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

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14 High-T

Superconductivity 807

is not significantly reduced below T

[217].The main

inﬂuence of agap functionwith nodesis again visible

in a non exponential decay of K(T)atlowtempera-

tures.

In Fig. 14.60 a selection of results of Knight shift

measurements involving nuclei on different lattice

sites of YBCO-123 is presented [218,219]. The strong

reduction of K in the superconducting state clearly

reﬂects spin singlet pairing. Together with the infor-

mation suggesting the existence of line nodes in the

gap function, there is strong evidence for a d-wave

type pairing being established in cuprate supercon-

ductors. Theoretical model calculations to account

for the experimental data gave better agreement,if d-

wave rather than s -wave pairing was assumed [191].

One such attempt is shown as the broken line in

Fig. 14.60.

(e) Angle Resolved Photoemission

As was mentioned above, significant progresses

in measurements employing photoemission spec-

troscopy techniques, in particular ARPES with high

Fig. 14.61. Angle dependence of the leading-edge shift in

ARPES spectra of Bi-2212, relative to the position of the

reference Fermi edge of Pt. The solid line represents a cal-

culation assuming a d-wavetypegapfunction(see[87])

energy and momentum resolution, were respon-

sible for the evaluation of the excitation spec-

trum of quasiparticles in cuprate superconductors.

While integrative photoemission has been able to

demonstrate the formation of a gap below T

[85],

ARPES also provides information concerning the k-

dependence of the gap. Since a whole chapter of

this compendium is dedicated to this type of exper-

iments, it suffices for our purposes to present the

results of ARPES measurements probing (k)(see

Fig. 14.61). The fourfold symmetry of the node con-

figuration expected for a gap function with d

−y

symmetryisverywellborneout[87].

(f) Phase Sensitive Measurements

The experiments cited and discussed above all con-

firm that the gap functions in the investigated

cuprate superconductors exhibit nodes, most likely

on lines, and that the pairing state of the quasiparti-

cles is of a spin-singletvariety.All this takentogether,

also taking into account the essential electronic wave

functions in the Cu-O planes and considering vari-

ous theoretical arguments based on phenomenolog-

ical and computational approaches [191] leads to the

conjecture that the pair state and hence the order pa-

rameter as well as the gap function exhibit a d

−y

symmetry. This implies that the amplitude of the or-

der parameter changes sign at the nodes or,in other

words, the phase of the order parameter changes dis-

continuously by . None of the above experiments

is sensitive to this phase change and other types of

measurements have to be invoked to settle this issue.

These measurements have to be devised such that

they probe the relative phase of the gap function on

different parts of the Fermi surface.

Since phase information is required, this type of

experiments is based on the macroscopic quantum

coherence in superconductors,most distinctly man-

ifested in the Josephson effects and the ﬂux quan-

tization. The concept for this kind of tests has first

been put forward by Geshkenbein and Larkin [220],

in connection with probing the unconventional na-

ture of superconductivity of heavy electron metals.

They suggested to investigate the phase coherence

of specially tailored superconducting loops contain-

ing a combination of the superconductor under in-

808 H.R.Ott

vestigation and conventional superconducting ma-

terials. This idea was later adapted by Sigrist and

Rice [221] for a proposal to test the d

−y

character

of the gap function of cuprate superconductors and

the schematic layout of the required superconduct-

ing loop is shown in Fig. 14.62.The idea rests on the

fact that at a Josephson contact between two super-

conductorswith opposite signs of their order param-

eter amplitudes, a so called -junction [222], where

the phase of the order parameter changes discontinu-

ously by ,is generated.It is well known thatthe mag-

netic ﬂux through the cavity of aclosed loopof a sin-

gle conventional superconductor is quantized [223];

theﬂux quantum is ¥

=hc/2e.Itturnsoutthatthere-

sponse to external magnetic fields of loops contain-

ing either an even (including zero) or odd number of

-junctions is significantly different.In the firstcase

the highest stability is achieved if an integer num-

ber of ﬂux quanta ¥ =  · ¥

is trapped in the loop.

In the second case,however,the stability criterion re-

quires that ¥ =( +1/2)·¥

.Essentially two different

types of experiments have been designed to obtain

the requested information. In this more general re-

view,noneof theintricaciesof theseexperimentsand

their analyses will be addressed in detail. This has, to

a large extent,been done in more specialized reviews

on this topic [224,225] (see below).

Fig. 14.62. Schematic layout of a superconducting loop for

phase sensitive tests of the order parameter symmetry of

cuprate superconductors, employing a SQUID type config-

uration (see [221])

(f1) SQUID-Type Experiments

Superconducting quantum interference devices

(SQUIDs) are based on the Josephson effect, i.e., on

the tunneling of pairs as a whole and usually contain

two Josephson contacts in parallel. As the name in-

dicates,SQUIDs offer the possibility of experimental

control of the phase of the macroscopic wave func-

tion describing the superconducting state. For our

purposes the chosen loop may be manufactured in

a SQUID configuration, as is schematically outlined

in Fig. 14.62. The design is such that one, two or

none of the contacts may form as a -junction.The

critical current I

, i.e., the maximum supercurrent,

varies periodically as a function of the total mag-

neticﬂuxinsidetheloop.Thelocationofthemax-

ima or minima of I

depends critically on the phase

coherence around the loop. If the SQUID loop con-

tains none or two -junctions, an I

(¥ ) pattern as

shown in Fig. 14.63a is expected. The I

(¥ )diagram

for a SQUID with only one -contact, however, is

shifted by  along the horizontal ¥ -axis, as shown

in Fig. 14.63(b). The first realization of this type of

SQUID experiments was accomplished by Wollman

and co-workers [226] by studying the critical cur-

rent behavior of SQUIDs that were fabricated from

single crystalline YBCO-123 and Pb.The Pb part was

in the form of a thin film, deposited in two differ-

ent configurations. In the first, the film made two

contacts around the corner of a single crystal, such

that one of them is along the a direction and the other

along the b direction of the crystal lattice.The second

configuration had both contacts onto the same edge

of the crystal, facing the same crystallographic di-

rection.The schematic sample configuration for this

pioneering work is shown in Fig. 14.64. Considering

the crystal structure of YBCO-123 and the expected

−y

symmetry of theorder parameter,the first con-

figuration is expected to contain one contact acting

as a -junction. For the second case either none or

two -junctions are likely tobe established.Measure-

ments of the dynamical resistances of these SQUIDs

as a function of applied magnetic field were used

to monitor the modulation of the I

(¥ )curves.The

analysis of the data indicated a tendency of a -shift

of I

(¥ ) for loops around the corner of the crystals

with respect toI

(¥ ) obtained for edge-typeSQUIDs.

14 High-T

Superconductivity 809

Fig. 14.63. Interference patterns of the critical current I

ofa

superconducting loop as shown schematically in Fig.14.62,

upon varying the magnetic ﬂux through the loop (a) with

an even or (b) an odd number of -junctions (see text)

Fig. 14.64. Schematic view of the experimental realization

of SQUID type configurations forming (a) acornerSQUID

and (b) an edge SQUID for phase aensitive tests of the or-

der parameter symmetry of Y-123 single crystals see [224]

The result was thus regarded as a confirmationof the

−y

character of the order parameter of this mate-

rial. In another approach [227], a direct comparison

of the field induced critical current variation of a hy-

brid SQUID fabricatedfrom single crystallineYBCO-

123 and Nb,expected to contain one -junction,with

(¥ ) of two control SQUIDs made of Nb alone was

made.This comparison of I

(¥ ) of the two configura-

tions, accomplished by simultaneous measurements

of the dynamical resistances, again gave evidence for

a -shift between the I

(¥ )curvesofthetwotypes

of SQUIDs.Additional experiments [228] ofthe same

typeconfirmedthemainresultof thequotedfirsttwo

investigations.

(f2) Single Junction Experiments

The magnetic field induced modulation of the criti-

cal current I

of a single Josephson junction between

aconventionals-wave superconductorand a cuprate

superconductor may also give information concern-

ing the order parameter symmetry of the latter. This

possibility has first been discussed and finally real-

ized again by Wollman and coworkers [229]. They

argued that the I

(¥ ) curves for single Josephson

contacts, either straddling the corner separating the

a and the b direction of a single crystal of YBCO-

123 or placed along the edge of a crystal,would again

differ significantly.They also pointed to some advan-

tages using this technique, avoiding some problems

that are related with the use of the SQUID technique

mentioned in (f1) above.

For extended Josephson junctions between two

conventional superconductors, the critical current is

strongly inﬂuenced by external magnetic fields and

it has since long been established that I

(¥ ),where ¥

is the magnetic ﬂux threading the barrier region of

the contact, exhibits a Fraunhofer type pattern (see

Eq.(14.25)),asshownin Fig.14.65.Forjunctions with

extensions that are still smaller than a characteristic

length, such that self-field effects can be neglected,

the resulting pattern has indeed the shape described

(¥ ) ∼



sin



¥



¥



sin



¥



¥



. (14.25)

810 H.R.Ott

Fig. 14.65. Fraunhofer type interference pattern of an ex-

tended Josephson junction as a function of magnetic ﬂux

through the junction (see Eq.(14.25))

For individual junctions based on contacts between

a conventional superconductor and a cuprate mate-

rial, essentially the same I

behavior is anticipated

for the above mentioned single edge contacts. In case

of an anisotropic order parameter with d

−y

sym-

metry, in contacts straddling a crystal corner in a

more or less symmetrical way, the compensation of

thepositiveandthenegativecurrentsisexpectedto

result in a zero critical current at ¥ =0(mod.)and

hence [229]

(¥ ) ∼



sin



¥

2¥





¥

2¥





sin



¥

2¥





¥

2¥





. (14.26)

This behavior is displayed in Fig. 14.66. Again, con-

tacts along a single edge ought to show a ﬂux depen-

dence according to Eq.(14.25).Since in practice these

corner contacts can never be made exactly symmet-

rical, some asymmetry in the patterns must be antic-

ipated. The key feature, however, remains whether a

dip or a peak in I

(¥ ) is observed for single corner

junctions at ¥ = 0. In Fig. 14.67 the result of one

of this type of experiments between Pb and an un-

derdoped single crystal of YBCO-123 is shown [230],

confirming the earlier results of [229], obtained on

optimally doped YBCO-123. The experimental at-

tempts based on the techniques brieﬂy addressed in

(f1) and (f2) have been reviewed in [224].

Fig. 14.66. Expected schematic interference pattern for a

single Josephson junction straddling the corner of a Y-123

single crystal, assuming a d-wave symmetry of the super-

conducting order parameter (see [229])

Fig. 14.67. Magnetic field dependence of the critical cur-

rent I

of a corner junction between an underdoped Y-123

single crystal and Pb.Theinsetsshow the schematic exper-

imental configuration and the ideal expected pattern. Note

that I

adopts a minimum at zero external magnetic field

(f3) Sc anning SQUID Magnetometry

For a simple closed superconducting loop, the max-

ima of I

at ¥ =  · ¥

imply that the loop may

trap ﬂux quantities given by an integer number of

ﬂux quanta,a situation with zero trapped ﬂux is also

possible.A loop with an odd number of -junctions,

however,will trap ﬂuxes ¥ =( +1/2)·¥

.Thismeans

that in order to stabilize, the loop will always gener-

14 High-T

Superconductivity 811

ate a spontaneous current, thus inducing a magnetic

moment corresponding to at least one half of a ﬂux

quantum if the external field is set to zero.

In order to test this consequence ofan oddnumber

of -junctions in a loop arrangement, Tsuei, Kirtley

and coworkers designed a really ingenious experi-

ment [231]. In their first attempts, they deposited

small ring-shaped films of YBCO-123 on suitably tai-

lored SrTiO

substrates. These substrates were com-

posed of differently oriented single crystals,mended

in such a way that would allow the preparation of dif-

ferent rings, epitaxially deposited on the substrates,

with either an even, including zero, or an odd num-

ber of -junctions. The amount of trapped ﬂux, in-

duced in different ways, was monitored with a scan-

ning micro SQUID probe.It was convincingly shown

that loops with an odd number of -junctions be-

haved differently from those with an even number

or none at all. The original layout of the samples is

schematically shown in Fig. 14.68.With this relative

orientationof thecrystals,therelative orientations of

the d

−y

lobes of the order parameter in the corre-

sponding segments of the YBCO-123 films as shown

in Fig.14.69, areanticipated.With these orientations,

the prediction is that only the loop with its center at

the location where all three crystals meet should ex-

hibit the trapping of half integer ﬂuxes.This is exactly

what the outcome of the experiment confirmed. A

number of additional experiments with either other

choices for the relative orientations of the substrate

crystals [232] or with films of other cuprate super-

conductors, suchas Tl-2201 [233] and Bi-2212 [234],

all confirmed the above result and helped to rule

out extrinsic causes for the observation of half in-

tegerﬂuxeswhereexpected.Thesametypeofex-

periments also settled an earlier debate concerning

the order parameter symmetry of electron doped

cuprate superconductors. For a while, based on pen-

etration depth experiments, such as mentioned in a

previous subsection, it was claimed that the super-

conducting state of this variety of copper oxides has

to be characterized by an s-wave type pairing. Phase

sensitive measurements using the scanning SQUID

method [235] have shown that also electron doped

cuprates adopt a superconducting state with a gap

of d

−y

symmetry. An extensive review covering

Fig. 14.68. Schematic view of superconducting loops of ring

shaped films of Y-123, deposited epitaxially on a tricrystal

substrate with specially chosen orientations. The central

ring is expected to have one -junction (see [231])

Fig. 14.69. Schematic relative orientation of the d-wave

type order parameter contour in the ring shaped films

(see [225])

the experiments based on the scanning SQUID tech-

nique is given in [225].

(f4) Related Experiments and Problems

In connection with these tunneling experiments is

should also be mentioned that not all of them give

straightforward evidences for the pairing and hence

gap anisotropies discussed above.Results of early ex-

periments probing the Josephson tunneling through

grain boundaries [236] were incompatible with d-

wave pairing in the sense that no complete suppres-

sion of the critical current in the relevant geometry

812 H.R.Ott

of two order parameter lobes with different signs op-

posing each other on either side of the grain bound-

ary could be achieved. It has been suggested [237]

thatinthelongjunctionlimit,thejunctionroughness

may play an important role and that the appearance

of local vortices at the grain boundary may weaken

the expected frustration and therefore allow for non

zero Josephson currents,i.e., currents at zero voltage,

in this case.

Somewhat more complicated is the interpretation

of tunneling experiments where the tunneling cur-

rent is directed along the c -direction of the crystal

lattice.Variousexperimentsinvestigating thistype of

arrangement with contacts between Pb and YBCO-

123 single crystals as well as epitaxial films have been

made [238–240], and non zero Josephson currents

through these junctions were observed. In case of a

strict d-wave anisotropy of the pairing in the plane,

such non zero currents should, of course, be absent.

This apparent inconsistency has been ascribed to the

orthorhombicity of the crystal lattice of YBCO-123

and the corresponding loss of the fourfold rotation

symmetry in the plane [241]. This situation allows

for a mixing of d and s symmetry in the pairing, a

scenario that was early suggested in [242].Therefore,

the observation of non zero Josephson currents may

be explained in this way. More detailed discussions

of these issues are available in [224] and [225].

Consequences of d-Wave Pairing

The now fairly well established dominant d-wave

pairing of quasiparticles in cuprate superconductors

doesnot automatically provide a clear cut conclusion

concerning the nature of the pairing interaction. An

unequivocalidentificationof the pairing mechanism

in cuprates is a formidable task, also in view of the

non Fermi liquid behavior. It was believed that, be-

cause of the appearance of gap nodes, at least an

electron phonon mediated pairing could be ruled

out. Recent Eliashberg-type calculations may indi-

cate that this may not necessarily be true [243].

Nevertheless, the earliest suggestions of a d

−y

type pairing symmetry were based on the assump-

tion that the pairing was achieved by exchange of

antiferromagnetic spin excitations. This work has

been reviewed in [191] and [244], see also chapters

by Chubukov et al. and Manske et al. in this volume.

Sinceinthiscaseinthegapequation

(k)=−£

V(k − k



)(k



)

2E(k



)

(14.27)

the interaction V(k − k



)isalwayspositiveandhas

its maxima at the (,)pointsoftheBrillouinzone,

the resulting gap function adopts a d

−y

-type sym-

metry of the form

(k)=

(cos k

−cosk

) . (14.28)

These simple considerations have been comple-

mented by extensive numerical studies cited in [191]

which, for this type of interaction, confirm the sta-

bility of a superconducting state with d

−y

pairing.

Among other consequences, the d-wave pairing

also has some direct inﬂuence on the configuration

of the vortices in the mixed state [245,246]. First

of all, singular vortices are no longer expected to

exhibit a circular cross section. Second, the vortex

lattice, exhibiting a hexagonal symmetry in conven-

tional superconductors, has been shown experimen-

tally [247,248] to adopt a rhombohedral symmetry

where the two unit cell vectors are of approximately

equal length, forming an angle of 77 degrees. One

may argue that this again is a consequence of the

d-wave pairing, considering both the distorted cross

section of the single vortices and possible resulting

changes in the interaction between the vortices.

In view of some isotope effect experiments, it

seems that interactions between itinerant quasipar-

ticles and the lattice are of some importance for the

superconductivity in cuprates as well. In general it

is assumed that electron phonon interactions are

not favorable for d-wave pairing. Therefore, it has

been proposed that the pairing or gap symmetry is

a combination mainly of d-wave and s-wave charac-

ter [242]. Even to date the situation is far from being

clear.

14.4.3 Coexistence of Supercond uctivity

and Magnetic Order

The phenomenon of coexistence of magnetic order

and superconductivity was long thought to be im-

14 High-T

Superconductivity 813

possible, especially in view of the pair breaking in-

ﬂuence of magnetic fields and magnetic impurities

established for conventional superconductors [249].

Nevertheless, the first examples of the coexistence

of antiferromagnetic order and superconductivity

were reported in the mid seventies in two classes of

compounds, the Chevrel phases (molybdenum sul-

fides and selenides) [250] and the Rhodium–Boron

compounds [251]. In these series, compounds with

rare-earth ionsoccupying regular latticesites may be

prepared. Because of the incompletely filled 4f elec-

tron shells, most of these ions carry respectable ionic

moments resulting from spin and orbital degrees of

freedom. Since these materials are metallic, it is no

surprise that these moments order spontaneously at

low temperatures via the Ruderman–Kittel–Kasuya–

Yoshida (RKKY) interaction mediated by the con-

duction electrons [252]. What is surprising here is

that both the N´eel temperature, fixing the onset of

antiferromagnetic order, and T

for superconductiv-

ity are of similar magnitude. This situation has been

studied in great detail for a number of different ma-

terials and reviews of the main results are available

in [253–255], see als chapters by Kuli´c and Buzdin in

volume one of this book.

As we mentioned in Sect. 14.2.2, some cuprates

with rare-earth ions on regular lattice sites may also

be synthesized in the classes of YBCO-123 andYBCO-

124 compounds.Most of the rare-earth elements may

be used to replace Y in these materials. At optimal

doping conditions, i.e., close to seven and eight oxy-

gen atoms per unit formula, respectively, the val-

ues of the critical temperatures of these rare-earth

cuprates are very close to those of the corresponding

Y compounds. In this sense there are no significant

pair breaking effects to speak of. The observation

of antiferromagnetic-type of ordering has been re-

ported for some of them. The ordering temperatures

are rather low and reach a maximum of approxi-

mately 2.2 K for fully oxygenated GdBCO-123 [256].

In most of theother of thesecompoundsthe ordering

sets in below 1 K [257].

It is most likely that both the coexistence phe-

nomenon and the low magnetic transition tempera-

tures are due to the rare-earth site in the YBCO-123

crystal lattice which is situated between the two es-

sential Cu-O planes (see Fig. 14.6(a)). These sites are

electronically quite well separated from the charge

carrier system in the planes and therefore have little

inﬂuence on superconductivity. Likewise the RKKY

interaction between these ions is, for the same rea-

son, rather weak. Together with the reduced num-

ber of available excited quasiparticles this implies a

priori a low ordering temperature. It turns out that

the concentration of itinerant charge carriers at the

rare-earth sites is so low that it is doubtful whether

the RKKY interaction is of any significance at all.

The N´eel temperature for antiferromagnetic order

in HoBa

is of the order of 0.1 K [258], most

likely induced by the hyperfine field of the Ho nu-

clei. This is no surprise since the CEF splitting of

the Ho

Hund’s rule J = 8 multiplet leaves a singlet

ground state and the ordered moments of approxi-

mately 2

/Ho ion turn out to be much reduced from

thefreeionvalue.

14.5 Physical Properties of Non-Cuprate

High-T

Superconductors

In this section, some selected physical properties of

superconductors with unusually high critical tem-

peratures are presented and discussed.None of them,

reaches values that exceed the boiling point of liquid

nitrogen. Nevertheless, the discovery of supercon-

ductivity of these materials and the magnitude of T

were and in some cases still are rather surprising,

and therefore of interest.

14.5.1 Ba-Based Oxides

BaPbO

, in spite of in principle compensating va-

lences of its chemical constituents, is metallic. The

electrical resistivity is almost temperature indepen-

dent and of the order of a few hundred §cm. Grad-

ually replacing Pb by Bi leads to a structural tran-

sition if about 10% of the Pb content is replaced

by Bi. For BaPb

1−x

, the electrical resistivity  in-

creases continuously with increasing x but still stays

more or less temperature independent [51]. Accord-

ing to measurements of the Hall constant R

[51],the

charge carrier concentration increases with x, pass-

ing through a maximum at x ≈ 0.25, which appears

814 H.R.Ott

to be the optimal doping situation for the occurrence

of superconductivity. At this x value, the density of

itinerant charge carriers and the sign of R

are com-

patible with one extra itinerant electron per Bi atom.

Conventionally viewed, this concentration of con-

duction electrons is rather low for a superconductor

with a T

that exceeds 10 K. In contrast to other ox-

ide superconductors, in particular also the cuprate

superconductors, the bulk magnetic susceptibility 

of the normal metallic phase of BaPb

1−x

is neg-

ative and largest in magnitude at x ≈ 0.25. This

suggests that the paramagnetic contribution to  is

weakestatthiscomposition.

Asshownin Fig.14.70,clearresistivetransitionsto

superconductivity are observed for 0.15 < x < 0.35,

but the sharpest transitions at maximum values for

are observed for x =0.25 [51, 259]. A further

increase of x provokes a reduction of T

and even-

tually leads, as indicated by both  (see Fig. 14.70)

and R

data, to a metal insulator transition which

is accompanied by the already mentioned structural

transition back to an orthorhombicsymmetry of the

crystal structure.

Investigations of the superconducting state [52]

revealed that (BPBO) BaPb

0.75

0.25

is an extreme

type II superconductor with a Ginzburg Landau pa-

Fig. 14.70. Variation of the temperature dependence of the

electrical resistivity of BaPb

1−x

(see [51])

rameter  of the order of 70÷80. It appears that the

electronic specific parameter  , estimated from a

barely resolvable specific heat anomaly at T

and us-

ing Eq. (14.6), is rather small, i.e.,  ≈ 1.5 mJ/molK

indicating a low density of electronic states D(E

)at

the Fermi energy, usually unfavorable for high crit-

ical temperatures for the onset of superconductiv-

ity.Tunneling data seem to indicate strong electron–

phonon interactions extending in energy into the

range of the superconducting gap in the excitation

spectrum [52,259].If strong coupling effects are im-

portant, the  value quoted above is significantly re-

duced and D(E

) turns out to be even smaller. From

the theoretical point of view [260] there is no con-

sensus, how the relatively high critical temperature

is to be explained, because the strong-coupling con-

cept is not compatible with band structure calcula-

tions[261] and their connection with the metal to in-

sulator transition mentioned above. In addition, the

ratio 2

(0)/k

has been estimated to be 3.5, prac-

tically identical with the weak coupling BCS value.

Some problems are also encountered in the inter-

pretation of measurements of the upper critical field

. In the dirty limit, a reasonable choice consid-

ering the resistivity data displayed in Fig. 14.70, the

limiting orbital critical field is given by [262]

(0) = −0.691



∂H

∂T



· T

. (14.29)

It appears that the extrapolated experimental limit

of H

(0), based on measurements of H

(T) [52],

shown in Fig. 14.71, by far exceeds the one that is

calculated from the measured initial temperature

dependence of H

just below T

, indicated by the

broken line in Fig. 14.71. A clear result is, however,

obtained from measurements of the isotope effect

on T

by exchanging

Oand

Oontheoxygen

sites [263]. The heavier oxygen isotope leads to a

significant reduction of the critical temperature and

the value of the parameter ˛ entering Eq. (14.10)

has been calculated to be ˛ =0.22. Although this

result seems to confirm that most likely the occur-

rence of superconductivity in the Pb/Bi oxides is

due to an electron phonon interaction, it has been

argued that electronic excitations involving atomic

displacements due to charge redistributions are me-

14 High-T

Superconductivity 815

diating the pairing interaction[263].This conclusion

is based on considering the relatively high value of

and indications that pairing in BPBO is a weak

coupling effect.

The observation of critical temperatures that are

more than twice as large as those of the Pb/Bi alloys

and comparable to some of the cuprate supercon-

ductors, i.e., of the order of 30 K, for Ba

0.6

0.4

BiO

(BKBO) [54,55], naturally led to additional experi-

ments probing the physical properties of this com-

pound [263, 264]. In particular, a comparison with

some of the properties of the cuprate superconduc-

tors was attempted. The total normal state magnetic

susceptibility again reveals a strong core diamag-

netism which, overall, masks a very weak paramag-

netic contribution[263,264].Normal state Hall effect

measurements [264] revealed that R

is weakly tem-

perature dependent and negative at all temperatures

below 300 K. This confirms that the Bi-based oxides,

without Cu,however,are electronically and magneti-

cally quite different species than the hole doped cop-

per oxides.

Fig. 14.71. Temperature dependence of the upper critical

field H

of BaPb

0.75

0.25

.Thebroken line is to guide the

eye (see [52])

The response of the superconducting state to ex-

ternal magnetic fields is rather similar to that of the

Pb/Bi alloys, leading to almost the same values of

(∂H

/∂T)

,oftheorderof−5kOe/K.Estimatesof

theGinzburg–Landau parametergive ≈ 45,again a

large value. As for the Pb/Bi alloys this is compatible

with a low density of itinerant carriers and conse-

quently a large penetration depth .Fromavailable

experimental results covering the upper and lower

critical fields,H

(T)andH

(T),estimates of theini-

tial temperature dependence of the thermodynami-

cal critical field (∂H

/∂T)

can be made. Finally, us-

ing the law of corresponding statesallowsto calculate

the electronic specific heat parameter  or effective

density of states D(E

). Although somewhat larger

than the corresponding value calculatedfor the Pb/Bi

alloy,also D(E

) for the Ba/K alloy is much too small

to account for a T

of the order of 30 K in any con-

ventional way. Nevertheless, also for this material, a

significant oxygen isotope effect confirms that some

type of electron phonon interaction must be impor-

tant for the onset of superconductivity [263]. The

values for the ˛ parameter differ somewhat in dif-

ferent reports; ˛-values between 0.21 and 0.35 have

beencalculated[263,264].Again,conventionalstrong

coupling effects have been ruled out.

For (Ba/K)BiO

, measurements aiming at prob-

ing the symmetry of the superconducting state have

been made [265].Corner-junctiontype experiments,

exploring the field dependence of the critical cur-

rent I

(¥ )(see Eq. (14.26)), indicate that the super-

conducting state of this alloy is characterized by a

nodeless gap function or order parameter.

14.5.2 C

-Based Materials

As we have brieﬂy mentioned in Sect. 14.3.3 the sim-

plest way to achieve a metallic state in bulk C

-based

material is via electron doping with atoms of the

alkaline metal elements. Most of the experimental

results that are brieﬂy discussed below have been

obtained for compounds of the type A

.Some

combinations of this type turn out to be electrical

insulators. Based on calculations of the widths W

of the essential electronic energy bands and the on-

site Coulomb correlation energies U,indicating that

816 H.R.Ott

Fig. 14.72. Temperature dependence of the electrical resis-

tivityof thin films of K

and Rb

(see [269])

the ration U/W significantly exceeds the value of

1, it is argued that the metallic species are on the

metallic side of a metal insulator transition of Mott–

Hubbard type and therefore, correlation effects have

to be taken into account [266,267].Anotherdifficulty

arises, because the energies of the relevant phonon

branches are not very different from W and hence,

invoking Migdal’s theorem [268] for justifying the

electron pairing via a retarded electron–phonon in-

teraction is questionable.

The absolute values of the electrical resistivities 

are rather high and correspond to electronic mean

free paths of the order or even less than the separa-

tion between single molecules. Examples of (T)of

superconducting A

films are shown in Fig. 14.72

[269]. Even at very low temperatures and for single

crystalline samples, the absolute values of  are of

the order of 1 m§ cm. In most cases the low temper-

ature electrical resistivities vary as T

[270] but also

linear-in-T or T-independent variations down to low

temperatures, depending on the sample configura-

tion (film, crystalline, granular) have been reported.

It is therefore clear that a theoretical treatment of this

transport property is difficult,even more so, because

it seems to matter whether (T) is measured at con-

stant volume or at constant pressure. Nevertheless,

calculations invoking the interactions between con-

duction electrons and various lattice modes may be

brought into reasonable agreement between exper-

iments, carried out under constant pressure condi-

tions, revealing  ∼ T

, and theory. A renormaliza-

tion of the same data to constant volume conditions

reveals that  varies approximately linearly with T

above 100 K [271]. This result has apparently not yet

found a theoretical explanation. Although the very

short mean free path at temperatures of the order of

300 K might suggest a saturation behavior in (T)at

even more elevated temperatures, this phenomenon

is, very much like in the case of the cuprates, not ob-

served. In spite of the similarity of these features in

(T) of the two types of materials, it seems quite un-

likely that the same scattering processes are the cause

for this observation.

Quite generally, the magnetic susceptibility  in

the normal state is positive and almost temperature

independent [272],as is also found for the resonance

shift in NMR measurements [273]. Comparing the

magnitude of experimental data with  values cal-

culated from band structure data suggests that a

rather substantial enhancement of  must be con-

sidered [272]. The corresponding density of states

D(E

) (see Eq. (14.4)) is larger by a factor of 3 than

the bare density of states D

) that follows from

the band structure, but in fair agreement with esti-

mates of D(E

) based on NMR spin-lattice relaxation

data [273].Asexpected formetals,the spin-latticere-

laxation rate T

−1

is approximately linear in T in the

normal state, however, the hyperfine coupling con-

stant is not known very well.

Apart from the clear manifestations of the super-

conducting transition in (T)and(T)thetransi-

tion is, as usual, also manifest in experimental data

of a number of other properties. This is certainly the

case for measurements probing the specific heat.Be-

cause of the fragility of the materials, especially with

respectto airexposure,thesemeasurements[272] are

technically not trivial.Asdemonstrated in Fig. 14.73,

the specific heat anomaly at T

is masked by a large

background of non electronic origin, but estimates

concerning its magnitude can still be made. For

, the ratio of C(T

)/T

is of the order of 70

mJ/mole K

, a rather small value but confirming the

rather low effective electronic density of states at E