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 777

Fig. 14.17.Variation of the critical temperature T

as a func-

tion of ı in YBa

6+ı

(see [66])

Bi-Based and Tl-Based Copper Oxides

Most of what is known about superconductivity of

the Bi-based compounds [67–69] is very similar to

what we have discussed in Sect. 14.3.2 concerning the

hole doped La-cuprates. Also here, underdoped and

overdoped regimes embrace the situation of optimal

doping, more or less independent of the maximum

value of T

which depends on the chemical compo-

sition of the compound, i.e., on the number of Cu-O

planes per unit cell.It is rather amazing that the max-

imum T

’s of the individual series of alloys ranges

from 15 to 125 K. Again it is observed that strong

overdoping eventually leads to the disappearance

of superconductivity. The corresponding data base

for superconducting Tl-based compounds [70–72] is

much less extensive but there is no evidence for a

significantly different behavior.

14.3.3 Fullerides

The secret of inducing superconductivity in these

types of materials is again doping, with electrons in

this case [73,74]. This is, e.g., achieved with the in-

troduction of various numbers, between 1 and 3, of

atoms of alkaline metal elements per unit cell. The

Fig. 14.18. Variation of the critical temperature T

of ful-

lerides as a function of the lattice parameter a,achieved

either by doping or by application of external pressure (p)

(see [45])

main result of numerous investigations is the dis-

tinct variation of the critical temperature with the

lattice parameter, seemingly a universal feature and,

with a few exceptions,independent of the chosen ma-

terial. An increase of the separation a of the atoms

leads to an enhancement of T

[75] and, vice versa,

areductionofT

upon applying external pressure

is observed [76]. This has been verified by changing

the lattice parameter both via variation of the chem-

ical composition and by applying external pressure.

A schematic representation of T(a) for a selection of

these materials is shown in Fig. 14.18. This plot sug-

geststhat an increaseof T

maybeachievedbysimply

enhancing the lattice parameter a.Slightdeviations

from a universal T(a) behavior under pressure have,

however, been reported [77,78]. Considering an ex-

tended set of data, and plotting T

max

versus the

concentration n of valence electrons, whereby T

max

is the maximum critical temperature in a single al-

loy series, it has been claimed that the maxima of

all the considered T

’s is reached if n =3andthat

no superconductivityis observed if either n < 2.5or

n > 4.0 [79].The maximum value of the criticaltem-

perature reported for bulk material is that of Cs

778 H.R.Ott

apparently a non cubic compound with a T

of about

40 K [80].

14.3.4 MgB

This compound does not seem to allow for large vari-

ations of the chemical composition without loosing

superconductivity [81]. So far it appears that the bi-

nary compound with the proper stoichiometry re-

veals the highest T

. Most attempts of alloying have

led to substantial reductions of T

and all other di-

borides with differentcationelementshave,so far,re-

sisted to exhibit superconductivity at similarly high

critical temperatures.An exception is Zn for which a

tiny enhancement of T

at low doping concentration

may have been observed [82,83].

14.4 Physical Properties of Copper Oxides

As we have indicated in the preceding section, the

general properties of the relevant cuprate com-

pounds and alloys depend crucially on the chem-

ical composition of the investigated materials. The

same is true for those materials that actually do

exhibit high-T

superconductivity.Already early re-

sults of measurements probing the physical proper-

ties of these materials indicated anomalous features

also in the normal state preceding the superconduct-

ing ground state and it soon became clear that the

occurrence of this superconductivity at unusually

high temperatures would not be explained without

a sound understanding of the character of the nor-

mal state [84]. It is for this reason that we also brieﬂy

discuss those normal state properties that are diffi-

cult to understand on the basis of the very successful

Fermi liquid concept that has, in the past, been used

for describing the characteristic electronic proper-

ties of common metals. To keep this review tractable,

only few examples that are believed to capture the

essential issues of both the normal and the super-

conducting state of this type of materials are pre-

sented and discussed. Most of these examples, if they

are not intended to emphasize the inﬂuence of the

crystal structure or the chemical composition, are

chosen to represent those materials with so called

optimal doping, i.e., with the composition for which

amaximumT

has been observed. In these cases,

some evidence for the existence of a boundary in the

quasiparticle excitation spectrum that may be iden-

tified as a Fermi surface has been obtained from both

conventional [85] and angle resolved photoemission

spectroscopy (ARPES) [86,87] with high energy res-

olution. A schematic view of the cross sections of

the 3D Fermi surfaces of Bi cuprates, deduced from

ARPES measurements, is shown in Fig. 14.19. This

evidence is more obvious for overdoped materials

than it is for optimally doped or underdoped com-

pounds although more recently it has been claimed

that also for underdoped material, resolution limited

signals, indicating well defined quasiparticle excita-

tions, have been obtained [88]. Measurements with

further improved energy resolution, however, seem

to reveal that quasiparticle-type signals are only well

identified if the investigated material is in the su-

perconducting state [89]. It thus appears that it is

still unclear whether well defined quasiparticles do

exist in the normal state. At any rate, their manifes-

tation certainly depends on the doping level of the

material. As we shall see below, the difficulties en-

countered with the spectra of underdoped cuprates

may arise because of the formation of a pseudogap

Fig. 14.19. Mapping of the Fermi surface of Bi-based super-

conducting cuprates by ARPES (see [86])

14 High-T

Superconductivity 779

in the quasiparticle excitation spectrum already in

the normal state of these materials.

In this context it is important to note that,because

of the particular crystal structures,featuringthe Cu-

O planes as the essential structural element, most

of the physical properties of these cuprates exhibit

distinct anisotropies.

14.4.1 Normal State Properties

The first indications of anomalous features of the

normal state of high-T

cuprates were provided by

measurements of the temperature dependence of

the electrical resistivity (T) [90] of some of these

materials, both parallel and perpendicular to the

Cu-O planes [91]. These early transport results were

confirmed in many cases and with improving sam-

ple quality, and complemented by additional data,

mainly concerning the thermal conductivity and the

Hall effect,both in the normal and the superconduct-

ing state.

Electrical Resist ivity

First it seems instructive to consider the varia-

tion of (T), measured for currents parallel to the

Cu-O planes, upon doping. A well studied case is

2−x

CuO

and some selected data are shown in

Fig. 14.20 [92]. What is amazing, is the strong tem-

perature dependence of the resistivity even at large

absolute values.The increasing concentration of hole

doping with increasing x is reﬂected in both the

magnitude of  at room temperature and the slope

∂/∂T. Another typical feature is the extended re-

gion of the linear variation of  with temperature for

x ≤ 0.18. For common metals, the linear in T vari-

ation of the resistivity is restricted to temperatures

around 200 K or above, but usually is not observed

much below 100 K.It may also be seen that for higher

doping levels, this particular feature is lost and the

change of (T) with increasing x is much less pro-

nounced than for lower values of x.Consideringthat

the maximum value of T

for this compound series

is reached at optimal doping with x ≈ 0.15, (T)is

clearly anomalous in what has been termed the un-

derdoped regime and progressively adopts a more

common behavior in the so called overdoped region.

This linear proportionality of the resistivity with re-

spect to the temperature is one of the common fea-

tures of the normal state of cuprate superconductors,

most clearly borne out in the underdoped regime.

An important case in this respect is the compound

CuO

(Bi2201) where the linear in T variation

of  may be observed from low temperatures, be-

cause T

is only of the order of 10 K,up to 700 K [93].

The persistence of this temperature dependence of 

or, equivalently, of the scattering rate 

−1

of the elec-

tronic quasiparticles down to T

,is taken as evidence

for an anomalous metallic state, not compatible with

expectations of the Fermi liquid model, where a T

dependence of 

−1

is expected. In particular, the ab-

sence of any onset of residual resistance, definitely

expected to be visible if the absolute magnitude of

the measured values of  are considered, is difficult

to understand.

Another unusual feature of the electrical resistiv-

ity is the generally observed large anisotropies and

Fig. 14.20. Temperature dependence of the electrical resis-

tivityof La

2−x

CuO

in polycrystalline form (solid lines)

and for currents in the basal (ab) plane of singlecrystalline

films (see [92])

780 H.R.Ott

Fig. 14.21. Temperature variation of the ratio of 

and 

for various cuprate superconductors. The general trend for

−1

and T

−2

behavior is also indicated (see, e.g., [94])

their temperature dependences if the resistivities are

measured either parallel or perpendicular to the Cu-

O planes [94].Asmay be expected, these anisotropies

are strongly inﬂuenced by details of the crystal struc-

ture and examples are shown in Fig. 14.21. These

large anisotropies imply that the motion ofthe quasi-

particles is confined to the Cu-O planes and that they

tend to localize in the motion perpendicular to it.

The localization of quasiparticles in only one direc-

tion cannot occur in a Fermi liquid and therefore

the observed anisotropies indicate the failure of the

Fermi liquid model for describing the metallic state

of cuprates above T

, at least in the underdoped and

optimally doped regime. Large anisotropies, as those

of the electrical conductivities,are also observed in

many other, physically relevant parameters. This has

led to the notion that at least electronically, these

cuprates should be regarded as quasi-two dimen-

sional systems. This view has guided many theoreti-

cal attempts to understand the physical properties of

these materials.

The Hall effect

Already early experiments [95] indicated anomalous

features of the Hall response in the normal state

of cuprate superconductors, exhibiting a strong in-

creaseof theHall coefficientR

with decreasing tem-

perature above T

. The Hall resistivity 

and the

Hall constant R

are related via



= R

· H , (14.1)

where His the externally applied magnetic field.Var-

ious later experiments confirmed this trend and ex-

periments made with single-crystalline samples of

YBa

7−x

(YBCO) revealed that the Hall con-

stant, for currents parallel to the Cu-O planes and

external magnetic fields oriented perpendicularly to

these planes, is positive and varies as T

−1

[96]. The

out of plane Hall constant, however, obtained for the

arrangement that both the current and the field di-

rection are in the plane but perpendicular to each

other, providing a Hall field E

along the c direction

of the crystal lattice, remained constant and nega-

tive between 400 K and T

[97]. As an example for

the distinct temperature dependence of the in plane

Hall constant we show, in Fig. 14.22, R

−1

versus T,

measured on single crystalline samples of YBCO-123

and Bi

CaCu

(Bi2212) [98].Equally intriguing

is the temperature dependence of cotŸ

= 

/

which, as demonstrated in Fig. 14.23, varies almost

as T

over a wide temperature range [97, 98]. The

in-plane Hall angle is defined as

= arctg







. (14.2)

These two phenomena, apparently quite universal

for high-T

copper oxides, were interpreted by An-

derson [99] to provide evidence for the anomalous

features of the electronic ensemble in these cuprates.

He argued that the electronic transport properties

appear to be those of a two-dimensional Luttinger

liquid,rather than those of a Fermi liquid. In essence

there are two different scattering rates, 

−1

and 

−1

14 High-T

Superconductivity 781

that have to be considered. The longitudinal trans-

port scattering rate is 

−1

∼ T,asimpliedby(T)

discussed aboveandfor the transverse scattering rate

Fig. 14.22. Temperature dependence of the in plane

Hall constant of single crystalline YBa

and

CaCu

in the form of R

−1

vs T

Fig. 14.23. TemperaturedependenceoftheinplaneHallan-

gle Ÿ of single crystalline YBa

and Bi

CaCu

in the form of cotŸ versus T



−1

, in this model representing the intrinsic scatter-

ing rate for quasiparticle-like excitations of the Lut-

tinger liquid, so called spinons, a T

dependence is

predicted. If, in addition, impurities are considered,

the model predicts that cotŸ

= aT

+ C,where

a is a constant and C increases with increasing im-

purity content. This predicted behavior has actually

also been verified experimentally, supporting again

the view that the normal state of these cuprates is

quite different from that of a common metal.

Magnetic Susceptibility

The temperature dependence of the bulk magnetic

susceptibility (T) is very much dependent on the

doping concentration. This may well be seen in

Fig. 14.24, where (T)ofaseriesofLa

2−x

CuO

compounds with varying x is displayed. While for

low doping, ∂/∂T is positive across the whole cov-

ered temperature range, we note the appearance of

a maximum in (T) with increasing Sr content x,

the maximum shifting to lower temperatures with

increasing x.Asimilarbehaviorof(T)withvary-

ing doping concentration was also observed for the

YBCO series, where also in the underdoped regime,

a strong decrease of  with decreasing temperature

Fig. 14.24. Temperature dependence of the normal state

magnetic susceptibility of La

2−x

CuO

(see [8])

782 H.R.Ott

was observed.Naturally there are many ways to com-

pare these experimental data with model calcula-

tions but it seems clear that a simple description in-

voking only a Pauli-type electronic magnetic suscep-

tibility,which is a measure of the electronic density

of states at the Fermi energy D(E

), usually inde-

pendent of temperature, is not appropriate. The low

temperature negative slope ∂/∂T indicates a loss of

magnetic degrees of freedom with decreasing tem-

perature and in retrospect, these measurements of

(T) gave the first hint of a phenomenon that is still

of high current interest: the opening of a pseudo-

gap in the quasiparticle excitation spectrum, possi-

bly due to the formation of pairs in some incoherent

fashion, at temperatures distinctly above T

.Aswe

shall discuss in more detail below, a lot of additional

experimental evidence for the formation of a pseudo-

gap or,in other words,a loss of degrees of freedom in

the quasiparticle excitation spectrum, has since been

accumulated.

NMR: Knight Shift, Spin-LatticeRelaxation

In common metals the NMR response in the normal

state is, to a large extent, dictated by the presence

of conduction electrons. First, they provoke a shift

of the resonance signal, the well known Knight shift

K =  H/H

,whereH

is the resonance field for

the nuclear spins in a non metallic environment, and

H is the shift in magnetic field if the resonance is

probed at constant frequency. In simple metals, K is

given by [100]

K =



8



· < |u

(0)|

> ·

, (14.3)

hence is related to the spin susceptibility of the con-

duction electrons



=2

D(E

) , (14.4)

and therefore expected to be essentially temperature

independent. Second, the conduction electrons also

dominate the spin-lattice relaxation process, leading

to a Korringa-type behavior of the relaxation rate

−1

,suchthat(T

−1

is approximately constant.

This so called Korringa constant is related to the

Knight shift via

−1

∼ K

. (14.5)

It is notat all surprisingthat distinct deviationsfrom

these types of behavior have been identified in NMR

studies of the cupratematerials.Complications arise,

particularly in the evaluation of the Knight shift, be-

cause in structurally and electronically complicated

materials, such as the cuprates, this shift is the result

of various different contributions,also involving or-

bital degrees of freedom. Since NMR is a local probe,

the response of different nuclei occupying different

lattice sites may be probed and all the results have to

be interpreted by taking into account the local envi-

ronment of chosen species of nuclei.For the cuprates,

Fig. 14.25. Temperature dependence of various

components of the magnetic Knight shift of Cu

and O NMR signals in YBa

(see [101])

14 High-T

Superconductivity 783

Fig. 14.26. Average temperature dependence of the mag-

netic shift of the NMR signals probing the nuclei on planar

Cu sites of YBa

(broken line)andYBa

(solid

line) (see [102])

the bulk of data has been obtained from nuclei of Cu

and O isotopes.

First of all, there are clear cases where the Knight

shift exhibits a strong temperature dependence, not

really surprising if we recall the behavior of (T)

mentioned above.Examples of different components

of K(T) are shown in Fig. 14.25. Here, these K com-

ponents have been evaluated for Cu and O nuclei

in the compound YBa

(YBCO-124) [101]. It

is obvious that these different shift components are

substantially reduced with decreasing temperature

below 300 K, i.e., at temperatures far above T

.Sim-

ilar observations were made in NMR experiments

probing underdoped YBCO and Bi-2212. The situa-

tion is quite different, if K is measured in optimally

or overdoped material. In Fig. 14.26 we show this

difference schematically for the planar Cu nuclei in

YBCO-123 and YBCO-124 [102]. It is now accepted

that the strong temperature dependence of K is re-

lated with the low doping of the corresponding ma-

terials and most likely also reﬂects the quasi-gap in

the excitation spectrum mentioned above.

It turns out that also the different experimental

data concerning the temperature dependence of the

spin-lattice relaxationrateT

−1

(T) are not compatible

with the simplest expectations for a metal. Examples

of experimental results [103] for the relaxation rate,

measured for

Cu nuclei on planar Cu sites in var-

Fig. 14.27. Temperature dependence of

Cu NMR spin lat-

tice relaxation rates for nuclei on planar Cu sites in differ-

ent copper oxide compounds. The solid lines are to guide

the eye and the broken lines indicate the Korringa-type

variation,depending linearly on T (see [103])

ious cuprates at temperatures above the respective

critical temperatures, are shown in Fig. 14.27. For

comparison, the data for superconducting and non

superconducting Cu oxides are shown. First we note

that the relaxation is much faster in those materials

which exhibit high-T

superconductivity. It may also

be seen that the relaxation in the non superconduct-

ing compounds is indeed of Korringa-type,atleast at

low temperatures. This is clearly not the case for the

superconducting compounds. At elevated tempera-

tures, the relaxation rate is rather well approximated

by T

−1

= D + cT, indicating at least two relaxation

channels. While the second contribution may be in-

terpreted as representing some sort of Korringa re-

laxation,the first and temperature independent term

has to be associated with another contribution,most

likely due to the dynamics of localized d-electron

spins on the Cu atoms. If the Korringa-type term is

subtracted from the experimental T

−1

values, the re-

sulting d-electron contributions,shownin Fig. 14.28,

areobtained[103].A temperatureindependent relax-

ation rate may be related with ﬂuctuations of local-

784 H.R.Ott

Fig. 14.28. Same data as in Fig. 14.27, after subtraction

of the Korringa type contribution varying linearly with

temperature, as indicated by the broken lines in Fig. 14.27.

These data are interpreted as representing the relaxation

due to Cu d electrons. The solid lines aretoguidetheeye

(see [103])

ized spins but may also be accountedfor byspin-den-

sity ﬂuctuationsin relation with a 2D Fermi surface.

Fig. 14.28, however, reveals that also this contribu-

tion suffers from a temperature dependent reduc-

tion, starting well above the corresponding critical

temperatures, at least in some cases. It seems quite

reasonable to ascribe this feature again to the pres-

ence of the pseudo gap discussed above. By now it is

clear that the onset temperature of this gap forma-

tion tends to grow with decreasing doping.

Speciﬁc Heat

Theoutstandingproblemin connection withthespe-

cific heat of copper oxide superconductors in the nor-

mal state is the evaluation of the electronic contribu-

tion to the total specific heat C

tot

(T), which in the

range of T

is completely dominated by the specific

heat due to lattice excitations.The lattice specific heat

(T) may be estimated by measurements of C

tot

(T) of one particular material in external magnetic

fields or, in an attempt to only change the electronic

contribution,by slightly changing the chemical com-

position, such as to vary the doping level. In both

cases it has to be assumed, that C

(T) is unaffected

by these measures, not easy to verify in a rigorous

way. The electronic part C

(T), i.e., (C

tot

− C

)(T)

is usually assumed to be proportional to T or, in

other words, it is assumed that the ratio C

/T is

agoodmeasurefortheelectronicspecificheatpa-

rameter  . For the formal representation of C

(T),

various approximation procedures have been sug-

gested and a number of relevant articles discuss this

issue [104,105]. For optimally doped YBCO-123 an

average and reasonable value for the electronic pa-

rameter in the normal state is  =25mJ/moleK

For Bi-based copper oxides, the corresponding mo-

lar values for  are somewhat smaller but still of the

order of 10 mJ/moleK

.Asweshallseebelow,thesit-

uation is more complicated in underdoped material

where C

/T is not a constant in the normal state. In

comparison with simple metals, the  value quoted

above seems fairly large but still, the corresponding

density of electronic states D(E

)isnotexception-

ally high in order to justify the high values of T

in a

straightforward manner.

A common way to estimate the  value is based on

the weak coupling BCS relation [106]

− C

)(T

)

=1.43 , (14.6)

where C

)= · T

, and experimental data con-

cerning the specific heat anomaly at T

. Since this

approachis validonly for mean-field type transitions

and the weak coupling limit, most of these calcula-

tions may serve as order of magnitude estimates at

most. As we shall see below, none of the transitions

in high-T

cuprates is really of mean-field type and

in most cases the characteristics of the transition are

really quitedifferent,mainly because of rather strong

ﬂuctuation effects.In addition and complicating the

situation,strong coupling effects are indicated by ex-

perimentally determined ratios of the  and T

and

therefore the use of Eq. (14.6) for estimating the 

parameter is more than questionable.

14 High-T

Superconductivity 785

Characteristics of the Normal State

From this brief compilation of some anomalies that

are encountered in the results of measurements of

some selected physical quantities, it may safely be

concluded, that very simple approaches, usually em-

ployed for the description of common metals, are

inadequate for treating the features of these cuprate

materials. Various theoretical attempts to cope with

these problems are the subject of other chapters in

this treatise and are discussed in more detail there.

Some of them start with the notion that the conven-

tional Fermi liquid model for describingthe metallic

normal state has to be abandonedhere and that other

quasiparticle-like excitations have to be introduced.

Many of the experimental facts mentioned above and

others, not considered here, seem to justify this view.

What is particularly hard to grasp, however, is the

fact that the character of these excitations seems to

vary with the doping level. There is not much doubt

that a common quasiparticle picture is adequate for

describing the properties of overdoped materials but

it is still much debated how the underdoped regime

should be treated. Here, the appearance of the of-

ten quoted pseudogap,to be discussed in some more

detail in the next subsection, adds another compli-

cation to the state of affairs. If finally it should turn

out that the electronic properties across the entire

doping range where superconductivity occurs may

still be understood on the basis of the Fermi liquid

model, a lot of additional input concerning details of

that model is probably needed.

Pseudogap, Additional Evidence

In this subsection we are brieﬂy discussing addi-

tional experimental evidence for the pseudogap in

thenormal stateof cupratesuperconductors.Wehave

chosen results of experiments whose interpretation

leads to conclusions that are somewhat controversial.

This should serve the purpose to indicate that these

issues have not yet been settled completely. To set

the stage we display a schematic phase diagram for

the cuprate superconductors, where also the doping

dependence of T

∗

, the temperature below which ex-

perimental evidence for the presence of a pseudogap

exists, in Fig. 14.29.

Fig. 14.29. Schematic low temperature phase diagram of

cuprate superconductors. Emphasized are the antiferro-

magnetically ordered phase at low doping, the supercon-

ducting phase and the approximate location of T

∗

(x), indi-

cating the pseudogap formation

First we mention the results of specific heat mea-

surements [107], covering the temperature range

above and below T

of a series of samples of

0.8

0.2

7−ı

with different oxygen contents,

i.e., varying parameter ı. This choice of material al-

lows to extend the overdoped regime for YBCO-123

type material; optimal doping conditions are reached

for ı ∼ 0.32. In Fig. 14.30 we show the temperature

dependence of theelectronic specificheatof different

samples in the form of C

/T. The electronic contri-

bution has been obtained by subtracting the lattice

contribution from the raw experimental data in an

apparently controlled way [108].In Fig.14.30a it may

be seen that in the normal state of overdoped ma-

terial, C

/T is temperature independent above the

respectivecritical temperatures and also does not de-

pend on ı. The upturns, appearing somewhat above

,most likely reﬂect ﬂuctuationcontributionsto C

At this point of our discussion we do not address the

temperature dependence of C

at very low temper-

atures, leaving this point to be addressed in another

section below. The situation is completely different

for underdoped material. Here, as may be seen in

Fig.14.30b, the different C

/T versus T plots start to

diverge above T

in a manner that C

/T decreases

with decreasing T if ı > 0.5,thus indicating a loss of

entropy already above the onset of superconductiv-

786 H.R.Ott

ity, distinctly different from the behavior displayed

in Fig. 14.30a. An additional clear difference in the

(b) panel of the figure is the sudden drop in the peak

height of C

/T for ı > 0.32, although no corre-

sponding large change in T

is observed. In [109], a

detailed discussion of the temperature dependence

of the related entropies and of similar data, obtained

for a series of La

2−x

CuO

compounds,leads to the

conclusion that the pseudogap phenomenon should

not be regarded as being due to a precursor forma-

tion of Cooper pairs and therefore is not related with

the gap opening in the superconducting state.In par-

ticular it is argued that the pseudogap related loss

of low-energy normal state spectral weight actually

weakens the superconducting condensate. To some

extent this is compatible with the fact that contrary

to the decrease of T

with decreasing doping, the on-

set temperature T

∗

increases.

A quite different view has been obtained from

ARPES experimentson single crystals of underdoped

Fig. 14.30. Temperature dependence of the electronic spe-

cific heat of Y

0.8

0.2

7−ı

in the overdoped (a) and

underdoped (b) regime (see [107])

Bi-2212. With this type of experiments it has been

possible to map the Fermi surface of this type of ma-

terials and, because of much improved energy res-

olutions, to verify the onset of the gap related with

the superconducting ground state. A detailed pre-

sentation and discussion of ARPES experiments on

high-T

superconductors is given elsewhere in this

compendium. Nevertheless we note here that mea-

surements of this type of material mentioned above

and as a function of temperature reveal that a gap

signature at E

in different directions of k space ap-

pears at different temperatures.In this sense,the data

indicate a gradual loss of Fermi surface with de-

creasing temperature, starting well above T

[110].

The evolution, ending in the gap features of the su-

perconducting state below T

appears to be smooth

and therefore, it is concluded that the gap forma-

tion in the normal state is most likely due to a pre-

cursor pairing of quasiparticles, i.e., definitely re-

lated with the appearance of superconductivity be-

low T

[111].A schematic view ofthe temperaturede-

pendent shrinking of the Fermi surface in k-space is

displayed in Fig.14.31,indicatingthat the pseudogap

in the normal state and the finally resulting gap of

the superconducting state reveal the same anisotropy

in k-space. A k-dependent incoherent pairing is, of

course, not compatible with conventional supercon-

ductivity, where the instability of the whole Fermi

surface occurs at once at T

. Results of the same kind

of experiments on some overdoped material were

interpreted as to confirm the previously established

view of the absence of a pseudogap above T

[88].

Traditionally, various kinds of tunneling exper-

iments have been instrumental for qualitative and

quantitative measurements of gaps in electronic

Fig. 14.31. Schematic representation of the growth of the

Fermi surface of Bi-2212 with increasing temperature, as

obtained from ARPES experiments (see [110])