Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors

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13 Unconventional Superconductivity in Novel Materials 741

tablished in ternary rare earth compounds such as

the rare earth molybdenum chalcogenides RMo

(X = S, Se) and the rare earth rhodium borides

RRh

. Since then about 20 different Ce, Pr, U, and

Pu-based heavy-fermion superconductors have been

discovered and the list will continue to grow. The

past few years have seen renewed interest in heavy-

fermion superconductors with the discovery of the

coexistence of ferromagnetism and superconductiv-

ity in UGe

, URhGe, and, possibly, ZrZn

.Theex-

citing discovery of superconductivity in PrOs

which contains two superconducting phases which

appear to have different order parameter symme-

tries, and may have a quadrupolar origin, suggests

a new route to unconventional superconductivity.

Heavy-fermion superconductors continue tobe a fer-

tile area of research and enhance our understanding

of superconductivity.

The organic metals and superconductors show a

rich variety of novel ground states and cover a broad

spectrum of tunable fundamental physical phenom-

ena which are of current interest. One of the most

studiedbut still very controversial issues is the nature

of the superconductivity.For the 1D Bechgaard salts

a number of recent experiments indicate a triplet

superconducting state. For the 2D organic super-

conductors, the experimental situation is unsettled.

NMR and thermal-conductivity experimentssuggest

that the superconducting order parameter is uncon-

ventional, whereas the measurements of the spe-

cific heat indicate a nodeless gap. In an attempt to

solve this puzzling situation, it was suggested that

an s-wave to d-wave gap-symmetry transition could

be induced by an in-plane magnetic field [538]. In

this theoretical treatment, phonon-mediated super-

conductivity with a small wavevector was assumed,

which could explain such a transition. Although this

proposal would resolve the controversial NMR and

specific heat results, other predictions of this the-

ory such as the orientation of the d-wave gap nodes

are at odds with recent experimental results [528].

Further studies, especially experiments that directly

probe the phase difference of the superconducting

wave function, are highly desirable.

Many of the high T

cuprate superconductors

are derived from antiferromagnetic insulating par-

ent compounds through chemical substitutions that

dope the CuO

planes with electrons or holes. It is

widely believed that the pairing mechanism in these

materials is associated with AFM spin ﬂuctuations.

The cuprates have a very rich and interesting T−x

phase diagram that contains AFM, pseudogap, and

superconducting phases. The pseudogap gap phase

and the superconductivity appear to be intimately

related and have energy gaps with the same symme-

try.The preponderance of evidence indicates that the

superconducting order parameter has d-wave sym-

metry. However, some experiments indicate that the

superconducting order parameter has s-wave sym-

metry or, at least, a significant s-wave component.

The pseudogap temperature T

∗

(x)mayvanishin

the superconducting phase at a QCP that may be

profoundly related to the high T

superconductivity

found in these remarkable materials. An interesting

feature whose role in the high T

superconductivity

of the cuprates is not yet known is electronic in-

homogeneity observed in various experiments (e.g.,

neutron diffraction, scanning STM). Other high T

cuprate superconductors will undoubtedly be dis-

covered, and it is not unreasonable that some will

have values of T

greater than the current maxi-

mum value of 133 K (at atmospheric pressure). The

theory of the normal and superconducting states of

the cuprate superconductors represents a significant

challenge. The compound Sr

RuO

exhibits uncon-

ventionalsuperconductivityand is oneof the leading

candidates for p-wave superconductivity.

13.6 Comparison of the Properties

of Different Classes of N ovel

Superconductors

In this chapter, we have focused on the uncon-

ventional superconductivity that is exhibited by

three classes of strongly correlated electron mate-

rials: heavy fermion f -electron intermetallic com-

pounds, organic compounds, and layered cuprates

and ruthenates. For all three classes of materials,

there is evidence that the unconventional supercon-

ductivity of many of these compounds is anisotropic

with an energy gap that vanishes at points or along

742 M.B. Maple et al.

Fig. 13.106. Schematic tem-

perature T

∗

-dopant con-

centration x and T

∗

-pres-

sure P phase diagram for

the cuprates (a);heavy

fermion f -electron mate-

rials (b)–(e), and organic

compounds (f)–(g)

lines on the Fermi surface. This indicates that the

superconducting electrons are paired in singlet or

triplet spin states with finite angular momentum

L > 0. The strongest evidence for superconduct-

ing electron pairing with L > 0 comes from experi-

ments on the layered perovskite like cuprate high T

superconductors that probe the magnitude as well

as the phase of the superconducting order param-

eter (k). These experiments generally indicate that

(k)hasd

−y

symmetry in the hole-doped cuprates,

although recent twist angle experiments mentioned

above suggest s-wave pairing. As discussed earlier in

this chapter, the situation with the electron-doped

cuprate high T

superconductors is less clear, with

13 Unconventional Superconductivity in Novel Materials 743

some experimentsindicating s-wave pairing andoth-

ers suggesting d-wave pairing; further research will

be required to clarify the situation in this complex

group of cuprate superconductors. For the heavy

fermion f -electron intermetallic compounds and the

organic compounds, a variety of measurements in

the superconducting state that probe the magni-

tude of the superconducting order parameter indi-

cate L > 0 pairing. These experiments are based on

power-law T-dependences of the physical properties

in the superconducting state.Since it has not yet been

possible to perform experiments that are sensitive to

the phase of the superconducting order parameter in

these materials, as was done in the cuprates, the case

for L > 0 pairing in these materials is not as strong

as in the cuprates.

In all three classes of materials, magnetically or-

dered ground states are found in close proximity to

the superconducting ground states.This is illustrated

by the temperature vs composition (T − x)andtem-

perature vs pressure (T − P) phase diagrams of the

high T

cuprate, f -electron heavy fermion, and or-

ganic superconductors shown in Figs. 13.106 (a)–(g).

Although these phase diagrams differ in the extent

to which the magnetic and superconducting phases

overlap,the overall similarity is compelling and sug-

gests that there is some kind of relationship between

magnetism and superconductivity in these materi-

als. In many of the compounds, the curve of T

x or P has an inverted parabolic shape and occurs

near a magnetic quantum critical point, where the

magnetic ﬂuctuations are expected to be especially

strong and extended in temperature and energy,and

are thereby effective in mediating superconducting

electron pairing. This is dramatically illustrated in

the phase diagrams of the heavy fermion compounds

CeIn

and CePd

as depictedin Fig.13.106(b).It is

interesting to note that electron tunneling measure-

ments on the heavy fermion f -electron superconduc-

tor UPd

have revealed a feature in the normal

state conductance at an energy where magnetic ex-

citations have been identified in inelastic neutron

scattering studies on this compound. In addition

to superconductivity, non-Fermi liquid behavior in

the physical properties is found in examples in all

three classes of these materials near magnetic quan-

tum critical points. (The situation is more complex

in the f -electron intermetallic compounds where

non-Fermi liquid behavior in some systems occurs

throughout a large range of concentration and ap-

pears to be associated with some type of unconven-

tional Kondo effect.) Experiments on the new heavy

fermion filled skutterudite compound PrOs

re-

veal unconventional superconductivity.In this com-

pound, the heavy fermion state and the supercon-

ductivity may arise from quadrupolar ﬂuctuations,

rather than magnetic ﬂuctuations that are believed

to be responsible for the superconducting electron

pairing in the unconventional superconductors con-

sideredin this chapter.Thepseudogapandsupercon-

ducting phases appear to be intimately related and

have energy gaps with the same symmetry (appar-

ently, d-wave symmetry). The pseudogap tempera-

ture T

∗

(x) may vanish in the superconducting phase

at a QCP that could be linked to the occurrence of

high T

superconductivity in these remarkable ma-

terials.An interesting featurewhose role,if any,inthe

high T

superconductivityof the cuprates has not yet

established is electronic inhomogeneity observed in

various experiments (e.g., neutron diffraction, scan-

ning STM).

Acknowledgements

We would like to thank N.P. Butch, N.A. Frederick,

T.A.Sayles,and J.D.Thompson forcriticalcomments

regarding this chapter. Invaluable technical assis-

tance in preparing this manuscript was provided by

C. Rosado and C. Rayt. Research support of the U.S.

Department of Energy, the National Science Foun-

dation, and the Deutsche Forschungsgemeinschaft is

gratefully acknowledged.

744 M.B. Maple et al.

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