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 817

Fig. 14.73. Anomaly of the specific heat at the supercon-

ducting transition of K

.Thesolid line is to guide the

eye and the broken line indicates the estimated background

(see [272])

Considering the electronic specific heat parameter

 that is related to C(T

)/T

and the enhancement

of the magnetic susceptibility,estimates of the elec-

tron mass enhancement (1 + 

) due to the electron

phonon interaction can be made. Depending on de-

tails of the approximations, 

values between 0,5

and 0.7 have been calculated and it was concluded

that the pairing via the electron–phonon interac-

tion could not be achieved by invoking low energy

phonon excitations. Instead a coupling to high en-

ergy phonons and an upper bound of 

≈ 1.0have

been postulated [272].

The participation of phonons in the pairing of

electrons is put in evidence by isotope-effect experi-

ments where

C is replaced by

C.Although there is

some scatter in the data, clear variations of T

upon

isotope exchange have been identified [274,275] and

˛-values between 0.3 and 0.4 seem to be the most

reasonable. The importance of the phonons related

with excitation modes of the C

molecules is empha-

sized by the absence of an isotope effect if the isotope

exchange involves the alkaline metal elements. This

indicates that apart from the direct inﬂuence on the

density of itinerant electrons via their ionicradii,the

alkaline-metal atoms have only a weak inﬂuence on

the magnitude of T

[276,277].

The onset of superconductivity is clearly reﬂected

in a distinct reduction of the NMR spin lattice relax-

ation rate T

−1

at low temperatures [273]. Results of

early measurements gave no evidence for a Hebel–

Slichter type enhancement of the relaxation rate

at temperatures close to but below T

,expectedif

the above mentioned coherence effects are effective.

Even for well known conventional superconductors,

the absence of this enhancement may be caused by

strong coupling and other trivial effects. However,

a later investigation clearly revealed an increase of

−1

with decreasing temperature below T

,asdis-

played in Fig. 14.74, providing convincing evidence

Fig. 14.74. Temperature dependence of NMR

spin lattice relaxation rates, plotted as (T

−1

vs T, in the normal and superconducting states

of Rb

CsC

0, providing some evidence for co-

herence effects just below T

≈31 K (see [278])

818 H.R.Ott

for coherence effects in a at most moderately coupled

superconductor [278]. The same type of feature has

been observed in SR experiments [279], and there-

fore it seems reasonable to qualify these alkaline-

metal doped C

materials as s-wave type supercon-

ductors with medium strength electron phonon cou-

pling.Another manifestation of the superconducting

stateis apparent in the reduction of the spin suscepti-

bility 

, which in principle can be deduced from the

temperature dependence of the spin part K

of the

resonance shift [273,278].Measurements of totalres-

onance shifts of nuclei occupying inequivalent lattice

sites made it possible to extract the temperature de-

pendence of theKnight shift K

(T) and the resultsare

compatible with expectations that are valid for weak

coupling superconductors ands-wave pairing,inpar-

ticular via the evaluation of the ratio 2

(0)/k

[278]. Different values of this ratio have been ex-

tracted from different NMR data sets, ranging from

3.0 to 4.3. Estimates of this ratio based on optical

experiments [280,281] gave values between 3.4 and

4.2. Somewhat more enhanced values of 6.3 and 6.4

have been extracted from results of tunneling experi-

ments[282,283].Thesevalueswould implya substan-

tial enhancement of the effective density of states and



≈ 3.0.This valueis far beyondany of theother es-

timates of this value, such that the results of the tun-

nelling experiments seem more than questionable.

Values forthe characteristic lengths of typical C

based superconductors have been evaluated from

SR [284, 285], NMR [273] and optical data [286],

as well as from results of measurements of the up-

per critical field H

[287]. Evaluations of H

(T)

from observing resistive transitions are hampered

byasizeableincreaseofthetransitionwidthinex-

ternal magnetic fields [269].The spread of the results

concerning 

and  is sizeable. For the penetration

depth 

(0), values between 4000 and 8000 Å have

been obtained.The effectivecoherence length 

,tak-

ing into account the very short electronic mean free

path, turns out to be of the order of 30 Å. The ratio

of these two quantities, defining the GL parameter

, indicates that these C

superconductors are also

oftheextremetypeIIvariety.Thelargevaluesof

the penetration depths are compatible with the pre-

vious indications for a low concentration of itiner-

ant charge carriers and a substantial enhancement

of their effective mass.

If all these experimental results providing infor-

mation on the superconducting state of these C

based bulk materials are taken together [45, 288],

it seems inevitable to conclude that one is dealing

with conventional superconductors of the weak, or

at best medium coupled variety invoking some sort

of electron–phonon interaction.

14.5.3 MgB

In the class of seemingly simple binary intermetallic

compounds, MgB

exhibits by far the highest onset

temperature for superconductivity.For comparison,

the cubic binary transition metal compounds of the

type A

B, crystallizing with the ˇ tungsten or A15

crystal structure, exhibit T

’s up to 18 K [289] and

only if they are prepared under special conditions

or in special forms, such as quench condensed films,

they exhibit superconductivity at temperatures ex-

ceeding the temperature of the boiling point of liq-

uid hydrogen,i.e., 20.4 K, by about 3 K [290] (see also

Fig. 14.1). The again completely unexpected discov-

ery of a critical temperature close to 40 K for a mate-

rial that simple,naturally raises the question whether

this feature may already be recognized in the normal

state properties of the compound. The magnitude

of T

also provokes experiments aiming at clarify-

ing the characteristics of the superconducting state

and, in particular, to identify the mechanism lead-

ing to the pairing of electrons at this relatively high

temperature. In this context also rather challenging

questions need to be answered by theorists. In addi-

tion, for MgB

, some properties that are crucial for

potential applications seem to show favorable behav-

ior and therefore this material is also of high current

interest for more technically oriented developments.

The first report on superconductivity of MgB

dates from mid-January of 2001 [11] and, although a

lot of subsequent work has appeared either in print

or in e-print archives, not all the open questions that

are indicated above have been answered up to now.

This brief review, by far not complete and aiming at

presenting typical features, considers work that has

become available until the end of July 2001. Samples

14 High-T

Superconductivity 819

for the many investigations were prepared in many

different forms, including single crystals, mostly of

rather small size, polycrystals, powder, films, wires

and tapes. A first brief review of various properties

of superconductingMgB

has been presented in [81].

The Normal State

The electrical resistivity below room temperature is

that of a typical metal with  of the order of 10 §cm

at room temperature and ∂/∂T > 0atalltemper-

atures between T

and 300 K. Resistance ratios RR

= (300)/(T

) of the order of 6 to 7 have been re-

ported [291]for single crystals forcurrents along the

basal plane of the hexagonal crystal lattice.An exam-

ple of (T),measured for a current in the basal plane

of a single crystal between 2 and 300 K is shown in

Fig. 14.75 [292]. Amazingly large ratios of up to 25

have been obtained for polycrystalline material and

even wires [293,294]. Since there are no reliable data

yet available for the electrical conductivity along the

hexagonal c-axis, it is not clear whether this differ-

ence might be due to a strongly anisotropic conduc-

tivity or, in other words, a much lower resistivity for

currents along the c-axis. In all cases, however, the

regime of residual resistivity, of the order of 1§ cm

or less, seems to be reached at T

. While practically

absent at temperatures above 200 K, a fairly large

positive magnetoresistance at T ≈ T

,withalmost

a doubling of the zero field resistance in an external

field of 10 T has been reported for polycrystalline

material [293].This effect is less pronounced for sin-

gle crystals, at least for  measured in the hexago-

nal basal plane [291,292]. Hall effect data are scarce

and they have not been obtained for well defined

geometrical configurations. Nevertheless, the avail-

able set of results [81,295] is entirely consistent with

holes being the itinerant charge carriers in MgB

No agreement has yet been established concerning

the question whether the positive Hall constant R

varies with decreasing temperature between room

temperature and T

. It is interesting to note [296]

that for other, isostructural metallic diboride com-

pounds, for which no occurrence of superconductiv-

ity has yet been reported, R

is negative [297,298].

Fig. 14.75. Temperature dependence of the in plane electri-

cal resistivity of single crystalline MgB

.Theinset empha-

sizes the superconducting transition region

Significant anisotropies have been revealed in

measurements of the lattice parameters, both as a

function of external pressure at room temperature

[299] and as a function of temperature at ambient

pressure [300].In both cases the response along the c

axis is much more pronounced than along the a axis,

confirmingthattheout-of-planeMg-Bbondsare sig-

nificantlyweaker than the in-plane Mg-Mg bonds.As

a consequence the c/a ratio decreases with increas-

ing pressure and with decreasing temperature. Band

structure calculations [49,301] indicate that the in-

plane B-B bonds are dominantly covalent.

As usual in cases where superconductivity sets in

at relatively high temperatures, it is not straightfor-

ward to obtain information on the normal state elec-

tronic properties,such as D(E

),from measurements

of the specific heat. This problem is also met in the

case of MgB

and several ways of solving it have

been followed. An analysis [302] of measurements

of C

(T) in external fields between 0 and 9 T pro-

vides the following important parameters: (i) the

electronic specific heat parameter  =2.6mJ/mole

and (ii) a Debye temperature Ÿ

= 1050 K. An-

other set of similar measurements, including results

obtained in magnetic fields up to 16 T give  =2.7

mJ/mole K

[303]. Since in this work, the lattice con-

tribution is ascribed to a continuum of modes, no

820 H.R.Ott

Fig. 14.76. Temperature dependence of the specific heat of

a pressed-powder sample of MgB

, plotted as C/T vs T.

The solid line represents an estimated normal state back-

groundC

.Thein set emphasizes the anomaly at and below

the transition temperature by plotting (C − C

)/T vs T

value for Ÿ

is provided. In yet another approach,

concentrating on a reliable estimate of the non-

electronic background contributions in zero field,

the same parameters turn out to be  =3.8mJ/mole

and Ÿ

= 750 K [304]. An example of C

(T) of

pressed non-commercial powder of MgB

below 60

K is shown in Fig.14.76 [305].The normal state NMR

response of MgB

, monitored on

Bnucleiiscom-

patible with the overall metallic character of thecom-

pound [306, 307]. The Knight shift K is very small

and positive at all temperatures. Different reports do

not agree with respect to the magnitude of the shift.

K(T) is approximately constant between 150 and 300

K. According to [307], K decreases steadily with de-

creasing temperature below 150 K and the decrease

continues smoothly through the transition at T

(H).

A distinct change in the slope of K(T)atT

has been

reported, however, in [308]. The spin-lattice relax-

ation is well described by assuming a Korringa type

relaxation via conduction electrons between room

temperature and T

with T

T ≈ 170 s, an average

value resulting from a number of different investiga-

tions [306–308]. The Korringa constant, incorporat-

ing the Knight shift, is substantially smaller than the

expected value for a relaxation via s-electrons and in-

dicates that orbital degrees of freedom are dominant

in the relaxation process [309].

Superconductivity

Rather narrow transition widths T have been ob-

served in resistivity measurements. For single crys-

tals, values of T of the order of 0.2 K have been

achieved [291, 292]. If RR, defined above, exceeds

the value of 5, the reported critical temperatures

only varywithina rangeofexperimentaluncertainty.

However, considerable variations of T

may be in-

duced by applying external pressure or by exchange

of isotopes, such as

B for

Bor

Mg for

Mg. At

this point we may also mention that many differ-

ent types of chemical substitutionshave been tested.

None of them, except one,have induced an enhance-

ment of T

.Sofar,theexceptionalcaseisZnforwhich

a small enhancement of T

at low concentrations of

Zn atoms has been reported [310,311].

As in most common superconductors, the appli-

cation of quasi-hydrostatic pressure causes a reduc-

tion of T

. We note a considerable scatter in the

available data set [81] which may be due to non hy-

drostatic pressure conditions in some of the experi-

ments, combined with an almost certain anisotropic

response of T

versus uniaxial stress. On the average

the pressure derivative of the critical temperature,

∂T

/∂p is of the order of 0.8 K/GPa. Clear evidence

for isotope effects have been obtained for isotope ex-

changes on both the B [312,313] and the Mg [313]

sites. As may be expected, the Boron isotope effect

is particularly prominent and from corresponding

experimental results, the measured shift of the mag-

netic transition is displayed in Fig. 14.77, the mass

exponent ˛

in Eq. (14.10) related with the isotopes

Band

B, is deduced to lie between 0.26 and 0.3.

This seems to confirm that lattice modes involving

B atoms are involved in interactions providing the

pairing of electrons. It is of no surprise that the

exchange of the two Mg isotopes mentioned above

leads to an only marginal shift of T

and hence the

Mgmassexponent˛

≈0.02.Whilethepressure de-

pendence of T

may partly be due to a change in the

electronic excitation spectrum, the isotope effect, al-

though reducedto belowitsclassical value[313],pro-

vides convincing evidence that the lattice and hence

theelectron–phononinteractionare essential for this

superconductivity at moderately high temperatures.

14 High-T

Superconductivity 821

Fig. 14.77. Boron isotope effect on the magnetically probed

superconducting transition of MgB

(see [312])

Fig. 14.78. Magnetization M(H) curve of a powder sample

of superconducting MgB

Simple magnetization M(H) curves (see

Fig. 14.78) [314] reveal at once that MgB

is a type

II superconductor,adopting a fairly large GL param-

eter . Many attempts to measure the temperature

dependence of the upper critical field H

(T) have

been made. The available data sets on H

(0) and

(∂H

/∂T)

cover a wide range of values for both

quantities [81]. It appears that the claimed H

val-

ues are systematically higher if measured on thin

film samples rather than on bulk crystalline speci-

mens and reported values for H

(0) vary between

15 and 40 T. Because of these uncertainties, which

partlymaybecausedbyanisotropyeffects,wecon-

centrate on the few experimental results that have

been obtained using single crystals [291,292].Unfor-

tunately these experiments, measuring the resistive

transition for currents in the ab plane and external

fields oriented either parallel or perpendicular to

the c-axis, have not been carried to very high fields

and therefore any extrapolation to low temperatures

is questionable. Nevertheless, a clear anisotropy is

discernible in these data. The destruction of the su-

perconducting state is more easily achieved,if the ex-

ternalmagnetic fieldis applied along the directionof

the c-axis of the crystal lattice. Overall,an anisotropy

ratio H

(H||ab)/H

(H||c) ≈ 2isobserved.Near

(T) derived from monitoring resistive transi-

tions,definitelyexhibits a positive curvature for both

field orientations. From these experimental data, the

corresponding coherence lengths have been esti-

mated using Eq. (14.8). The respective values are



(0) = 65 Å and 

(0) = 25 Å. A comparison with

the electronic mean free path of the order of 500 to

700 Å, that may be estimated from the normal state

transport properties, indicates that MgB

may be

treated as a clean-limit superconductor.Various esti-

mates [81,303] of average penetration depths 

(0)

at zero temperature range between 1200 and 1850 Å.

Estimates of the anisotropy of superconducting

properties using electron spin resonance (ESR) and

magnetization data from measurements on powder

samples indicate an anisotropy parameter  between

6 and 9, i.e., much larger than what is quoted above

from measurements of the upper critical fields on

single crystals [315].

The bulk transition to the superconducting state

has also been verified by measurements of the spe-

cific heat. Distinct anomalies indicating the discon-

tinuity C = C

− C

at T

have been reported

[302–304]. In magnitude these discontinuities are of

the order C/T

≈ 3mJ/moleK

,avaluethatis

822 H.R.Ott

distinctly smaller that what is expected if the weak

coupling BCS limit is considered. Electron phonon

coupling parameters  between 0.6 and 1.8 have

been derived. These values are of the same order of

magnitude as results from band structure calcula-

tions [316]. Most estimates of the ratio 

(0)/T

im-

ply that MgB

should be regarded as a weakly coupled

superconductor,rather surprising in view of the rel-

atively high critical temperature. Specific heat data at

temperatures well below T

have been used to reach

conclusions concerning details of the gap function.

Whilesomereportsclaimthatthedataareconsistent

with a common s-wave type pairing and a gap with-

out nodes [304], a more detailed analysis suggests

that at least two gap values, differing by a factor of

4, have to be considered [302,303]. In [303], the C

data obtained at very low temperatures are claimed

to suggest line nodes of the gap and also the field de-

pendence of C

indicates a d-wave pairing.Similarly

inconclusive are available tunneling data [317,318].

In this context,however,a first principles calculation

of the electron phonon coupling in MgB

leads to

the conclusion that two gaps, related with different

coupling strengths on different parts of the Fermi

surface, should develop [319].

The NMR response in the superconducting state

has also been investigated by a number of groups.

The Knight shift data [307,308] exhibit a reduction

of K below T

but a rigorous analysis is very much

hampered by the small size of the shift in the first

place [309]. In one of the studies the observation of

a coherence induced enhancement of the spin lattice

relaxation rate T

−1

below T

and a subsequent dras-

tic reduction of T

−1

has been reported [307]. The

result was interpreted as reﬂecting a strong coupling

s-wave type superconducting state of MgB

.Inother

work[308,309],noevidencefora coherencepeakwas

identified and only minor reductionsof T

−1

belowT

were observed in rather low fields. It cannot be ruled

out that problems with sample purities and inhomo-

geneities are responsible for these discrepancies.

First reports on probing the temperature depen-

dence of the occupied electronic states close to

above and below T

using photoelectron spec-

troscopy on polycrystalline material [320] indicate

that the gap formation is close to isotropic in k-

space, consistent with s-wave pairing. The temper-

ature dependence of the gap amplitude is well repro-

duced with the original BCS expression of (T)and

2(0)/k

=3.0, a value indicating weak coupling.

At this point is seems fair to say that the situation

concerning the basic aspects of superconductivity

of MgB

has not yet been settled. Although there is

some consensus that the electron pairing in the su-

perconducting state of MgB

is, in view of the high

critical temperature, caused by an amazingly weak

electron phonon interaction, many details need to be

clarified. To some extent this situation is due to the

present lack of experimental data obtained from sin-

gle crystalline specimens. In particular, any aspects

concerning the anisotropy of normal state and su-

perconducting properties of this material must be

treated by using single crystals.

Polycrystalline or powder samples of high purity,

however, may be used for exploring the potential of

MgB

forapplications[293].Inthiscontextsomevery

encouraging results have been obtained [81]. Mea-

surements of critical current densities j

as a func-

tion of temperature and magnetic field in samples of

different configurations have beenmade.Contrary to

the Cu-oxide materials, the critical currents in MgB

seem to be controlled by the pinning strength for

trapping vortices and not by effects of weak links at

grain boundaries [321].Inzero magnetic field,j

val-

ues of the order of 10

A/cm

have been achieved in

films [322] and about an order of magnitude less in

bulk samples,wires andtapes[81].In wires andtapes,

the critical current values are reduced rather rapidly

in external magnetic fields, even at 4.2 K, much more

than for the technically important superconductors

of Nb-Ti alloy and Nb

Sn, which exhibit much lower

critical temperatures. It has been established, how-

ever, that at the same fields, MgB

films may sustain

much higher current densities and, in the range of

liquid Helium temperatures, perform equally well as

commercial Nb-Ti and Nb

Sn wires [322]. This nur-

tures hopes that also the performance of wires and

tapes may considerably be improved. Last but not

least,the fabrication of superconducting MgB

-based

conductors appears to be a rather low cost affair.

14 High-T

Superconductivity 823

14.6 Final Remarks

The field of superconductivity has experienced a

number of unexpected discoveries during the last 20

years. Only some of them are discussed in this chap-

ter, others are discussed in different sections of this

book.

The appearance of superconductivity at unprece-

dented high temperatures has not only triggered re-

search activities worldwide, but has also inﬂuenced

the thinking with respect to the common under-

standing of the metallic state in solids in general.

While there is consensus that the cuprate materials

exhibit some unconventional superconductivity, itis

yet not clear what is the pairing mechanisms. Note,

recent ARPES experiments [321,323] observe inter-

esting changes in the dispersions of some high-T

cuprate superconductors.

There is little doubt that electron phonon interac-

tions are essential for superconductivity in the con-

sidered non-cuprate oxides,the fullerides and MgB

Nevertheless, there are still open questions with re-

gard to the characterization of the superconducting

state in these materials.

Acknowledgement

I thank H. Thomas and S. Sigrist for their help in

preparing the manuscript.

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