{Cu}}_2{\mathrm{Ge}}_2$; 1996, $\mathrm{Ce}{\mathrm{Pd}}_2{\mathrm{Si}}_2$ and $\mathrm{Ce}{\mathrm{Ni}}_2{\mathrm{Ge}}_2$; 1997, $\mathrm{Ce}{\mathrm{In}}_3$; 2000, $\mathrm{Ce}\mathrm{Rh}{\mathrm{In}}_5$ and $\mathrm{U}{\mathrm{Ge}}_2$; 2001, $\mathrm{Ce}\mathrm{Ir}{\mathrm{In}}_5$, $\mathrm{Ce}\mathrm{Co}{\mathrm{In}}_5$, and URhGe; 2002, $\mathrm{Pu}\mathrm{Co}{\mathrm{Ga}}_5$ and $\mathrm{Pr}{\mathrm{Os}}_4{\mathrm{Sb}}_{12}$; 2003, ${\mathrm{Ce}}_2\mathrm{Rh}{\mathrm{In}}_8$ and $\mathrm{Pu}\mathrm{Rh}{\mathrm{Ga}}_5$; 2004, $\mathrm{Ce}\mathrm{Ni}{\mathrm{Ge}}_3$, ${\mathrm{Ce}}_2{\mathrm{Ni}}_3{\mathrm{Ge}}_5$, UIr, $\mathrm{Pr}{\mathrm{Ru}}_4{\mathrm{P}}_{12}$, $\mathrm{Ce}{\mathrm{Pt}}_3\mathrm{Si}$, $\mathrm{Ce}\mathrm{Ir}{\mathrm{Si}}_3$, and $\mathrm{Ce}\mathrm{Rh}{\mathrm{Si}}_3$; 2005, $\mathrm{Pr}{\mathrm{Ru}}_4{\mathrm{Sb}}_{12}$; 2007, UCoGe, $\mathrm{Np}{\mathrm{Pd}}_5{\mathrm{Al}}_2$, $\mathrm{Ce}\mathrm{Co}{\mathrm{Ge}}_3$, and $\mathrm{Ce}{\mathrm{Pd}}_5{\mathrm{Al}}_2$.

Figure 2

(Color online) Variations in the tetragonal $\mathrm{Ba}{\mathrm{Al}}_4$ crystal structure. The $\mathrm{Th}{\mathrm{Cr}}_2{\mathrm{Si}}_2$ structure is frequently found among $\mathrm{Ce}{M}_2{X}_2$ compounds reviewed in Sec. 2A1. The $\mathrm{Ba}\mathrm{Ni}{\mathrm{Sn}}_3$ crystal structure, which lacks inversion symmetry, is typical of the $\mathrm{Ce}M{\mathrm{T}}_3$ compounds reviewed in Sec. 4A. Among $f$-electron systems with the noncentrosymmetric $\mathrm{Ca}{\mathrm{Be}}_2{\mathrm{Ge}}_2$ structure no compounds are known that exhibit superconductivity. From 339.

Figure 3

(Color online) Temperature vs pressure phase diagram of $\mathrm{Ce}{\mathrm{Cu}}_2{\mathrm{Si}}_{2-x}{\mathrm{Ge}}_x$. The pressure dependence of the superconducting transition temperature, here denoted $T_c$, in S-type $\mathrm{Ce}{\mathrm{Cu}}_2{\mathrm{Si}}_2$ exhibits a plateau between 20 and $70\mathrm{kbar}$ (black dots). Weak impurity scattering in moderately Ge-doped $\mathrm{Ce}{\mathrm{Cu}}_2{\mathrm{Si}}_2$ decomposes the superconductivity into two domes, one at the border of antiferromagnetism and the other at a presumed valence transition. From 750, representing a compilation of several studies described in the text.

Figure 4

Combined temperature vs pressure phase diagram of the isostructural isoelectronic pair of systems $\mathrm{Ce}{\mathrm{Pd}}_2{\mathrm{Si}}_2$ and $\mathrm{Ce}{\mathrm{Ni}}_2{\mathrm{Ge}}_2$, where superconductivity is observed at the border of antiferromagnetism and at low and high pressures, respectively. From 216.

Figure 5

(Color online) Depiction of the structural series ${\mathrm{Ce}}_n{M}_m{\mathrm{In}}_{3n+2m}$. The infinite-layer system $\mathrm{Ce}{\mathrm{In}}_3$ is shown on the left, the single-layer systems are shown in the middle, the double-layer systems, which are intermediate between the infinite- and the single-layer systems, are shown on the right. Also indicated are typical muon stopping sites in the double-layer system. From 489.

Figure 6

Temperature vs pressure phase diagram of $\mathrm{Ce}{\mathrm{In}}_3$. $T_M$ denotes the coherence maximum in the resistivity, $T_N$ denotes the Néel temperature, $T_c$ denotes the superconducting transition temperature, and $T_1$ denotes the upper boundary of the regime with Fermi- liquid resistivity. From 352.

Figure 7

(Color online) Temperature vs pressure phase diagram of $\mathrm{Ce}\mathrm{Co}{\mathrm{In}}_5$. From the resistivity a “pseudogap” at $T_{\mathrm{pg}}$ is inferred that merges with the maximum in the onset of the SC. At high pressures the superconductivity condenses out of a Fermi liquid temperature dependence of the resistivity below $T_{\mathrm{FL}}$. From 684.

Figure 8

(Color online) Temperature vs magnetic field and pressure phase diagram of $\mathrm{Ce}\mathrm{Rh}{\mathrm{In}}_5$ as reported by 567. Detailed studies of the specific heat suggest a well-defined line of quantum criticality, separating a regime where superconductivity and antiferromagnetism coexist with a regime of homogenous bulk superconductivity. From 567 as shown in 631.

Figure 9

(Color online) Compilation of the evolution of superconductivity and antiferromagnetic order in the series $\mathrm{Ce}M{\mathrm{In}}_5$, where $M=\mathrm{Co}$, Rh, and Ir. Note the continuous evolution of the superconducting transition temperature when going from Ir to Co despite the presence of disorder. This continuous evolution is interrupted by an antiferromagnetic dome in the series Co→Rh→Ir. From 561 as shown in 631.

Figure 10

(Color online) Evolution of the superconducting transition temperature as a function of $c∕a$ ratio for the lattice parameter in the series $\mathrm{Ce}M{\mathrm{In}}_5$ (left hand and bottom axis) and $\mathrm{Pu}M{\mathrm{Ga}}_5$ (top and right-hand axis). From 631.

Figure 11

Temperature vs pressure phase diagram of $\mathrm{U}{\mathrm{Ru}}_2{\mathrm{Si}}_2$ inferred from various experimental probes. The onset of the hidden order $T_0$ is weakly pressure dependent. The hidden order changes to large-moment Ising antiferromagnetism above 7 kbar without pronounced effect on the evolution of $T_0∕T_N$. Superconductivity vanishes with the appearance of the antiferromagnetism. HO, hidden order; AF, large moment antiferromagnet; and SC, superconductivity. From 17.

Figure 12

Temperature vs Th concentration in ${\mathrm{U}}_{1-x}{\mathrm{Th}}_x{\mathrm{Be}}_{13}$. The upper curve corresponds to the onset of superconductivity in the resistivity. In the range $2\%<x<4\%$ a second transition is observed in the specific heat, which may be related to magnetic order and/or another superconducting phase. From 423.

Figure 13

(Color online) Ferromagnetism and superconductivity in $\mathrm{U}{\mathrm{Ge}}_2$. (a) Pressure vs temperature phase diagram of $\mathrm{U}{\mathrm{Ge}}_2$. Superconductivity is observed well within the ferromagnetic state in the vicinity of transition between a large-moment and a small-moment ferromagnet. (b) Magnetic field vs pressure phase diagram of $\mathrm{U}{\mathrm{Ge}}_2$. The ferromagnetic transitions at $p_x$ and at $p_c$ are both first order as seen by first order metamagnetic transitions at $H_x$ and $H_m$. From 578.

Figure 14

(Color online) Temperature vs magnetic field phase diagram of URhGe for magnetic fields in the $b\text{−}c$ plane. The critical end point of the reorientation transition of the magnetic order is surrounded by a dome of superconductivity (S2). From 396.

Figure 15

Qualitative depiction of chiral exchange splitting of a spherical Fermi surface by Rashba spin-orbit interactions. Also shown are Cooper pairs that may form under such an exchange splitting, notably a mixed singlet with triplet state. From 190.

Figure 16

Key properties in the series $\mathrm{Ce}M{X}_3$ ($M$=Rh, Ir, Co; $X$=Si, Ge). (a) Néel temperature $T_N$ vs unit cell volume $V$ in the series $\mathrm{Ce}M{X}_3$ ($M$=Rh, Ir, Co; $X$=Si, Ge). (b) Sommerfeld coefficient $\gamma$ of the specific heat vs unit cell volume $V$ in the series $\mathrm{Ce}M{X}_3$. From 320.

Figure 17

(Color online) Magnetism and superconductivity in $\mathrm{Ce}\mathrm{Ir}{\mathrm{Si}}_3$. (a) Temperature vs pressure phase diagram of $\mathrm{Ce}\mathrm{Ir}{\mathrm{Si}}_3$. At the border of antiferromagnetic order a wide superconducting dome emerges. Note that the pressure axis begins at 19 kbar. (b) Superconducting specific heat anomaly as a function of pressure. (c) Extrapolated zero-temperature upper critical field. From 658.

Figure 18

Temperature vs pressure phase diagram of UIr. Three ferromagnetic phases have been identified. Superconductivity is only observed at the border of the FM3 phase with very low superconducting transition temperatures. From 356.

Figure 19

(Color online) Comparison of superconducting transition temperature with the characteristic spin fluctuation temperature. The latter is essentially a bandwidth and may be insensitive to the precise microscopic nature of the correlations. From 136 as shown in 631.

Figure 20

(Color online) Magnetic field vs temperature phase diagram of $\mathrm{Pr}{\mathrm{Os}}_4{\mathrm{Sb}}_{12}$. In high magnetic field an ordered state is stabilized that is driven by the level crossing of the crystal electric fields under magnetic field. From 26.

Figure 21

(Color online) Superconducting phases of $\mathrm{U}{\mathrm{Pt}}_3$ as a function of magnetic field. The insets show the nodal structures of the $E_{1u}$ representations proposed on the basis of small angle neutron scattering of the flux line lattice. From 273.

Figure 22

(Color online) Superconducting phase diagram of $\mathrm{Pr}{\mathrm{Os}}_4{\mathrm{Sb}}_{12}$. From 223.

Figure 23

(Color online) Superconducting phase diagram of $\mathrm{Ce}\mathrm{Co}{\mathrm{In}}_5$. In the low-temperature limit a body of evidence suggests the formation of a FFLO state (shading). From 441.