Superconductivity at the Border with Magnetism

Publisher Summary This chapter discusses ternary magnetic superconductors, kondo impurities, and ferromagnetic superconductors. Ginzburg was the first to point out that superconductivity and ferromagnetism were incompatible. The depression of T c experiments included a study of superconductivity and ferromagnetism in the binary La–Gd alloy system, which appeared to be consistent with this claim. A number of intermetallic systems were subsequently investigated similarly via alloying between isostructural superconducting and ferromagnetic end members, where the magnetic exchange coupling appeared to be very weak but without any convincing evidence of the coexistence of superconductivity and long-range magnetic order. Ferromagnetism in materials such as ErRh 4 B 4 caused the reentrance of a normal state at temperatures below the ferromagnetic Curie temperature. Various theoretical arguments suggested, however, that in some cases ferromagnetic order might be compatible with superconductivity with superconductivity occurring near the quantum critical point at which the ferromagnetic Curie temperature is manipulated to approach T = 0 K.

)> 1. HISTORY lt was known weil before the Bardeen, Cooper, and Schrieffer (BCS) theory of superconductivity that magnetic impurities in superconducting materials could cause large depressions in superconducting transition temperatures. But the fust experirnents where the systematics of the depression of superconductivity by magnetic impurities could be quantified were only carried out in 1958 by Matthias et al. in rare-earth-doped fcc lanthanum (La) (1) (Fig. 3.1). Rare earths carry a 4flocal momem given by Hund's rules. lt had been supposed prior to the experiment that the depression of Tc might vary with the effective moment carried by the rare-earth impurity. This proved not to be the case. Rather the depression followed the spin S of the 4f ground state (Fig. 3.2), varying as the square of its projection on the total angular momentum J of the 4f ground state (J'S) 2 , as recognized immediately by Suhl.
This result had a simple explanation within the then recent BCS theory of superconductivity. If the interaction between a conduction electron spin s and the 4f local moment spin on the impurity atom S is supposed of the form H; 11 ,=-2];ntS·s=-2(g-1)]; 0 J·s, where the Landeg-factor provides the projection of S on J, then the local moment spin S will interact with opposite sign on the two spins of a Cooper pair, depressing Tc in proportion   to the so-c~ed deGennes factor (g-1)2.JU + 1). The detailed .theory was worked out by Abrikosov and Gor'kov [2], and this theory when account is taken of crystal 6.eld effects gives a good description of experiment for the entite superconducting range of compositions in the numerous intermetallic superconductors where this has been studied. Representative data are shown in Fig. 3 Matthias et al. in 1958 included a study of superconductivity and ferromagnetism as weil (Fig. 3.4) in the binary La-Gd alloy system, which appeared to be consistent with this claim.
A number of intermetallic systems were subsequently investigated sirnilarly via alloying between isostructural superconducting and ferromagnetic end members where the magnetic exchange coupling J;ng appeared to be very weak., but without any convincing evidence of the coexistence of superconductivity and long-range magnetic order. But the study of the coexistence of superconductivity and magnetic order took a new turn with the discovery of superconductivity in the rare-earth temary compounds RMo0s (X= S, Se; R =rare earth) (1975,1976) [5], known as Chevrel phases, and RRh~4 (1977) [ 6), compounds which possess a stoichiometric sublattice of rare-earth ions. Matthias had earlier (1972) reported the discovery of superconductivity at 13.7K in PbMo 6 S 8 [7], but it was very much a surprise when Fischer et al. reported superconductivity in a nwnber of the isostructural materials with rareearth atorns in place of Pb. While the Tc's found were in the range of 1 K without sign of magnetic.order as measured by magnetic susceptibility down to  this temperature, one nevenheless wondered what was happening to the rareearth 4f magnetic moments, parricularly in the cases where K.ramer's theorem told you that the crystal field ground state of the 4f electrons could not be a singlet. lt was soon discovered that a number of the rare-eanh Chevrel phases as weil as the rare-eanh rhodium borides supponed coexisting superconductivity and antiferromagnecic order ( Fig. 3.5). Funherrnore, a reentrant superconductivity was discovered in HoMo 6 S 8 and in ErRh.iB 4 ( Fig. 3.6) where a higher temperature superconducting state was quenched by a lower Tc ferromagnetic ordering. The experimental observation that the establishment of ferromagnetic order quenched superconducting order in both HoMo 6 S 8 and ErRhJ3 4 was consistent with tbe prediction of Ginzburg. That antiferromagnetic order could coexist with superconductivity could be rationalized with the observation that the antiferromagnetic exchange field could average to zero over the superconducting coherence length. And the antiferromagnetic ordering was not without effect on the superconducting state, there being clearly observed effects on the upper critical fields of these type II superconductors.
In a certain sense, these temary magnetic superconductors are like dilute alloys in that the weak conduction electron--4f local moment interaction allows one to think of the 4f moroents as a weak perturbation to the conduction electron system. The imponant aspect of these compounds, however, is 14 13 o Tc 12    that the rare earths sit on a chemically ordered sublattice that can support a lowtemperature magnetic state with long-range order. In addition to the Chevrels and rare-earth rhodium boride materials discussed above, a number of other similar systems have been discovered, most importantly the rare-earth RNi2B 2 C compounds, which can be grown f.iirly easily as high-quality single crystals fi:om Ni-In molten fl.ux and which have enabled as a result of careful investigation of important details regarding the interaction of magnetism and superconductivity not easily pUl'Slled in polycrystalline material [9].

)> 3. KONDO IMPURITIES
The original experiments on the depression ofTc by rare-earth impurities in La (Fig. 3.1) were shown tobe consistent with a conduction electron-local moment interaction ]:,,, 0 which varied only slowly with rare earth provided account was taken of crystal field effects on the Hund's rule 4f ground state, with one exception, Ce. Ce is seen to have an anomalously large depression of Tc relative to its position in the rare-earth series. Ce is the first rare earth with an occupied 4f level, and it is not surprising that the situation here might be somewhat different since the 4f shell is only just becoming stabilized.
lt tums out that the physics of Ce is quite different than that of the heavier rare earths, that it alone among the rare earths dissolved ·in La has ];nc < 0, an antiferromagnetic interaction, whereas all the other rare earth have];nc > 0, a ferromagnetic interaction. This was shown in experiments by Sugawara and Eguchi (1966) [10] where Ce-doped La was shown to exhibit an electrical resistivity minimum at low temperatures, consistent with the then very recent theoretical explanation of resistivity minima in dilute magnetic alloys by Kondo (1964) (11]. Kondo provided an answer to a long-standing puzzle in the low-temperature electrical resistivity of alloys with magnetic impurities. He showed that the resistivity rninimum observed in a number of transition meta! alloys could be explained in second Born approximation if ]inc was antiferromagnetic, predicting a low-temperature rise in the electrical resistivity b.p--ln T as observed. While this theory was developed with transition metal impurities in simple metal hosts in mind, it was evident from the experiments of Sugawara and Eguchi that Ce impurities in La were to be classed as Kondo ions. The reason that Ce differs from the other rare-earth impurities dissolved in La is a result of the unst.able nature of the 4f 1 state. Friede! [12) had developed the concept of the virtual bound state to explain va~ous magnetic and electrical properties oftransition meta! ions, the idea being that the d level of a t:ransition metal atoni dissolved in a metallic host might have this level coincidem in energy with conduction electron states of the host metaL Since a bound d state cannot exist in such a case, Friede! suggested thae the impurity d level could be thought of as a resonant staee thae had been broadened by hybridization with conduction electron states. If this broadening was not too great, some atomic-like properties might persist. Anderson developed subsequently an impurity model which formalized the ideas of Friedel (13). Schrieffer and Wolff [14] then showed how in the limie of weak hybridization, one could derive the effective conduction electron-local momem spin Hamilt~nian H;nc =-2]incS·s, where Jinc= 1 Vkd l 2U/(ed (c:d + U)) < 0. In chis expression, Vicd is the hybridi.zation matrix elemem between the conduction eleccrons and the d level at energy c:d with respect to the Fermi level and U is ehe Coulomb repulsion between ehe two elecrrons in the 4flevel, and is large and positive. These are the parameters in the Anderson impurity model which determine whether an impurity is magneric. This theory applies equally well with 4f electrons in a virtual bound state. For the case of welllocalized 4f elecerons where the 4f energy lies weil below the conduceion band, ]inc is given by the Coulomb exchange integral and this is always positive. lt is the closenes.5 of the energy of the Ce 4f level to the Fermi level, its weakly bound characeer, thae gives rise to ics Kondo behavior.
The Kondo impurity presented a difficult theoretical problem which was finally formally solved by Wilson using renormalization group methods (15). For ];nt < 0, the conduction elecerons form ineo a collective singlet wich the local moment below a characteristic Kondo eemperature kB T K -(1/ p)exp(l/ P.JinJ, where p is ehe electronic density of states at the Fermi level. As temperature decreases below T K• the entropy associaeed with the local momem spin degrees of freedom become shared with the conduction eleccrons and give rise to an enhanced conduction eleccron mass reflected in a contribution to the low-eemperature eleccronic specific heat coefficient / -kB ln 2/ T K per impurity. Over this same range of temperature, the higheemperature Curie-Weiss magnetic suscepeibility characteristic of local momencs evolves ineo a temperature-independent Pauli paramagnetism with magnitude consiseent with the electronic specific heat /. Following the work of Sugawara and Eguchi, a number of ineermetallic superconductors were investigaeed where it was found that rare-earth addirions depressed Tc in similar fuhion as in elemental La, with again Ce often being anomalous and associ.ated with Kondo behavior (Fig?.  nonmagnetic behavior of the impurity ion. lt tums out that one can accomplish this using applied pressure in the case of Ce impurities. The simple picture for this is that the 4f level of Ce is an irmer level with small radial extent relative to the outer valence electrons of Ce. This means that an f electron screens quite effectively one nuclear charge, the outer valence electrons acting as if they were in an atom with one less proton in the nucleus. This makes the effective metallic radius of the Ce ion larger than it might be if ehe flevel was unoccupied. The effece of applied pressure is then eo try eo push out ehe 4f electron from ehe aeomic core, demagnetizing Ce as ie were. Such an effece is seen in the experiments by Maple et al. (Fig. 3.9) on Ce-doped La.
Here for 2 ae.% Ce in La one sees that ehe pair breaking in ehe 10-kbar range has become sufficiently large to kill Tc, and thae Tc recovers at higher pressures where Ce no langer has a local moment. Wich increasing pressure, a Kondo description of ehe Ce ion will no langer be appropriate, where the spin and charge aspects of the 4f level both become relevant to ehe physical properties as opposed to the spin-only physics of the Kondo effect. The regime where Tc and TK become comparable has also received special attention, where it is found that materials can go from normal to superconducting to normal as a function of temperature in some cases.  persisted into the chemically fully ordered Ce intermetallics in many cases. This is evident in the electrical resistivity, as seen ( Fig. 3.10), for example, in CeB 6 , the x = 1 limit of the Kondo system Y 1 -xCexB 6 ( Fig. 3. 7). A clearer picture conceming what was going on in the dense Kondo materials sÜrted to emerge in 1976 with the discovery that the low-temperature specific heat    [19].

61
that of a typical metal that is in the 1-10 mJ!mol K 2 range [18). Furthermore, the value was in reasonable accord with a free electron estimate of the measured Pauli-like paramagnetic suscepribility, suggesting that the lowtemperature properties of CeAl 3 resembled those of a free electron metal with an enormous enhancement of the conduction electron effective mass. Typical of dense Kondo materials, the electrical resistivity of CeAl 3 ( Fig. 3 state. Yb (4f 0 ) also is often a Kondo ion, and here the Hund's rule ground state is the eightfold degenerate J= 7 /2. The crystalline environment of the 4f ion lifts the degeneracy of the Hund's rule ground state, typically in the case of Ce into three doublets (four doublets typically in the case of Yb), so that one is dealing at low temperature with a ground state doublet often weil separated (but not necessarily so) from the high er lying doublets. The doublet carries a magnetic moment, and it is not surprising that Ce dense Kondo materials can order magnetically, particularly if the lattice coherence temperature, below w hich the magnetic degrees of fi:eedom become entangled with those of the conduction electrons, is low. lt is common in such cases to see magnetic order happening at a peak in the electrical resistivity, since the lattice coherence has not yet set in. Superconductivity might appear to be a less likely occurrence even when coherence is fully established, given the rnagnetic parentage of the coherent ground state. So it came as a shock when Steglich showed in 1979 that the dense Kondo material CeCu 2 Si 2 was a bulk superconductor (Fig. 3.13) at 0.5 K [20]. One sees the speci.fic heat jump characteristic of a second-order superconducting phase transition occurring with gap opening in a very high density of states band, C/T-0.75J/mol-CeK 2 at Tc. The electrical resistivity of this material (Fig. 3 .14) is typical of a dense Kondo compound, showing Kondo features supecin?posed on changing populations of the crystal 6eld levels, with coherence setting in below app~oximately 10 K.
There are several points to make about this result. The fust is that the superconductivity had been seen in a number of earlier studies of this material but not believed because of its implausibility, rather thought tobe due to some impurity phase. The compound is difficult to prepare as a bulk superconductor for metallurgical reasons, and it was essential to get this metallurgy correct to establish bulk superconductivity fi:om speci.fic heat data. Furthermore, a number of what were believed to be regular Ce-based BCS superconductors were already known, such as CeRu 2 (Tc= 7 .0 K) and CeCo 2 (Tc = 1.5 K). The thinking for these was that Ce was in the tetravalent state or, equivalent at the time, the collapsed nonmagnetic state. There was no hint of high-temperature 200 300 T(K) Figure 3.14 Temperature variation of electrical resistance (in arbitrary units; room temperature resistivity of UBe 13 is 100µ.Qcm) of the heavy fermion superconductors CeCu 2 Si 2 (triangles), UBe 13 (squares), and UPt 3 (circles) (19). magnetic behavior in the magnetic susceptibility of these compounds, neither were ehere any Kondo-type resistivity fearures. Finally, ehe very !arge value of-y suggests that very narrow bands are present at the Femli surface with attendant streng electron correlations. The question ehen arises as to how Cooper pair formation can be favorable. Higher angular momentum pairing seerns possible, which would have a wave function node at r= 0, but such pairing was thought eo be highly unlikely except in extremely clean metals, due eo impurity quenching of ehe angular momenrum. In fact, in ehe early days following BCS such superconductivity was looked for and not found in super pure samples of the almest magnetic metal Pd.
The label heavy Fennion superconductor was applied to CeCu 2 Si 2 , coming about from a disagreement between Steglich and his postdoctoral advisor Wohlleben who was particularly distressed by this development in the field of superconductivity. No further Ce-based dense Kondo superconductors were immediately forthcoming despite a large effort to find ehern. Unexpectedly, the next development in heavy fennion superconductivity occurred not in 4f materials but in ehe Sf materials UBe 13 and UPt 3 , with T/s of0.9 and 0.5 K, respectively. lt is perhaps not so Strange that superconduccors in this dass should be found among Sf materials. There are many similaricies between 4f and Sf materials, but the Sf electrons are considerably more extended in space,  (22). Bucher had not measured the specific heat, and, as with CeCu2Si 2 , superconductivity seemed so unlikely in this material. Large single crystals of UBe 13 proved to be easily grown from Al flux, and specific heat measurements (Fig. 3.15) showed unequivocally that the material was a buJk superconductor with the gap opening in a very high density of states band. The electrical resistivity shown above in Fig. 3.14 has remarkable similarities to that of CeCu2Si 2 .
To come to grips with this unusual superconductivity, much effort went into trying to find ways in which it showed differences with standard BCS behavior, for example, how did impurities influence Tc? These data were surprising (Fig. 3.16). Not only did impurities carrying moments (Gd) not have effects much different from those without moments (La, Lu), but Th impurities produced a nonmonotonic depression, with a sharp cusp in Tc: at 1.7 at.% Th (25]. Specifi.c heat measurements then found (Fig. 3.17) that two phase transitions existed in samples with Th concentration greater than x = 0.017. The lower transition was found not to be a magnetic transition but rather a · second superc;onducting transition. This had not been observed in any previous homogeneous materials and represented something quite new. While it seems possible that such a double transition might result from superconducting transitions on separate pieces of Fenni surface, this had never been known to occur, and a more likely explanation was that the superconducting order parameter is complex with more than one component. T his was the indication that the superconductivity in these heavy fermion matetials was of a new kind and immediately experiment was directed toward establishing this. Theorists had already been thinking about this in the context of the superfluidity of 3 He, where more than one superfluid phase was known to exist. T he simple possibilities that presented themselves were spin triplet p-wave and spin singlet d wave, with various possibilities within these categories. A partial signature of these exotic pairings was possible nodes of the superconducting gap on the Fermi surface, either points or lines depending on the particular state, and such nodes would lead to power-law temperature dependences in various physical properties below TC> such as specific heat, for example, in cont.rast to the exponential variation expected weil below Tc in ehe case of BCS. Power laws in specific heat were observed (Fig. 3.18) as weil as in other properties below Tc such as NMR relaxation rate 1/T 1 Tand acoustic attenuation. Also observed were behaviors right at Tc not seen previously in superconductors  having to do with the so-called coherence factors, inclicative of a non-BCS pairing. For example, a peak in acouscic attenuation at Tc was seen.
Shortly after the discovery of heavy fennion superconductivity in UBe 13 a second U -based heavy fem1ion superconductor was cliscovered, UPt 3 with Tc= 0.5 K [27]. This material fom1s in the same crystal structure as the dense Kondo compound CeA1 3 , the hexagonal stacking of Cu 3 Au, this being one reason why it was chosen for study. lt grows as needle-like single crystals from Bi flux with residual resistivity ratios weil in excess of 100, enabling extensive Femu surface invesrigations, and shows well-characterized Landau Fenni liquid behavior. The initial specific heat measurements establishing bulk superconductivity showed a dome-like anomaly at TC> which subsequent measurements on higher resistance ratio crystals established to be due to two closely spaced superconducting transitions (Fig. 3.19). As with UBe 13 , power-law temperacure dependences below Tc in various properties suggested that the superconducting gap had nodes on the Fenni surf.ace. Extensive ultrasonic and thermodynarnic measurements in applied rnagnetic field were conducted on high-quality single cryscals, escablishing a superconducting phase diagram w hich bears some similarities to what is seen in 3 He (Fig. 3.20), wich the presence of multiple superfluid phases. Three superconducting phases appear in the H-T phase diagram. The low-temperacure phase found in zero field has a line of nodes of the gap on the Femu surf.ace in the basal plane and poinc nodes along the  hexagonal z-axis. The detailed description of these phases still rernains to be completely worked out. While the early attention in heavy fermion superconductivity centered on CeCu2Si 2 , UBe 1 3, and UPt 3 , there remained in the background a mysterious superconducting antiferromagnet, UR.u 2 Si 2 , crystallizing in the same structure as CeCu 2 Si 2 . Anriferrornagneric order sets in below 17 .5 K, followed by superconductivity below 1.2 K. The antiferromagnetic order, however, involves only a sublattice magnetization of 0.03µ 6 , in spite of a large heat capacity anomaly at T N· This has led to the suggesrion that the upper 17 .5 K transition involves some kind ofhidden order, which the small ordered moment is a symptom o( Extensive experimental work has been clone on this system, including pressure studies (Fig. 3.21) which find that the hidden order gives way beyond 0.7 GPa to a rnagnetically ordered large moment (0.4µB) phase, with supcrconducrivity only coexisting with the hidden order phase. The electronic specific heat 'Y ~ 180 mJ/mol-U K 2 at TN, with integrated entropy up to TN of ßS~0.2Rln2. At the superconducting Tc, /::::::;60mJ/mol-UK 2 . The 'Y value at TN clearly places this material in the heavy fermion category, and the HO ordered moment in light of the substanrial fraction of R ln 2 developed by T N points to physics differing from that seen in usual antiferromagnetic ordering. One viewpoint on the coexistence of superconductivity with the hidden order is that these two phases are competing for the Fermi surface. This is a quite different viewpoint than that which appears appropriate for the Chevrel phases and rare-earth rhodium borides, where local moment magnerism is thougbt to coexist with superconductivity of conventional nature.
Two other heavy fennion U-based materials supporting both antiferromagnerism and superconductivity occur in the hexagonal CaCu 5 structure: UNi2Al3 (TN=4K, Tc= 1 K) [31)and UPd2Al3(TN=14.5 K, Tc=2K) (32). Their respective 1's at TN are 150 and 125 mJ!mol-UK 2 , respectively. Both appear to have cylindrical pieces of Fenni surface, characteristic ·of 2D behavior. In UNi 2 A1 3 the magnetic order is an incommensurate spin density wave with amplitude 0.2µ 5 ; in UPd 2 A1 3 it is a co mmens~rate antiferromagnetic ordering with ordered moment 0.8µa . In the Ni compound, the superconducting pairing is believed to be of triplet type, in the Pd material singlet.
The idea slowly developed during the 1990s that the heavy fennion superconductors all had a nearby magnetic phase. This idea came from thinking about the effects of pressure on Ce-based materials in the context of the Doniach phase diagram (Fig. 3.22) [33). The coupling of conduction electrons to Ce local moments with strength P]inc produces two energy scales in the dense Kondo lattice, the Kendo scale kaTK-p-1 exp(-1/p]ind and the Ruderman-Kittel-Kasuya-Yoshida (R.KKY) 4f moment-moment interaction ka T RKKY -p .fin~F determining magnetic ordering temperature, where F contains factors specific to the lattice and not dependant on ]inc· For small p};nc the RKKY dominates and magnetic order is the ground state. Beyond some critical value of (p]itic) the Kondo scale will dominate. The simple idea was that superconductivity might be favored in the critical region and that one could access this region in Ce-dense Kondo systems using applied extemal pressure (Fig. 3.23). This proved to be the case. The fust example was the finding of superconduccivity under pressure in the 35 K antiferromagnet CeRh 2 Si 2 below 400 mK near 0.9 GPa, where the Neel temperature had been suppressed close to 0 K (34]. More detailed scudies followed on Celn 3 and CePd 2 Si 2 (Fig. 3.24) (34], supporting the hypothesis that probably all ehe heavy femuon superconductocs might lie in ehe region indicaced in Fig. 3.23. This line of thinking also encouraged the idea that ehe pairing mechanism in heavy fennion superconductivity was of magnetic origin. lt is worth nocing that establishing the pairing mechanism is not directly accessible to experiment but that establishing the superconducting gap structure over the Fermi surface is. The search for superconduccivity near the magnecic quantum critical point in the Doniach phase diagram increased substantially the number of known heavy fennion superconductors. At about the same time, a new syscem of heavy fermion materials was uncovered, ehe isostructural, isoeleccronic sequence CeColn 5 (Tc = 2.3K), CeRhln 5 (TN"."3.9K), and Celrlns (Tc= 0.4 K) [36]. These tetragonal materials are quite easily grown as single crystals from In flux wich very high residual resiscance ratios, allowing extensive detailed studies. Single crystals can also be grown of mixed members of the series (Fig.3.25), and superconductivity is found to coexist wich antiferromagnecism in some ranges of solid solution. Exactly how eo describe chis coexistence is not known, but NMR measurements suggest ehe coexiscence is homogeneous. An interesting observation g .....  [37).
concerning this alloy phase diagram is that there is no coexistence when superconducrivity occurs first on cooling.
Extensive studies have been made on the 3.9-K antiferromagnet CeRhlns as a funcrion of pressure (Fig.3.26) [38]. Heavy fennion superconducrivity develops with pressure, and as in the alloy case, coexists with antiferromagnetism uncil the Tc becomes larger than the T N near 1.8 GPa. Tc . just beyond 2.0 GPa increases to a maximurn of2.3 K, the same Tc as found in CeColn 5 at ambient pressure. One can extrapolate to a T N = 0 K quantum critical point in the phase diagram P2 = 2.3 GPa. And de Haas-van Alphen measurements find that the Fenni surface of CeRhln 5 changes beyond this pressure into a larger one that is similar to that found in CeCoin 5 , from a Fenni surface that is similar to that of LaR.hln 5 and LaColns [39}. The implication of this is that the Ce 4f electron has delocalized and been incorporated into the Fenni surface of both CeCoins at ambient pressure and CeRhln 5 beyond P2 = 2.3 GPa. In this regard, we noce that these materials have a quasicylindrical piece of Fenni surface suggestive of 2D electronic character, plus other 3D Fenni surface pieces. Neutron-scattering measurements do not find characteristics of 2D magnetic correlations in CeRhln 5 . It has been proved possible with doping on the In sites in CeCoin 5 to induced antiferromagnetic order (Fig. 3.27). There are two inequivalent In sites in CeCoin 5 , but X-ray Absorption Fine Structure (XAFS) measurements have shown that Cd substitutes on both these sites. These data demonstrate that superconducting CeCoin 5 lies in fact very close to a magnetically ordered ground state. Also interesting is that pressure can exactly undo the effect of Cd doping (Fig. 3.28). A simple conjectured explanation of this is that Cd substituted for In in CeCoin 5 induces the 4f electrons of its near neighbor Ce ions to becomes more localized, and this more localized Ce is enough to shift ehe balance to a magnetic ground state. Applied pressure will have the largest effect on these more compressible Ce ions, so that pressure is able to easily reverse the effect of Cd doping. What is particularly interesting is chat it is possible wich a simple shift of the pressure axis co superimpose the temperature-pressure phase diagram of pure CeR-hin5 on that of Cd-doped CeColn 5 (40}.  Single crystals of CeColn 5 can be grown with residual resistivities in the few ~ens of the n.0-cm range. This fact has encouraged a search for a modulated superconducting state in applied magnetic field known as the Fulde-Ferrel-Larkin-Ovshinikov (FFLO) (41)state which has never been observed. Theory suggests that it could only be observed in extremely clean material. Experiments in CeCoin 5 (Fig. 3.29) have looked for this state and discovered evidence for a new phase within the superconducting phase. lt is not clear at present whether this is the FFLO phase. In particular, exactly what the definitive diagnostic for the FFLO phase is in a real material is not settled.
A number of actinides fonn compounds with the tetragonal 115 structure. None of the U-based materials wich this strucrure were found to be superconducting ( Fig. 3.30), but remarkably, the first known Pu superconductors were found in it with the astonishing · Tc's of 18.6 K for PuCoGa 5 and 8.5 K for PuRhGa 5 (43]. Subsequently, the first N~-based superconductor was found in a related tetragonal structure with Tc= 4. 9 K, NpPd 5 A}i (44]. These three materials all are moderately enhanced heavy fennion materials with 1-100 mj/ mol K 2 for the Pu materials, 200 mj/ mol K 2 for the Np compound.
One further Ce-based superconductor is CePt 3 Si (45}. This compound is a T N = 2.2 K antiferromagnetic with coexisting superconductivity below Tc=0.75 K. What is novel here is that this is the first example of heavy fem1ion superconductivity in a noncentrosymmetric crystal structure. A number of noncentrosy1runetric superconductors are known that are not heavy fennion materials, and in fact one of them is the isostructural analog of CePt 3  importance of the interplay of noncentrosymmetric ccystal structure and heavy fennion character in this superconductivity. > 6. FERROMAGNETIC SUPERCONDUCTORS W e have seen earlier that ferromagnetism in materials such as ErR.hJ3 4 caused the reentrance of a normal state at temperatures below the ferromagnetic Curie temperature. Various theoretical arguments suggested, however, that in some cases ferromagnetic order might be compatible with superconductivity with superconductivity occurring near the quantum critical point at which the ferromagnetic Curie temperature is manipulated to approach T= 0 K. A search was undertaken to explore this possibility, resulting in the discovecy of two U-based intermetallics where this was found, UGe 2 [46] and isostructural UR.hGe [47].
The fust successful study came from Lonzarich and collaborators on UGe 2 , following an unsuccessful search for superconductivity under pressure in MnSi. The low-temperature pressure/temperature phase diagram for this orthorhombic compound is shown in Fig. 3.31. UGe 2 is an itinerant ferromagnetic with a Curie temperature of 53 K at ambient pressure. The rather modest pressure of 1.6 GPa is sufficient to drive the Curie temperature to 0 K. What is interesting is that the superconductivity shows up inside the ferromagnetic state near where ehe Curie temperature vanishes under 40 ... applied pressure and does not extend beyond the ferromagnetic boundary at 1.6GPa.
In the subsequencly studied isostructural compound URhGe, superconductivicy was found to coexist below 0.25 K with small ordered moment (0.4µB) ferromagnetism with Curie temperature 9.6K. The suggestion here for establishing a congruence between ehe behavior ofURh.Ge and UGe 2 is that UR.h.Ge is at · ambiem pressure close to its magnetic quantum critical point. A comparison of ehe properties of URh.Ge at ambient pressure and UGe 2 at 1.5 GPa makes this argument reasonable. The effect of pressure ( Fig. 3.32) on URhGe is opposite to that of UGe 2 with the Curie temperature increasing with applied pressure. W e note that the electronic specific heat coefficients of UGe 2 and UR.h.Ge are 120 and 160 mJ!mol K 2 , respectively, in the lower heavy ferrnion range. This is still a relatively unexplored area of the magnetic/ superconducting boundary and awaits further experiment.