J000

Temperature (K)

Fig. 15.6. Temperature dependence of the shielding fraction xsh for an Rb,C6Q powder sample [15.44].

temperature dependent, as shown in Fig. 15.6 [15.44], The value for xsh that is quoted when the sample characterization specifications are given is xsh(T -> 0). Many of the superconducting fullerene samples that were prepared initially had low values of xsh (< 0.5), although more recent samples typically have higher values, such as xsh > 0.6, reported for K3C60, CsRbjQo, and Cs2RbC60 [15.45]. Much effort continues to be given to improving synthesis techniques for preparing M3C60 compounds with high values of xsh, high Tc values, and small transition widths ATC [15.46-48],

Referring to the BCS formula [Eq. (15.1)], if the dominant phonon frequency responsible for the electron-phonon coupling is known, then \ep can be found from Eq. (14.5), assuming that the fullerene superconductors are in the weak coupling limit. However, if the fullerene superconductors are strong coupling superconductors, we should instead use the McMillan solution [15.49] of the Eliashberg equations ha>ph r -1.04(1 + \ep)

where aJph denotes the dominant phonon frequency for the electron-phonon coupling, and where the effective electron-electron repulsion is denoted by /i* and is related to the short-range Coulomb repulsion ¡x by

for /X < 1. In Eq. (15.4) the Coulomb interaction may be reduced because the screening of one electron by another is faster than the effective vibrational frequency of the electron-phonon coupling. Most superconductors have values of /a* between 0.1 and 0.3, and values of /a* in this range have also been suggested for fullerene-based superconductors [15.50].

To gain further understanding of the McMillan formula with regard to doped fullerenes, numerical evaluations of Xep and fi* have been reported by several groups based on phonon mode and electronic structure calculations, since it is difficult to calculate Tc from first principles [15.50,51]. Referring to Eq. (15.1), one problem that is encountered in estimating \ep = VN(Ef) for crystalline C60 is the large range of experimental values for N(Ef) (see §14.5) and V in the literature. Referring to Eq. (15.3), there is also a large range of values in the literature for wph (see §14.2.2). Thus the theoretical understanding behind the relatively high Tc values in the doped fullerenes is still in a formative state. We summarize below the range of values in the literature for V and wph.

In §15.7, results are given for a number of models for calculating V, the pairing interaction energy in Eq. (15.1) arising from the electron-phonon interaction. Assuming that the electron pairing interaction is via intramolecular phonons, the summary of values reported in the literature (between 32.2 and 82.3 meV [15.52]) indicates the need for further systematic work, especially because of the high sensitivity of Tc to V through the exponential relation in Eq. (15.1). A value of V ~ 50 meV provides a rough estimate for this interaction energy [15.3].

In §15.7, some discussion is given of the various viewpoints on the specific phonon modes that are most important in coupling electrons and phonons in the pairing mechanism. As noted in §15.7, wph is determined by Eq. (14.7) from the contribution to kep from each vibrational mode. A better determination of a>ph awaits a definitive answer about the modes that contribute most importantly to Xep, and this subject is also considered further in §15.7.

15.3. Magnetic Field Effects

The most important parameters describing type II superconductors in a magnetic field are the upper critical field Hc2 and the lower critical field HcV The lower critical field denotes the magnetic field at which flux penetration into the superconductor is initiated and magnetic vortices start to form.

The lower critical field //cl(0) in the limit T 0 is also related to the Ginzburg-Landau coherence length £0 and the London penetration depth \L through the relation

Experimental values for the macroscopic parameters of the superconducting phases of KjC«, and Rb^C^.

Parameter

l^Qo

RbjC«,

fee a0 (A)

14.253»

14.436"

Tc (K)

19.7"

30.0"

2A(0)/kTc

5.2C, 4.0', 3.6s, 3.6h

5.3d, 3.1', 3.6^, 3.0«, 2.98h

(dTc/dP)p=0 (K/GPa)

-7.8'

-9.7'

Hcl(0) (mT)

13'

26', 19*

Hc2(0) (T)

26', 30', 29m, 17.5"

34', 55', 76"

Hc(0) (T)

0.38'

0.44'

Jc (106 A/cm2)

0.12'

1.5'

So (nm)

2.6', 3.1', 3.4", 4.5"

2.0', 2.0", 3.0"

A,, (nm)

24Qi, 480°, 600', 800«

168', 37iy, 460", 800«, 210*

K = KU 0

92'

84' , 90*

dHJdT (T/K)

-1.34", -3.5r

-3.8"

foo (nm)

9.5", 12.0', 15.01

4.0-5.5"

I (nm)

3.1", 1.0'

0.9"

"Ref. [15.30]; "Ref. [15.50]; CSTM measurements in Ref. [15.53]; 'STM measurements in Ref. [15.54]; 'NMR measurements in Ref. [15.55,56]; f /xSR measurements in Ref. [15.57]; sFar-IR measurements in Ref. [15.58]; "Far-IR measurements in Ref. [15.59]; 'Ref. [15.17]; 'Ref. [15.60]; 'Ref. [15.44]; 'Ref. [15.61]; mRef. [15.62]; "Ref. [15.63]; "Ref. [15.64]; 'Ref. [15.55]; «Ref. [15.65,66]; 'Ref. [15.67]; 5Ref. [15.68]; 'Ref. [15.69],

"Ref. [15.30]; "Ref. [15.50]; CSTM measurements in Ref. [15.53]; 'STM measurements in Ref. [15.54]; 'NMR measurements in Ref. [15.55,56]; f /xSR measurements in Ref. [15.57]; sFar-IR measurements in Ref. [15.58]; "Far-IR measurements in Ref. [15.59]; 'Ref. [15.17]; 'Ref. [15.60]; 'Ref. [15.44]; 'Ref. [15.61]; mRef. [15.62]; "Ref. [15.63]; "Ref. [15.64]; 'Ref. [15.55]; «Ref. [15.65,66]; 'Ref. [15.67]; 5Ref. [15.68]; 'Ref. [15.69], where <f>0 is the magnetic flux quantum. The value of Hcl(0) is typically very low for fullerene superconductors (see Table 15.1).

The upper critical field denotes the magnetic field above which full magnetic flux penetration takes place and a transition from the superconducting to the normal state occurs. The determination of Hc2(0) in the limit T —>■ 0 is also of importance, since Hc2(0) is related to the coherence length i0 of the superconducting wave function through the relation

The temperature dependence of the resistivity of a superconductor in a magnetic field provides one method for studying the magnetic field penetration in the superconductor. Such data are shown in Fig. 15.7 for a single crystal of K3C60 (where the resistivity is normalized to its room temperature value) for various values of magnetic field (up to 7.3 T) [15.70], The results of Fig. 15.7 show that the application of a magnetic field decreases Tc and increases the transition width A Tc, as is also observed for high-Tc cuprate superconductors. One can obtain the T dependence of Hc2(T) from p/p0

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