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 787

quasiparticle excitation spectra, especially in con-

nection with superconductivity [112]. With respect

to the present problem we quote the results of some

recent measurements employing the technique of

scanning tunneling microscopy (STM),providing the

possibility of local tunneling experiments. Informa-

tion about gap features is obtained from measure-

ments of the tunneling current I as a function of

the applied voltage V and by plotting the tunneling

conductance ∂I/∂V versus V. The voltage depen-

dence of the conductance is, to some extent, repre-

senting the energy dependence of D(E

), the density

of states for quasiparticle excitations at the Fermi en-

ergy. Early STM experiments [113], again on single

crystals of Bi-2212 compounds, confirmed the pres-

ence of a pseudogap in the normal state, but now

it was claimed that this feature was present for un-

derdoped and overdoped material [114] and that the

temperature dependence of the experimental ∂I/∂V

Fig. 14.32. Temperature dependence of the normalized con-

ductance vs. voltage of overdoped Bi-2201,as measured by

STM.The broken line represents the background at 82 K to

which the curves are normalized. The gap feature persists

to temperatures substantially exceeding T

(see [115])

versus V curves was such that the pseudogap had

to be a precursor of the genuine superconducting

gap. More recent experiments on the single Cu-O

layer material Bi-2201 confirmed this view [115]. In

Fig.14.32we show the reported resultsof the temper-

ature evolution of conductance versus voltage curves

for overdoped Bi

2.1

1.9

CuO

6+d

. It may well be seen

that the reduction of the conductance at low energies

that is believed to indicate a gap feature persists to

temperatures well above the critical temperature of

the bulk. An intimate relation between the two gaps

in either the normal or the superconducting state is

also deduced from the fact that the gap width is more

or less temperature independent [113]. In addition it

has been pointed out that the gap-width variations

from one material to the other is the same for the su-

perconducting and the normal state, again suggest-

ing a common origin of the two features. The data

imply a filling of the gap with increasing tempera-

ture, rather than the commonly expected narrowing.

Aspointedout above,the STM results seem to con-

firm some of the conclusions that are drawn from

the ARPES data but in addition and unlike a number

of other experiments, they claim the persistence of

the normal state pseudogap well into the overdoped

regime.

14.4.2 Superconductivity

General Remarks

The unexpected occurrence of superconductivity in

these cuprate materials at unprecedented high tem-

peratures almost immediately raised questions with

regard to possible mechanisms that might trigger

these transitions. Most researchers would agree that

this issueis not yet definitely settled, as also the theo-

retical chapters of this treatise may demonstrate.The

main controversy addresses the question whether

lattice degrees of freedom are in any way involved in

theoccurrence of superconductivity in copperoxides

or whether this phenomenon is entirely based on in-

teractions within the,however anomalous,electronic

subsystem. Early ﬂux-trapping experiments [116]

made it clear, at least, that the objects responsible for

the superconducting state carry a charge of twice the

charge of a singleelectron,i.e.,2|e|,justastheCooper

788 H.R.Ott

pairs in conventional superconductors. As has long

been known, of course, pairing states need not be

of the simplest variety with zero orbital momentum

and a spin singlet configuration. Although of little

interest at first, the pairing configuration became an

issue in the wake of some theoretical debates,andwas

finally addressed experimentally by a number of dif-

ferent types of measurements.Another phenomenon

that is intimately related with superconductivityand

is not observed for normal conductors, is the screen-

ing of static external magnetic fields H from the in-

terior of a singly connected superconducting speci-

men, i.e., the decay of H over a characteristic length,

the London penetration depth 

.Earlyexperiments

using muon-spin rotation (SR) techniques estab-

lished beyond any doubts that also for the cuprate

superconductors this field expulsion, the Meissner–

Ochsenfeld effect, is present [117].

The major difficulty, both in experiment and in

theory, however, is still the unequivocal identifica-

tion of the objects that are to be paired. At least

part of our discussion concerning the normal state

properties have made it clear that these objects are

not yet really well defined. Because of the structural

characteristics of the cuprates and the correspond-

ing anisotropies,alarge part of the discussion of their

superconductivity concentrated on the properties of

the Cu-O planes and the possible inﬂuence of the

two-dimensionality of these systems.In what follows

we try to achieve a dense presentation of some of

the most significant aspects concerning supercon-

ductivity in Cu oxide materials, now concentrating

on the characterization of the superconducting state.

A lot of experimental evidence has been accumulated

that the superconductivity of cuprates is of uncon-

ventional nature.Above we have already emphasized

the importance of the concentration n

of itinerant

charge carriers, dictated by the degree of doping the

electrically insulating parent compounds and its in-

ﬂuence on the magnitude of the critical tempera-

ture. That the behavior of T

) is, to some extent,

quite universal,as may be seen in Fig.14.33,has been

pointed out by Uemura[118].Themagnitude of T

is,

to a large extent, also dictated by the number n of Cu-

O planes per unit cell.Itturns out that for one class of

materials, T

may be enhanced by enhancing n,but

Fig. 14.33. The variation of the critical temperature T

cuprate superconductors vs the low temperature SR re-

laxation rate  .Theclosed and open triangles arefor La-214

compounds, the closed and open circles are for Y-123 ma-

terial, the crosses represent data of Bi-2212 material and

the closed diamonds are for Bi-2223 compounds.The solid

lines are to guide the eye (see [118])

Fig. 14.34. Variation of the critical temperature T

of mer-

cury based superconductors HgBa

n−1

2n+2+ı

as a

function of n, the number of Cu-O planes (see [39])

14 High-T

Superconductivity 789

this trend is usually only obeyed up to n =3.This

latter trend has very nicely been demonstrated for

Hg-based cuprates,as may be seen in Fig. 14.34 [39].

Inspecting the behavior of electron or hole doped

bulk cuprate superconductors,it seems obvious that

hole doping of insulating cuprates results in signifi-

cantly higher values of T

than electron doping. This

naturally forces the question in what way the degree

of the occupation of electronic states contributing

to electronic conduction inﬂuences the magnitude

of the critical temperature for the onset of super-

conductivity in general. In this context it is at least

amusing to note that even in the case of supercon-

ductivity of common metal elements most of them

have a normal-state Hall constant with a positive

sign, which is typical for hole dominated conduc-

tivity.Atheoretical model,emphasizing“hole super-

conductivity”, has been promoted by Hirsch and co-

workers [119].

As most non-elemental superconductors, the

cuprates are type II superconductors with extremely

large ratios H

between the upper and the

Fig. 14.35. Negative magnetization of a ceramic sample

of superconducting YBa

7−ı

as a function of applied

magnetic field up to 100 kOe. The arrows indicate the di-

rection of the field variation

lower critical fields or, in other words, resulting in

a very large value of the Ginzburg–Landau param-

eter  = /. This may qualitatively be recognized

from magnetization curves M(H), such as the one

displayed in Fig. 14.35 for YBCO-123 [120]. Taking

into account that by definition



4



ln  (14.7)

and

2

, (14.8)

the shape of the curve readily implies that the pen-

etration depth 

for magnetic fields is much larger

than the superconducting coherence length .For

various reasons, the experimental evaluations of

these critical fields of cuprate superconductors are

not very precise or even impossible.In particular,H

is often regarded as a crossover field only, without

manifesting a true phase transition. Because of these

uncertainties we refrain from presentations and dis-

cussions of critical field data and their temperature

dependence. However, both the magnetic-field pen-

etration depth 

and the coherence length  may be

evaluated with an adequate certainty from M(T, H)

data[121,122].Itturnsoutthatthesetwo characteris-

tic lengths are very anisotropic, of no surprise when

considering the structural properties of these mate-

rials. This also means that the magnitudes of the crit-

ical fields for magnetic-field orientations parallel or

perpendicular to the Cu-O planes are very different.

In most cases, the in-plane components of the two

lengths that are of interest.Since the values for the in-

plane penetration depths are between 1000 and 3000

Å for most of these cuprates [123], it is clear that the

in-plane coherence lengths are of the order of a few

in-plane interatomic distances, i.e., extremely short.

The situation is even more extreme for the direction

perpendicular to the Cu-O plane.Here,

may of the

order of a few 10

Åand is barely as long as an

out of plane interatomic distance or even less. These

anisotropies, often expressed by the ratio  =(m

)

1/2

between the effective masses entering the

penetration depth equation, and the short coherence

lengths create a situation that is very different from

that encountered in conventional superconductors

and, as we shall see, provoke special features in some

790 H.R.Ott

of the observations to be discussed below. Because

of the short coherence lengths, the coherence vol-

ume is rather small and may host only a low number

of Cooper pairs. This situation favors large ﬂuctua-

tion effects; their importance is discussed in a special

chapter of this book.

Pressure Effect, Isotope Effect

So far we have only considered T

variations induced

by changes of the chemical composition of the mate-

rials. Other external inﬂuences, apart from applying

external magnetic fields, that may affect the critical

temperature, are the application of external pressure

or replacing part of the atoms with other isotopes of

the same element. Historically, the strong enhance-

ment of T

of the first La-based copper oxide super-

conductors of the 214 family upon applying exter-

nal pressure [124] led eventually to the discovery of

the YBCO-123 superconductor, by following up the

idea that the smaller ionic radius of the Y

ions

might enhance the internal pressure of La

-based

compounds and thus lead to higher critical tempera-

tures [5]. Since the reduction of the volume by exert-

ing external pressure may affect the spectra of both

the lattice and the electronic excitations, direct and

chemical pressure may not always induce equivalent

variations of physical properties. In most common

superconductors, external pressure usually reduces

and this trend is interpreted as being due to a

pressure induced weakening of the electron phonon

coupling constant 

which in essence determines

the magnitude of T

of materials where supercon-

ductivity is due to the electron phonon interaction.

It has been shown [125] that to a good approximation



(MŸ

)

, (14.9)

whereAisaconstantforagivencrystalstructure,

M is the ionic mass and Ÿ

is the Debye tempera-

ture. The pressure induced stiffening of the lattice

and the corresponding enhancement of Ÿ

thus ac-

count forthe observed decrease of T

with increasing

pressure. Exceptions from this trend of ∂T

/∂p < 0

areonlyafew,themostprominentbeingthatofLa

metal where T

increases considerably with increas-

ing pressure [126],and always have been considered

as indicating some anomalous pressure response of

the electronic subsystem.Thus,in relation with com-

mon superconductors, the generally negative inﬂu-

ence of external pressure on T

is usually interpreted

as confirming the importance of the electron phonon

interaction for triggering the phase transition.

For optimally doped Cu-O superconductors, the

pressure effect on T

is, in most cases, negligible.

Prominent exceptions are YBCO-124 and Hg-1223,

the compound with the highest yet achieved criti-

cal temperature under ambient pressure conditions.

An unusually strong enhancement of T

from 80 K

at ambient pressure to 108 K at 10 GPa has been re-

ported for the YBCO-124 compound [127] and was

interpreted as being due to a sizeable shortening of

the Cu2-O1 bond, leading to a charge transfer into

the Cu-O planes. Also for Hg-1223, the application

of external pressure raises T

considerably.Although

different claims for the T

enhancement, up to 160

K at pressures between 150 and 300 kbar, have been

made [128–130], it is undisputed that the pressure

induced enhancement is significant and can appar-

ently not be achieved by means of variations of the

chemical composition. It has been argued that the

three Cu-O planes per unit cell of this compound are

not all optimally doped at ambient pressure and that

the situation can only be improved by a pressure in-

duced transfer of charge carriers between the planes,

not achievable by chemical means. This automati-

cally means that the large positive value of ∂T

/∂p is

most likely not a lattice effect and certainly cannot

be used to make a case for a significant inﬂuence of

an electron phonon interaction in the creation of a

superconducting state.

Much more clean in regard of identifying the sig-

nificance of lattice effects being involved in the su-

perconductivity of any material is the controlled ex-

change of isotopes of some or all the atoms involved

in the chemical composition of a superconductor,be-

cause the different ionic masses will only affect the

lattice properties of the material. Indeed, measure-

ments of the isotope effect on T

gave the first un-

equivocal experimental evidence for the importance

of the lattice dynamics on superconductivity in sim-

14 High-T

Superconductivity 791

ple metals [131, 132]. For elements the situation is

particularly simple because

∼ M

−˛

(14.10)

and in first approximation [133] it is expected that

˛ =1/2. This result has been verified in a number

of cases [134], but deviations from this most simple

behavior, the most drastic for ˛-U, implying neg-

ative values of ˛ [135], have been identified even

for elemental superconductors. The interpretation

of isotope effect experiments is increasingly more

complicated, if the chemical composition of a mate-

rial involves four or more constituents, as is the case

for many high-T

superconductors. Nevertheless,the

presence of an isotope effect on T

almost inevitably

indicatesthe inﬂuence,evenif oflimitedsignificance,

of the crystal lattice on the mechanism triggeringthe

superconductivity.

Reliable results on the isotope effect on T

cuprates are difficult to achieve.This is because apart

from exchanging the chosen isotope,anyother varia-

tionsin the composition ofthematerial and its struc-

tural properties must be avoided. In copper oxide

compounds, the element of choice for a controlled

isotope exchange is, of course, oxygen. Careful work

in exchanging

Owith

O and back has been made

on La

2−x

CuO

with 0.1 ≤ x ≤ 0.15 [136]. From

these experiments it is concluded that the compound

with

O exhibitsa distinctlylower T

than if the oxy-

genation is accomplished with

O. While the anal-

ysis of the measurements indicates that there is no

isotope effect on the charge carrier concentration, as

one might expect, a significant shift of the effective

mass of the carriers, which are identified as being of

polaronic character, is deduced. Since similar results

have been obtained in a number of other analogous

investigations [137, 138], it has to be accepted that

at least in some of the high-T

superconductors, the

isotope effect is not negligible.

More recently it has been claimed that iso-

tope variations are also affecting the onset tem-

perature T

∗

of the pseudogap formation discussed

above. The corresponding experiments involved the

measurements of linewidths of crystal field exci-

tations probed by inelastic neutron scattering on

HoBa

[139]. While the isotope effect on the

critical temperature is, in accordance with other

work, found to be rather small with ˛ ≈ 0.05,

the variation of T

∗

with the exchange of

Oand

O is considerable and of opposite sign, i.e., T

∗

the

O variety of the compound is distinctly higher

than T

∗

of the

O material. The corresponding ˛

parameter in T

∗

∼ M

−˛

is therefore negative. This

again seems to indicate that the lattice dynamics are,

to a certain extent, involved in the formation of the

pseudogap of the normal state. It may well be that

understanding the significanceof the sign difference

in the ˛ parameters for T

and T

∗

will also clar-

ify the relation between the pseudogap of the nor-

mal state and superconductivity below T

.Inviewof

these experimental results it seems difficult to deny

that superconductivity in copper oxides involves, to

some extent, the lattice dynamics of these materials.

In this regard it has been argued [140] that these

lattice dynamics lead to structural inhomogeneities

in these materials, better known under the names of

phase separation and stripe formation. Microscopic

evidence for stripe formation in some of these ma-

terials has definitely been obtained [141, 142], but

it is not yet clear to what extent this phenomenon

is essential for the occurrence of superconductivity

at elevated temperatures. In interpreting experimen-

tal data obtained with different techniques it should

be kept in mind that the process of stripe forma-

tion may also be highly dynamical in the sense that

the structural inhomogeneities are not static. This

might explain some controversies that arise in com-

paring experimental data from measurements that

probe in different energy or time windows. It seems

clear that structural inhomogeneities will also gen-

erate electronic inhomogeneities and it has not yet

been worked out how these have to be taken into

account in the interpretation of some of the anoma-

lous features of the normal state properties discussed

above.

Phase Transition Asp ects

The short coherence lengths, even within the Cu-O

planes, of the cuprate superconductors, play an es-

sential role in all experimental verifications of the

superconducting phase transition. This is mainly be-

792 H.R.Ott

cause even in zero magnetic field,ﬂuctuation effects

are non negligible to an extent that distinct devia-

tions from the mean field behavior,usually observed

for clean common superconductors, are observed.

The fundamental parameter governing the strength

of these ﬂuctuationsis the Ginzburg number [143]

Gi =



(0)





, (14.11)

where 

and 

are the zero temperature coher-

ence lengths parallel and perpendicular to the Cu-

O planes, respectively. This number, which is obvi-

ously large if the  values are small, fixes the temper-

ature range around the critical temperature where

ﬂuctuation effects are significant, i.e., critical, via

|T − T

| < T

·Gi. For common superconductors,

Gi is of the order of 10

−6

or less and therefore, de-

viations from mean field behavior are usually not

important. The situation is completely different in

cuprates and further complications arise,because of

quasi two-dimensional ﬂuctuation effects in partic-

ularly anisotropic cuprate systems. In that case the

relevant Ginzburg number is [143]

√

2 · "d

, (14.12)

where " =(¥

/4)

and d is the thickness of one

layer. Fluctuation effects in the temperature depen-

dences of the electrical resistivity (T) and the mag-

netization M(T) are somewhat annoying because

they mainly hamper an exact evaluation of the zero

field critical temperature. In Fig. 14.36 we show an

Fig. 14.36. Temperature dependence of the electrical re-

sistivity of Bi-2212. Note the growing slope of ∂/∂T far

above T

example of (T) of Bi-2212 which demonstrates the

gradual decrease of  above T

.Athoroughdiscus-

sion of ﬂuctuation effects on the electrical trans-

port is beyond the scope of this more general re-

view. Instead we brieﬂy consider the temperature

dependence of the specific heat C(T)inzeromag-

netic field and in the vicinityof T

.As an example we

first present the result of a relevant experiment on

single crystalline YBCO-123 in Fig. 14.37, revealing

the anomaly in C(T) at the transition [144]. As may

be seen from the scale on the left vertical border of

the figure, this anomaly rides on a large background

which at temperatures above the boilingpoint of liq-

uid nitrogen, is due to the specific heat of the lattice.

In order to achieve the experimental resolution that

is demonstrated here and which is necessary to ob-

tainany reliablecomparison withtheoretical models,

special methods of measurement have to be involved.

One particular approach,a differential thermal anal-

ysis (DTA) adapted for measurements on small sin-

gle crystals,is described in [145].Fromthe data close

to T

both in the normal and the superconducting

state, clear deviations from ideal mean field behav-

ior may be recognized.The solid line in Fig. 14.37 is

a fit on the basis of the 3D XY model for phase tran-

sitions [146]. A detailed discussion of the relevance

of this model with respect to the critical regime of

the zero field phase transition of cuprate supercon-

Fig. 14.37. Specific heat anomaly at the superconducting

transition of single crystalline YBa

7−ı

in zero mag-

netic field.The solid line is a fit based on the 3D XY model,

as explained in the text

14 High-T

Superconductivity 793

ductors has been given in [147]. Here we only note

that the experimental data can reasonably well be

described on the basis of this model. The analysis is

by no means straightforward. The universal temper-

ature dependence C(T)isgivenby

C(T)=C

lattice

(T)+C + A

−

ln |





−1|,

forT < T

= C

lattice

(T)+A

ln |





−1|,

forT > T

. (14.13)

The parametersA

andA

−

are related via the univer-

sal ratio A

−

∼

1.05. First of all, the lattice part

of the specific heat has to be evaluated with some

degree of confidence. This can be achieved by ad-

ditional measurements in external magnetic fields.

Second it has to be tested to what extent the values of

the fit parameters depend on the temperature win-

dow chosen for the fits.In the case presented here,the

fit quality suggests that the critical regime appears to

be extended over a range of ±8KaroundT

.Without

going into further details, we note that attempts with

fits on the basis of other approaches for dealing with

ﬂuctuations in the critical region around T

gave less

convincing results.

As may be expected, the specific heat anomaly

is considerably reduced upon the application of an

external magnetic field. This is demonstrated in

Fig. 14.38, where C(T, H) data, measured on the

sameYBCO-123 sample in various external magnetic

fields, oriented perpendicularly to the Cu-O planes,

are displayed.In viewof the success in describingthe

critical ﬂuctuations on the basis of the 3D XY model

in zero field, it seems justified to test a related scal-

ing law that is inherent in this model for non zero

magnetic fields. The relevant scaling relation is

[

C(T, H)−C(T, 0)

]

2

= C



−1



−1

2



(14.14)

where C(x) is the unknown scaling function [148].

The parameters ˛ ∼0.007 and  ∼ 0.669 are the 3D

XY criticalexponents of the specific heat and the co-

herence length,respectively.In Fig.14.39,this scaling

is tested by plotting the appropriately scaled quanti-

Fig. 14.38. Inﬂuence of external magnetic fields on the spe-

cific heat anomaly at the superconducting phase transition

of single crystalline YBa

7−ı

Fig. 14.39. Specific heat anomalies of YBa

7−ı

plotted

in variables that are consistent with 3D XY-scaling (see

Eq. (14.14)). B = 

794 H.R.Ott

Fig. 14.40. Specific heat of textured Bi

CaCu

in the

temperature range around T

= 91 K in different magnetic

fields.The plots C

(T)/T are for 

H =0, 0.1, 0.5, 1, 2, 3, 4

and 7T from top to bottom. Note the crossing point of all

the curves at T ≈ 87 K

Fig. 14.41. Magnetic field dependence of the specific heat

anomalyof superconducting Bi-2212if plottedin the form

(0) − C

(H)]/T

ties.It may be seen,that the scaling is not well obeyed,

indicating that the use of the 3D XY model for de-

scribing thecritical ﬂuctuationsin external magnetic

fields exceeding 1 T,is inappropriate.Results of other

measurements on other single crystals of YBCO-123

are in conﬂict with this conclusion,claiming a 3D XY

type scaling to be valid in fields up to 14 T [149,150].

Our second example is C(T) of a Bi-based copper

oxide materialwhoseanisotropyis farmoreexcessive

than that of YBCO-123.For the latter, the anisotropy

parameter  is estimated to be less than 10; for su-

perconducting Bi-2212,  ∼ 350. In that case it may

be expected that ﬂuctuation effects are drastically

inﬂuencing the temperature dependence of the spe-

cific heat in the vicinity of the transition. That this

is indeed the case may be seen in Fig. 14.40, where

C(T, H) of Bi-2212 is shown for a set of different

fields H. Even in zero external field no trace of a dis-

continuity at T

,still discernible in the zero field data

of YBCO-123, may be recognized. More instructive

is the plot of [C(0) − C(H)]/T versus T,shownin

Fig. 14.41. It may be seen that the anomaly adopts

the shape of a triangle whose apex is, amazingly,

not shifted in temperature with increasing magnetic

field. These data cannot be brought into agreement

with predictions of the 3D XY model [150,151].

From our brief description of specific heat data of

two typical cuprate superconductors with very sim-

ilar critical temperatures but differing a lot with re-

spect to their anisotropy, it is clear that the char-

acter of the phase transition depends very strongly

on the anisotropy parameter  . For small values of

 , i.e., less than 10, the phase transition seems, at

least in zero magnetic field, to belong to the 3D XY

universality class. This is certainly not the case for

strongly anisotropic, quasi-2D superconductors. Un-

fortunately, YBCO-123 is probably the only cuprate

superconductor with a low value of  ;mostofthe

others are far more anisotropic. It has been argued

that this trend might indicate that in the quasi-2D

systems, the bosonic pairs are preformed and simply

condense at T

,thus implying an altogether different

universality class of the phase transition.

The characteristics of the phase transition may

also be discussed on the basis of magnetiza-

tion M(T, H) data. A particularly intriguing phe-

nomenon has been observed in M(T, H)ofcuprate

superconductors with a large anisotropy parameter

 . A data set of the temperature dependence of the

magnetization of a single crystal of Bi-2212, mea-

sured in different external magnetic fields [152], is

shown in Fig. 14.42. It may be seen that the M(T, H)

curves all cross in a single point at a temperature

< T

,implying that the magnetization is field in-

dependent at this single temperature T

.Inpractice

14 High-T

Superconductivity 795

it is found that T

is always a few degrees below the

zero field T

.The same phenomenon is also observed

for Bi-2201 [153], another extremely anisotropic su-

perconductor, but not for YBCO-123. This crossing

feature indicates that the field dependence of the

magnetization ∂M/∂H changes sign at T

,certainly

not expected for the usual magnetization of a super-

conductor in the mixed state. The sign change may

be understood, however, by considering two contri-

butions to the magnetization, such that

M = M

+ M

ﬂ

, (14.15)

where M

is the usual field dependent magnetization

of the mixed state, calculated using a modified Lon-

donequationwiththeappropriateapproximations,

and M

ﬂ

is due to thermal ﬂuctuations of the vortices

belowT

[154].Usinga modifiedLondon model,also

ﬂ

(T) has been calculated with different approxi-

mations in high and low fields. Inserting the appro-

priate expressions forM

and M

ﬂ

into Eq.(14.15)and

setting ∂M/∂lnH ≡ 0, one obtains

≡ M(T

)=−







˛

√



(14.16)

Fig. 14.42. Temperature dependence of the magnetization

of single crystalline Bi-2212 in the range of the supercon-

ducting transition temperature in various external mag-

netic fields. Note the crossing of all curves at T

∗

=86.4K

and



32k





)



, (14.17)

where k

is the Boltzmann constant and s is taken as

the interlayer distance, which is of the order of 1 nm.

The numerical values of  and ˛ are not easy to eval-

uate andoften it is assumed that ln(˛/

√

e) = 1 [155].

Forthisreasonitisdifficulttocheckthequantitative

agreement between the experimental and calculated

values of M

and T

The crossing of the M(T, H)curvesatasingle

temperature, experimentally observed to occur in an

unexpectedly wide range of magnetic fields [152],

indicates that some scaling should be obeyed. This

is indeed the case, as may be seen in Fig. 14.43.

The analysis is based on scaling functions that have

been deduced from a Ginzburg–Landau free en-

ergy functional for quasi-2D type II superconduc-

Fig. 14.43. Scaling of the magnetization of Bi-2212at differ-

ent temperatures and in different magnetic fields.The scal-

ing variables of the main frame are from a 2D Ginzburg–

Landau, lowest Landau level (GL–LLL) type approxima-

tion. In the inset,the same data are plotted assuming a 3D

version of the GL–LLL scaling (see text)

796 H.R.Ott

Fig. 14.44. Scaling of the magnetization of Y-123. The scal-

ing variables in the main frame are compatible with the 3D

XY model while those in the inset are consistent with 3D

GL-LLL scaling

Fig. 14.45. F

(T), as defined in Eq. (14.18), vs T. Open tri-

angles: 

H =0.75 T. Full squares, 

H = 15 T. The plot

indicates that the critical exponent  is not the same above

and below T

(see text)

tors [155]. In this case, the scaling function variable

is [T − T

(H)]/(TH)

1/2

. The corresponding plot of

the renormalized magnetization in Fig.14.43 reveals

that the scaling, based on ﬂuctuations in two dimen-

sions,is rather well obeyed.In addition,the same plot

for 3D scaling, shown in the inset of Fig. 14.43, im-

plies that the corresponding approach is not valid

in this case. A 3D scaling of this same type has,

however, been verified [156] for analogous data of

YBCO-123 where, as mentioned above, the crossing

phenomenon is absent. A scaling plot, based on the

scaling variable valid for the 3D situation is shown in

Fig. 14.44.

Another possibility for checking the scaling of

M(T, H) in the critical regime is offered in the fol-

lowing way. The unknown scaling function may be

eliminated by evaluating



∂M(T,H)

∂H

−

M(T,H)





∂M(T,H)

∂T



≡ (T

− T)/ , (14.18)

Fig. 14.46. Temperature dependence of the specific heat of

single crystallineY-123 in various external magnetic fields

oriented along the orthorhombic c-axis. Note the sharp

anomalies at decreasing transition temperatures upon in-

creasing magnetic field (see [162])