Superconductivity

CHAPTER 15

Superconductivity The first observation of superconductivity in doped C60 attracted a great deal of attention because of the relatively high transition temperature T~ that was observed, the first observation being in the alkali metal-doped K3C60 with a Tc of 18 K [15.1]. Ever since, superconductivity of doped fullerenes has remained a very active research field, as new superconducting fullerene materials were discovered and attempts were made to understand the unusual aspects of superconductivity in these fascinating materials and the pairing mechanism for the electrons. At the present time, superconductivity has been reported only for C60-based solids, although transport properties down to -~1 K have been measured on alkali metal--doped C70 [15.2] and perhaps higher-mass fullerenes as well. In this chapter the experimental observations of superconductivity in various fullerene-related compounds are reviewed, followed by a discussion of critical temperature determinations and how T~ relates to the electronic density of states at the Fermi level. Magnetic field phenomena are then summarized, followed by a discussion of the temperature dependence of the energy gap, the isotope effect, and pressure-dependent effects. The leading contenders for the pairing mechanism (the electron-phonon and electron-electron interactions) are then discussed, although it is by now generally agreed that the electron-phonon interaction is the dominant pairing mechanism, giving rise to a mostly conventional Bardeen-CooperSchrieffer (BCS) fullerene superconductor [15.3]. 15.1.

EXPERIMENTALOBSERVATIONS OF SUPERCONDUCTIVITY

The first observation of superconductivity in a carbon-based material goes back to 1965, when superconductivity was observed in the first-stage alkali 616

617

15.1. ExperimentalObservationsof Superconductivity

metal graphite intercalation compound (GIC) C8K (see w [15.4]. Except for the novelty of observing superconductivity in a compound having no superconducting constituents, this observation did not attract a great deal of attention, since the superconducting transition temperature Tc in CsK was very low (--~140 mK) [15.5]. Later, higher Tc values were observed in GICs using superconducting intercalants (e.g., second-stage KHgC8, for which T~ = 1.9 K [15.6, 7]) and by subjecting the alkali metal GICs to pressure (e.g., first-stage CENa, for which Tc ~ 5 K) [15.8]. As stated above, the early observation of superconductivity in a doped fullerene solid attracted interest because of its relatively high Tc (18 K in K3C60) [15.1]. This work was soon followed by observations of superconductivity at even higher temperatures: in RbaC60 ( T c -- 30 K) [15.9, 10], in RbCsEC60 ( T c -- 33 K) [15.11, 12], and in Cs3C60under pressure (T~ = 40 K) [15.13]. It is interesting to note that if superconductivity had been found in these alkali metal-doped C60 compounds prior to the discovery of superconductivity in the cuprates [15.14], the alkali metaldoped fullerenes would have been the highest-Tc superconductors studied up to that time. At present, the highest-To organic superconductor is an alkali metal-doped fullerene, namely Cs3C60 under pressure (--,12 kbar) with Tc ~ 40 K [15.13], breaking a record previously held by the tetrathiafulvalene derivative (BEDT-TFF)ECuN(CN)2C1 with T c = 12.8 K under 0.3 kbar pressure [15.15]. Table 8.3 lists the transition temperatures for various fullerene-based superconductors. For the alkali metal-doped fullerenes, a metallic state is achieved when the tlu-derived level is approximately half-filled, corresponding to the stoichiometry M3C60 for a single metal (M) species or MxM~_xC60 for a binary alloy dopant, as discussed in w Increases in Tr relative to that of K3C60 have been achieved by synthesizing compounds of the type MxM~_xC60, but with larger intercalate atoms, resulting in unit cells of larger size and with larger lattice constants as seen in Table 8.3 and in Fig. 15.1. As the lattice constant increases, the intermolecular C60-C60coupling decreases, narrowing the width of the qu-derived LUMO band and thereby increasing the corresponding density of states. Simple arguments based on BCS theory of superconductivity yield the simplest BCS estimate for T c kB T~ = 1.13~ph exp (

1)

N(EF)V

'

(15.1)

where ~ph is an average phonon frequency for mediating the electron pairing, N ( E F ) is the density of electron states at the Fermi level, and V is the superconducting pairing interaction, where the product N ( E F ) V is equal to Aep , the electron-phonon coupling constant. The Debye temperature as measured from a specific heat experiment for the case of fullerenes refers

618

15. Superconductivity

l

i

'

i

Rb 2 Cs 30 -

9 Rb 3

9 Rb2K # 2 9

25--

9

K1.5Rbl. 5 Rb2K #1

K2Rb 20 9 =

(a)

K3 l

I

I 14.4

14.3

14.5

ao

40 35 30

RbCs 2 b2Cs

25 20

r

K=Csy KxRby

K3

15

Fig. 15.1. (a) Early reports on the dependence of Tc for various MxM~_xC60 compounds on the fcc lattice constant a 0 [15.16]. (b) More complete summary of the dependence of Tc on a0 [15.12], including points provided by pressure-dependent studies of T~ [15.17].

Na2Cs LizCs

10 &

5 0, 13.9

(b)

Na2Rb _C60 Na~K~, , 14.1

14.3

14.5

Lattice constant a (A)

14.7

15.1. ExperimentalObservationsof Superconductivity

619

to the intermolecular degrees of freedom, while the pairing is generally believed to be dominated by high-frequency intramolecular vibrations, although the detailed pairing mechanism still remains an open question at this time (see w Some typical values for the parameters in Eq. (15.1) are ~ph ~ 1200 cm -1, V ~ 50 meV, and N ( E F ) ~ 9 states/eV/C60/spin which are consistent with Tc "~ 18 K [15.3]. The BCS formula for T c [Eq. (15.1)] suggests that higher T~ values might be achievable by increasing N ( E e ) . A high density of electron states in the doped fullerenes occurs primarily because of the highly molecular nature of this solid, resulting in a narrow width (0.5-0.6 eV) for the electronic LUMO bands in K3C60 and Rb3C60. Two main approaches have been used experimentally to increase T c. The first approach has been directed toward extensive studies of the MxM~_xC60 alkali metal system to maximize the lattice constant while maintaining the face-centered cubic (fcc) phase, and the second has been directed toward the introduction of neutral spacer molecules such as NH 3, along with the dopant charge transfer species, to increase the lattice constant. The first step taken toward increasing T~ was the experimental verification that T~ could be increased by increasing the lattice constant. This was done in two ways, the first being a study of the relation between Tc and the lattice constant a0 within the fcc structure, and the second involved study of the effect of pressure on Tc. The ternary MxM~_xC60 approach [15.11, 16, 18-21] provided many data points for evaluating the empirical relation between Tc and a0 (see Fig. 15.1). Early work along these lines validated the basic concept [15.16] and led to the discovery of the highest present T c value for a binary alloy compound, RbCs2C60 [15.12, 18]. Studies of the pressure dependence of Tc for K3C60 and Rb3C60 (see w provided further confirmation of the basic concept and provided points (denoted by open symbols) for the plots in Fig. 15.1 [15.17,22,23]. The crystal structure of the highest-Tc doped fullerene Cs3C60under pressure (Tc = 40 K at 12 kbar) is not fcc, and is believed to be a mixed A15 and body-centered tetragonal (bct) phase [15.13], and for this reason cannot be plotted on Fig. 15.1, which is only for the fcc structure. Much effort has gone into discussing deviations from the simple empirical linear relation between Tc and a0 for small a0 values and strategies to enhance Tc. Regarding the low a0 range, it was found that the introduction of the small alkali metals Li and Na for one of the M or M' metal constituents lowered a0 below that for undoped C60, indicating the importance of electrostatic attractive interactions in the doping process. Some alkali metal MxM~_xC60 compounds with these small a 0 values exhibit superconductivity with low T~ values and nevertheless tend to fall on the solid curve of Fig. 15.1(b). Other compounds show anomalously low T~ values relative to their lattice constants, as also indicated in the figure.

620

15.

Superconductivity

A systematic study of the relation between Tc and the lattice constant for CSxRb3_xC60was carried out [15.12] in an effort to maximize To, since Cs2RbC60 is the fcc alloy superconductor with the highest T~ for the M3C60 series. Direct measurement of T~ vs. alloy composition for this family of compounds presented in Fig. 15.2 [15.12] showed that the maximum T~ occurs near the stoichiometry Cs2RbC60. This work eventually led to the discovery of the Cs3C60phase under pressure with a higher Tc value but having a different crystal structure [15.13]. It should be noted that as the lattice constant increases, the overlap of the electron wave function from one C60 anion to that on adjacent anions decreases, and this effect tends to lower T~, in contrast to the effect discussed above. Thus there is a limit to the maximum Tc that can be achieved merely by increasing the lattice constant. A vivid demonstration of the effect of the increase of lattice constant a0 on T~ is found in the uptake of NH 3 by Na2CsC60 to form the ternary intercalation compound (NH3)4Na2CsC60, thereby increasing the lattice constant from 14.132 /~ to 14.473 /~ while increasing the Tc from 10.5 to 29.6 K [15.24]. We thus see that the Tr value of 29.6 K achieved for (NH3)4Na2CsC60 fits the empirical T~ vs. ao relation of Fig. 15.1 and suggests that the insertion of spacer molecules to further increase a0 within the fcc structure [15.12] may lead to further enhancement of T~. However, the results of Fig. 15.2 for the CsxRb3_xC60 system suggest that further enhancement of Tc by simply increasing a0 may not be rewarding. Structural studies for the (NH3)4Na2CsC60compound show that all of the (NH3) groups and half of the Na ions go into octahedral sites in the form of large Cs~Rb3-,C6o 33

J

/ /

32

/

%

f

I l

/

,,O

31 S /

3O

/ /

~u 29

D.

I Il l I

q, 1 1

Fig. 15.2. Dependence of Tc on Cs concentration for CsxRb3_xC60. The discontinuity in the dashed curve at x = 2.7 indicates a change in crystal structure from fcc for x <__2.7 to bcc close to x - 3 [15.12].

I l I I

3

15.1. ExperimentalObservations of Superconductivity

621

(NH3)4Na cations [15.24]. The remaining bare Na and Cs ions are found in tetrahedral sites [15.24]. This is somewhat unusual, since in the pristine NazCsC60 the Cs ion occupies an octahedral site while the Na ions preferentially occupy tetrahedral sites. The (NH3)nNa2CsC60 compound is analogous to ternary graphite intercalation compounds of the type M-C6H 6 GICs, MTHF GICs, or M-NH3 GICs, where M denotes the alkali metal. Such compounds provided a powerful vehicle for tailoring the physical properties of GICs [15.25,26]. Ternary fullerene compounds of this kind may likewise open up new possibilities for modifying the properties of fullerene host materials. Since the addition of (NH3) 4 to Na2CsC60 was effective in increasing T c, ammonia was added to K3C60 in an attempt to enhance Tc further. A stoichiometry of (NH3)K3C60 was achieved upon NH 3 addition, without additional charge transfer, but this compound was not found to be superconducting [15.27] even though the unit cell volume was almost the same as that for RbECsC60 with a Tc = 31.3 K (see Table 8.3). The introduction of (NH3) produced a 6% volume expansion of the unit cell, accompanied by a distortion of the crystal structure away from cubic symmetry. The absence of superconductivity in NH3K3C60 was attributed to either departures from a cubic crystal structure or charge localization on the C~o anions due to correlation effects [15.27]. Since the Tc of a superconductor such as KxRb3_xC60 varies continuously with x [15.18], it is believed that the superconductivity in alkali metal alloy systems M3_xM'xC6ois not sensitive to superconducting fluctuations [15.20]. This is in contrast to organic superconductors such as (BEDTTrF)EX, where a mixture of two different ions suppresses superconductivity [15.28,29]. It is generally believed that a relation between T~ and the electronic density of states at the Fermi level N ( E F ) is more fundamental than between T~ and the lattice constant a0, and efforts have been made to present the literature data in this form (see Fig. 15.3) [15.22]. On the other hand, there is considerable uncertainty presently regarding the magnitude of the experimental electronic density of states for specific compounds (see w while the lattice constants can be more reliably measured. It is for this reason that plots of Tc vs. lattice constant a0 are more commonly used at the present time by workers in the field. Superconductivity has also been observed in alkaline earth-doped superconductors with Ca, Ba, and Sr. Although the T~ value for these compounds is much lower than for alkali metal dopants, the superconductivity in these compounds is interesting in its own right and merits attention. Since the alkaline earth ions tend to have smaller ionic radii than the alkali metal ions (see Table 8.1), they would be expected to result in less lattice expansion

622

15. Superconductivity

30

~"

Fig. 15.3. Dependence of Tc on the density of states at the Fermi level for K3C60(closed circles) and Rb3C60(open circles) [15.22] using pressure-dependent measurements of T c on these compounds [15.17,23].

20

10 20

25

30

DOS ( eV "1. C61 )

and therefore lower Tc values generally, and this is indeed found experimentally [15.30-32]. Also because of their smaller size, multiple alkaline earth metal ions can enter the larger octahedral sites for the case of the fcc structure. In fact, the stoichiometries that exhibit superconductivity from magnetic susceptibility and microwave measurements are Ca5C60 [15.30], Ba6C60 [15.31], and Sr6C60 [15.33], and neither Ba6C60 nor Sr6C60 exhibits the fcc structure. Shown in Fig. 15.4 are measurements of the magnetization vs. temperature for Ca5C60 in a field-cooled (FC) and zero-field-cooled (ZFC) situation, with the intersection between the two curves yielding the Tc value. Two aspects of the superconducting alkaline earth compounds bear special attention. The first relates to the difference in crystal structures relative to the alkali metal compounds and the greater variety of crystal structures 0.3

0.0

:-0-.'.-=-.'-

Ul u) :::) -0.3

-0.6

Fig. 15.4. Magnetization vs. temperature for field-cooled (FC) and zero-field-cooled (ZFC) cycles for Ca5C60 [15.30].

-o.g 4

I 6

T(K)

' 8

10

623

15.1. Experimental Observations of Superconductivity

that support superconductivity in the alkaline earth-doped C60.The second relates to the divalence of the alkaline earth ions, where up to two electrons per dopant atom are donated to the C60-derived states. The tlu-derived band is filled at three alkaline earth dopants per C60 unit, and upon further metal ion addition, the tlg-derived bands start to fill, reaching a metallic phase when they are about half-filled at a stoichiometry between five and six alkaline earth ions per C60 unit (see w since the metal-derived bands significantly hybridize with the C60 bands above the tlu-derived band. Figure 15.5 shows that the T~ for the fcc superconducting alkaline earth Ca5C60 follows the same relation between T c and the C60-C60 separation that is found for the alkali metal M3C60 compounds, whereas the bodycentered cubic (bcc) alkaline earth compounds Sr6C60 and Ba6C60 also show correlation between Tc and the C60-C60distance, but show a different functional relation relative to the alkali metal compounds. Regarding t h e issue of crystal structures, superconductivity in the alkali metal MxM~_xC60 compounds corresponds to the fcc F m 3 m space group, while superconductivity in the alkaline earth compounds corresponds to the following structures: Ca5C60 has a simple cubic structure where each of the four C60 anions of the fcc structure is distinct, while Ba6C60 and 8r6C60 both have the bcc I m 3 crystal structure (which is also the structure exhibited by the semiconductors K6C60 and Rb6C60) (see w and w Since the superconductivity for all of these compounds is similar, we conclude that the superconductivity is predominantly influenced by the C60-derived states and to a lesser degree by the dopant species. With regard to band filling, the observation of superconductivity in Ca5C60, Ba6C60, and 8r6C60 indicates a transfer of electron charge carriers to C60, beyond the six electrons needed to fully occupy the t~u bands. These observations indicate that superconductivity is possible for both a half-filled tlu band and a partially (perhaps half) filled tlg band. Symmetry arguments (see w [15.30] suggest that if the

30

20 u I--

10 Sr s C s o j ~ J Ba e C so ,

,

,

,

.

,

,

,

i

0 9.4

9.6

9.8

10.1

Bali.Ball Separation (A)

,.

10.3

Fig. 15.5. Plot of the superconducting transition temperature Tc dependence on the intermolecular C60-C60 separation of fcc superconductors (including Ca5C60) is extended to include the bcc superconductors Ba6C60 and 5r6C60. The bcc-based superconductors also exhibit a positive slope in the Tc vs. C60-C60 plot, but with a different magnitude for the two slopes

[15.33].

15. Superconductivity

624

electron-phonon interaction is the dominant electron-pairing mechanism, then the electron-phonon matrix elements would be expected to be similar for electrons in the tl~ or tlg bands (see w It is also interesting to note that alkali metal-doped C70 [15.34--38] and alkali metal-doped higher-mass fullerenes C76 [15.38,39], C78 [15.38], C82 [15.40], C84 [15.38,41,42], and C90 [15.38] do not show superconductivity. Further, the addition of C70 to C60 results in a rapid decrease in T~ for the Mx(C60)l_y(C70)y compounds [15.38, 43]. In fact, doped C70 never becomes fully metallic, showing activated conduction even for the concentration K4C70 where the magnitude of the conductivity is a maximum (see w Furthermore, it is not expected on the basis of curvature arguments that doped carbon nanotubes would be attractive as a high-Tc superconducting material (see w It has been argued that the lower curvature of the higher-mass fullerenes suppresses the electron-phonon interaction and thus suppresses the occurrence of superconductivity. It would thus appear that doping might enhance the electron-phonon coupling for the lowermass fullerenes more than for C60. But at the same time, the smaller lattice constant would be expected to lead to a lower density of states at the Fermi level. Thus it is not clear whether higher T~ values could be achieved by the doping of smaller-mass fullerenes. 15.2.

CRITICALTEMPERATURE

The onset of superconductivity is typically measured in one of three ways: (1) by the loss of resistivity in temperature-dependent p ( T ) curves (see Fig. 14.3), (2) the temperature where the field-cooled and the zero-fieldcooled magnetic susceptibilities merge (see Fig. 15.4), and (3) the onset of the decrease in microwave losses in temperature-dependent surface resistance plots R~(T). In the initial report of superconductivity in K3C60, evidence for superconductivity by all three methods was reported [15.1]. From measurements using one or more of the three techniques mentioned above, T~ is found for specific doped fullerene compounds, and the results for a variety of doped fullerene compounds are given in Table 8.3. A plot of the field-cooled and zero-field-cooled magnetic susceptibility vs. T is probably the most convenient method for the measurement of T~ (see Fig. 15.4 [15.30]), since the slope of the magnetization curve M ( H ) also gives the shielding fraction or superconducting fraction X~h through the relation X~h= --4"rr(~

,

(15.2)

H~0

and an ideal type I superconductor is a perfect diamagnet (i.e., X~h = 1). Since the shielding fraction is dependent on the flux exclusion, X,h is

625

15.2. Critical Temperature 0.12 ! O.lO

o.08

Xsh

o.o6

0.04

0.02

0.00 ' 1 ! 0.00

i,

i1,11

ii'lwl

! / i'1 Iiii

I'1111

! illl

i,!1

t illliili,

! [11

i i,

ii1111

5.00 10.00 15.00 20.00 25.00 30.00

T e m p e r a t u r e (K)

Fig. 15.6. Temperature dependence of the shielding fraction Xsh for an Rb3C60 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, CsRb2C60, and Cs2RbC60 [15.45]. Much effort continues to be given to improving synthesis techniques for preparing M3C60 compounds with high values of X~h, 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 Aep 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 h~'oh Zc -- 1.45k'~ e x p

{ --1.04(lq-Aep) } 0.62Aep) )~ep -

t"$*(1 -~"

(15.3)

where tO---phdenotes the dominant phonon frequency for the electronphonon coupling, and where the effective electron--electron repulsion is denoted by/~* and is related to the short-range Coulomb repulsion/z by ~* =

/x 1 +/z ln(Er/~oh)

(15.4)

for/z < 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

626

15. Superconductivity

have values of/z* between 0.1 and 0.3, and values of ~* 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 i~ep and/z* have been reported by several groups based on phonon mode and electronic structure calculations, since it is difficult to calculate T c from first principles [15.50,51]. Referring to Eq. (15.1), one problem that is encountered in estimating i~ep "-- VN(EF) for crystalline C60 is the large range of experimental values for N ( E F ) (see w and V in the literature. Referring to Eq. (15.3), there is also a large range of values in the literature for ~ph (see w Thus the theoretical understanding behind the relatively high T c values in the doped fullerenes is still in a formative state. We summarize below the range of values in the literature for V and ~ph. In w 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 w 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 w ~ph is determined by Eq. (14.7) from the contribution to Aep from each vibrational mode. A better determination of ~ph awaits a definitive answer about the modes that contribute most importantly to '~.ep, and this subject is also considered further in w

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 Hcl. 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 Hc~(0) in the limit T ~ 0 is also related to the Ginzburg-Landau coherence length ~:0 and the London penetration depth AL through the relation ~b0 ln(AL/~0). H~,(0) = 47rA2L

(15.5)

15.3.

Magnetic Field Effects

627 Table 15.1

Experimental values for the macroscopic parameters of the superconducting phases of K3C60 and Rb 3C60. Parameter

K3C60

Rb3C6o

fcc a 0 (A) T~ (K)

14.253 a 19.70 5.2 ~, 4.0 e, 3.6 g, 3.6 h -7.8 i 13J 26 j, 30 t, 29", 17.5 b 0.38' 0.12' 2.6', 3.1 t, 3.4", 4.5 b 24@, 480 ~ 600 p, 800q 92J -1.34 b, -3.5 r 9.5 b, 12.0 t, 15.1Y 3.1 b, 1.0 t

14.436 a 30.0 b 5.3 d, 3.V, 3.6Y, 3.0 g, 2.98 h -9.7' 26J, 19k 34J, 55 t, 76 b 0.44' 1.5J 2.0', 2.0 b, 3.0" 168J, 370 r, 460 p, 800q, 210k 84/, 90k -3.8 b

2A(O)/kTc

(dL/aP).=o (K/GPa)

H~I(0 ) (mT) Hc2(0 ) (T) He(0 ) (T)

Jc ( 106 A/cm2) ~0 (nm) a L (nm) K = AL/ ~0 dHcE/dT (T/K) ~:00 (nm) (nm)

4.0-5.5 b 0.9 b

aRef. [15.30]; bRef. [15.50]; cSTM measurements in Ref. [15.53]; aSTM measurements in Ref. [15.54]; eNMR measurements in Ref. [15.55,56]; //zSR measurements in Ref. [15.57]; gFar-IR measurements in Ref. [15.58]; hearIR measurements in Ref. [15.59]; iRef. [15.17]; JRef. [15.60]; kRef. [15.44]; tRef. [15.61]; mRef. [15.62]; "Ref. [15.63]; ~ [15.64]; PRef. [15.55]; qRef. [15.65,66]; rRef. [15.67]; SRef. [15.68]; 'Ref. [15.69].

where 4,0 is the magnetic flux quantum. The value of H c l ( 0 ) is typically very low for fuUerene 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 H c 2 ( 0 ) in the limit T --~ 0 is also of importance, since Hc2(0) is related to the coherence length ~:0 of the superconducting wave function through the relation He2(0) =

4'0 2zr~:2"

(15.6)

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 ATc, as is also observed for high-T~ cuprate superconductors. One can obtain the T dependence of H~2(T) from P/Po

628

15. Superconductivity

0.5

' ' '

'J

i

'

''i

9

~

,

'

'

',,r',

9

,

'

,o~ 0

~o0.3

.

9

0

9

0

0

9

O

9

!

9 o

Q..

'

K3C60 crystal

0.4 Fig. 15.7. Normalized resistivity of single-crystal K3C60 near Tc for varying values of the applied magnetic field: O, 0.0 T; i-1, 1.6 T; A, 3.1 T; 0, 4.6 T; II, 6.1 T; and O, 7.3 T. P0 is the room temperature value and the inset shows the zerofield temperature dependence of the normalized resistivity [15.50, 70].

'

13

.

I).2

-

9

0

9

o

9

0

~.o a.

0

0.1

"

0 10

,

0

9 o o 11_~ . ' ~ . ,

,

14

18

9

_e,.

0

1....

22 T(K)

l O0

200

TIK) l

t

26

,

,

30

vs. T data such as those shown in Fig. 15.7. For a given H, the temperature for which (P/Po) ~ 0 is used to obtain Hc2(T ). Results of this data analysis

for K3C60 are shown in Fig. 15.8. Assuming the validity of BCS theory, one can obtain the zerotemperature value of the critical field Hc2(0 ) from the slope of the Hc2(T) data near Tc by means of the Werthamer, Helfand, and Hohenberg (WHH) formula [15.71]" Hc2(0) = 0.69[~/~ 2] T c. Tc

(15.7)

According to the WHH formula values of He2(0 ) at T = 0 can be estimated from H~2 measurements predominately near Tr The results presented in Fig. 15.8 show that (dH~2/~T) is constant near T c, and Hc2 has a linear T dependence near T~ in accordance with Ginzburg-Landau theory. Justifi0 ...,...,...,...,... ~16 m

~12

'*8 Fig. 15.8. Temperature dependence of the upper critical field He2 for K3C60. The dots are experimental points taken at relatively low magnetic fields and the solid curve is a fit of these data to the WHH model (see text) [15.70].

4 0

4

8 12 T(K)

16

20

15.3. MagneticField Effects

629

cation for use of the WHH formula for fullerene superconductors comes from measurements on powder samples of K3C60 and Rb3C60 extending to very high magnetic fields and including Pauli-limiting spin effects [15.61]. An independent determination of the dHc2/dT has been made from analysis of the measured jump in the specific heat at Tc, yielding a value of dHc2/dT = -3.5 4- 1.0 T/K (see {}14.8.3) [15.67]. The high-field determinations of Hc2(0) yielding 30 T and 55 T for K3C60 and Rb3C60, respectively, represent the most detailed experimental determinations of Hc2(0) that are presently available, although the high-field measurements were not made on high-quality single crystals. Although prior work suggested departures from the WHH formula on samples taken to high fields [15.62] due to sample granularity and Fermi surface anisotropy, it seems to be generally believed on the basis of more detailed high-field measurements [15.61] that the WHH model provides a good first approximation for describing the functional form of Hc2(T ) for fullerene superconductors. Generally, Hc2(0 ) for single-crystal Rb3C60 is reported to be higher than that for an Rb3C60 film [15.17,23, 61, 72], although the opposite trend was reported for K3C60 (see Table 15.1). Further systematic studies are needed to clarify the dependence of the superconducting parameters on grain size and microstructure. Typical values for He2(0) are very high, as given in Table 15.1 for powder, film, and single-crystal K3C60 and Rb3C60 samples. Also given in Table 15.1 is the range of values for so0,which are typically found from H~2(0) [see Eq. (15.6)] and therefore reflect the corresponding range of values for Hc2(0). Also shown in Table 15.1 are values for the London penetration depth AL which is measured through magnetic field penetration studies. Microwave experiments for K3C60 and Rb3C60 have yielded values near 2000 ,/k [15.60, 73], while muon spin resonance studies [15.57,64,74] have yielded values of AL about a factor of 2 greater (-~4000 A), and far-infrared optical experiments have reported even larger values of AL (-~8000/~) [15.65]. The reason for this discrepancy in the value for AL is not presently understood, and this remains one of the areas where further systematic work is needed. Measurements for Hc2(0) show on the basis of Eq. (15.5) that so0 is only slightly larger than a lattice constant for the fcc unit cell of the fullerene superconductors. In contrast, the measured AL values are very much larger than both so0 and g (see Table 15.1), especially near Tc. Magnetization studies on powder samples of Rb3C60 show that the temperature dependence of the London penetration depth is well fit by the empirical relation [15.44] AL(T) = AGL(0)[1 -

~c]-1/2,

(15.8)

630

15. Superconductivity

where hCL(0) is the Ginzburg-Landau penetration depth at T = 0. The values of AL in Table 15.1 correspond to extrapolations of the measured AL(T ) to T --+ 0, i.e., to A~L(0), if Eq. (15.8) is used to fit the data. Measurements of the lower critical field He l, where magnetic flux penetration initially occurs, show that He1(0) for superconducting fuUerene solids has a very small value [15.17]. Microwave measurements indicate that n c l ( T ) follows the simple empirical formula [15.17,73]

[Hcl(T)/Hcl(O)] = 1 -- (T/Tc) 2.

(15.9)

Values for Hcl(0 ) are also given in Table 15.1, and through Eq. (15.5) the values for Hcl(O ) provide consistency checks for determinations of AL and so0. More detailed studies of the superconducting state for the M3C60 type II superconductors reveal a phase diagram akin to that of high-T~ cuprates, consisting of a Meissner phase (with no vortices), a vortex glass phase with pinned vortices, and a vortex fluid phase [15.75]. The inset to Fig. 15.9 shows the temperature dependence of the magnetic susceptibility x ( T ) for a powder sample of K3C60 (Tc = 19.0 K) under zero-field-cooling (ZFC) and field-cooling (FC) conditions for an applied field of 10 G. Comparing .

.

.

.

I

.

.

.

.

l

~"'

-

.

.

"!

.

.

.

.

i

" -

~

9

-

!

.

.

.

.

~ ~ O & O A O & O 6 0 & O & O 6 0 Q

oeme oommeoolmomoa~

K3Cso

8"------

o

____..6

o, &O&O&O&060O 5 ~''"

H=500

Oe

O

04

m E

-0

"

oO/

01

-

..__._-...3 9 6060 & o 6060&0~

U

2*--'---'----

-

r

-"'

i

'"-

]

-

i

-

3

9

O 0

E ft.)

0 o

I:: (J

-0.02

o

X

-0.02

o 1

:}

E (D

-0.04

H-tO

, -0.08

ooO ~

" 0

l 15

"

' 10

"

'

15

~

'

Oe "

;!0

' 25

" 30

T (K) -0.03

.... 0

'. . . . . . . . . . 10

5

'' 15

20

25

30

T (K)

Fig. 15.9. Temperature dependence of the magnetic susceptibility of K3C60measured at H = 500 Oe. Open circles for heating (numbered arrows: 1, 3, 4, 6, 7, and 9); open triangles for cooling (numbered arrows: 2, 5, and 8). The inset shows field-cooled (FC) and zero-fieldcooled (ZFC) susceptibilitydata for K3C60measured at H = 10 Oe [15.75].

631

15.3. Magnetic Field Effects

the measured low-temperature ZFC and FC values of X to the ideal X value of -1/47r yields values of 85% for the shielding fraction (ZFC) and 19% for the Meissner fraction (FC), consistent with an irreversible ZFC diamagnetism that is more than a factor of 4 greater than the reversible FC diamagnetism [15.75]. Referring to the main plot in Fig. 15.9 for the same K3C60 sample but now in a field of 500 Oe, we see irreversible behavior for the 1 ---> 2, 4 ~ 5, 7 ~ 8 segments and reversible behavior for the 2 ~ 3, 5 ---> 6, 8 ~ 9 segments. The intersection points between the reversible and irreversible behaviors converge to a value T* for a given H [i.e., each x ( T ) diagram such as Fig. 15.9 yields a single ( T * , H ) pair], which determines a point on the phase boundary between the vortex glass and vortex fluid phases shown in Fig. 15.10; the entire phase boundary in Fig. 15.10 is constructed by measuring T* for a range of H values [15.75]. The results in Fig. 15.10 show similarities in the superconducting phase diagram for K3C60 and Rb3C60, except for differences in scale arising from the different Tc values for these compounds. The phase boundaries for both K3C60 and Rb3C60 are fit by the relation H 9 .

A

(~

4

(a) ~~ .

.

!

."."..

!

. . . .

l,,~,

T*(H)/T*(O)] ~'

-

(15.10)

K3C6o .

.

.

!

9

VORTEX I .~_ FLUID I

"\'"

VORTEX GLASS

s

Ho[1

=

I

MEISSNER

k

. .

STATE

~

Hct

I

I

~

0 '" .-',-i~. : , { : . ' : ,-.~'N~...I , 9 0 5 10 15 20 6 ..... ....~, 9 9 . . , .... , 9 "~ 4

(b)

E~ V' -r-

~ C s ~ Rb: ,

VORTEX GLASS ~

\

VORTEX FLUID I

~

2

i / Hc=

MEISSNER~L STATE. 0

.

10

, .--.'1

~ i l ~CS

.-v..--.-T

15

___ .m,

20 T

(K)

T

.--~ ~

25

p m~.

i

30

,

Fig. 15.10. Magnetic phase diagrams determined for (a) K3C60 and (b) Rb3C60. Solid lines are fits to the de AlmeidaThouless relation [Eq. (15.10) with 7 = 3/2], which represents the demarcation between vortex glass and vortex fluid phases. Dashed lines indicate schematically the critical fields Hc](T) and Hc2(T )

[15.75].

632

15. Superconductivity Table 15.2

Parameters for the phase boundary between the vortex fluid and vortex glass phases" [15.75]. Materials

K3C60 Rb3C60

7

T* (K)

H 0 (kOe)

1.41 + 0.10 1.64 4- 0.14

19.2 + 0.1 29.1 + 0.1

12.2 4- 0.07 27.2 -4- 1.7

aValues for T* and H0 are calculated using Eq. (15.10).

which defines a second-order phase transition. Values of the parameters 7, T*(0) and H 0 of Eq. (15.10) are given in Table 15.2 for K3C60 and Rb3C60. The values of 3' found experimentally are consistent with the de AlmeidaThouless relation [15.76] given by Eq. (15.10) with 3' = 3/2. The vortex pinning mechanism for K3C60 and Rb3C60 is associated with a large density of grain boundaries in these powder samples. While T* coincides with Tc for H = 0, the value of H0 differs from Hc2(0), since H0 relates to the boundary between the vortex glass and the vortex fluid, while H~2(0) relates to the boundary between the vortex fluid and normal phases. The granular nature of many of the M3C60 films that have been studied gives rise to much vortex pinning, which often results in an overestimate of Hcl. The thermodynamic critical field He(0 ) listed in Table 15.1 is found from the relation

H~(O) = H~,(O)H~2(O)/ In K

(15.11)

K = -~L -.

(15.12)

where

~:o

The large values of K (see Table 15.1) indicate that M3C60 compounds are strongly type II superconductors [15.17]. Reports on values for the mean free path e vary from --~2 to 30/~ and e is often reported to be smaller than the superconducting coherence length ~:0 for single crystals, films, and powder samples. The small value of e is mainly attributed to the merohedral disorder, which is present in all doped fullerene samples, including the best available single crystals (see w In addition, e is smaller for films and powders than for single crystals, since e is highly sensitive to carrier scattering from crystal boundaries. For some single-crystal samples, e is comparable to, or greater than, the diameter of a C60 molecule. But since e << ~:0 in most cases, or at best e -~ ~:0, fullerene superconductors are considered to be in the dirty limit defined by e < ~:0.

15.4. Temperature Dependence of the Superconducting Energy Gap

633

This conclusion, that the M3C60 compounds are dirty superconductors, is supported by the vortex glass behavior [15.75]. The coherence length ~:0, which enters the formulae for H~(0) and He2(0), is the Ginzburg-Landau coherence length, and ~:0 is sensitive to materials properties. The Pippard coherence length ~:00in the clean limit is an intrinsic parameter, not sensitive to materials properties, and is related to ~0 by ~0 ~" (~00e) 1/2,

(15.13)

and the BCS theory provides the following estimate for s%0: ~:oo =

1 hv e

rr A

(15.14)

yielding so00 in the 120-130 A range for the superconducting fullerenes [15.51, 69], consistent with estimates using Eq. (15.13) and mean free path values e in the 5-10 ,~ range (see Table 15.1). Available data thus suggest that a fullerene superconductor may be described as a strongly type II superconductor, which is in the dirty limit and has weak to moderate electron-phonon coupling.

15.4.

TEMPERATURE DEPENDENCE OF THE SUPERCONDUCTING ENERGY GAP

Several different experimental techniques have been applied to measure the superconducting energy gap for M3C60 superconductors, including scanning tunneling microscopy, nuclear magnetic resonance, optical reflectivity, specific heat, and muon spin resonance measurements. The various results that have been obtained are reviewed in this section. It would appear that scanning tunneling microscopy should provide the most direct method for measuring the superconducting energy gap. Using this technique, the temperature dependence of the energy gap has been investigated for both K3C60 and Rb3C60 [15.53,54]. Figure 15.11 shows d I / d V , the derivative of the tip current I with respect to bias voltage V, plotted vs. bias voltage for an Rb3C60 sample with a 60% Meissner fraction at 4.2 K. The sample in this study was characterized by zero-fieldcooled and field-cooled magnetization measurements at very low magnetic field ( ~ 5 0 e ) [15.53]. The experimental data for ( d I / d V ) are compared in Fig. 15.11 to the expression for ( d I / d V ) d I / d V = Re{(eV - iF)/[(eV - iF) 2 - A2]1/2},

(15.15)

634

15.

Superconductivity

2.5 2.0 1.5 1.0 ~"

0.5 )

:3

Fig. 15.11. Plot of d I / d V vs. bias voltage V (meV) for an Rb3C60 sample at a temperature of 4.2 K [15.53]. The experimental data for the conductance (solid circles) were calculated numerically from I - V measurements. The data in (a) are fit with the expression dl/dV = e V / [ ( e V ) 2 - A2]1/2 (solid curve) with A -- 6.8 meV and assuming no broadening. The data in (b) are fit with the expression given in Eq. (15.15) which includes a phenomenological broadening parameter F (A = 6.6 meV and F = 0.6 meV).

0.0 -40

~

2.5

' -30

-30

I

-20

, m=~l " I 0 - 0--

10

-20

-10

10

20

30

40

b

2.0 1.5 1.0 0.5 0.0

-40

i

0

i

20

i

30

40

which is plotted in Fig. 15.11 for the superconducting state normalized to that in the normal state [15.54]

(di/dV)N = Re [(E -

iF) 2 - A21i/2

,

(15.16)

where E = eV is the energy of the tunneling electrons relative to EF, 2A is the superconducting energy gap, and F is a phenomenological broadening parameter. A good fit between the experimental points and the model is obtained. From data such as in Fig. 15.11, the temperature dependence of the superconducting energy gap A(T) was determined and a good fit for A(T) to BCS theory was obtained (see Fig. 15.12) [15.53] for A(0) = 6.6 meV and F = 0.6 meV, from which (2A(O)/kBTc)was found to be 5.3 for K3C60 and 5.2 for Rb3C60, well above the value 3.52 for the case of a BCS superconductor. These results were interpreted to indicate that the M3C60 compounds are strong coupling superconductors [15.54]. The discrepancy between the experimental points and the theoretical curves in Fig. 15.11 has been cited as an indication of systematic uncertainties in the data sets, and a reexamination of the determination of 2A by tunneling spectroscopy seems necessary. Also, other experiments, discussed below, yield values of 2A(O)/kaTc close to the BCS value.

15.4. Temperature Dependence of the Superconducting Energy Gap

1.0

:I! v,

635

--

0.8 A

0.6

0.4

0.2

0.0 0.0

i

02

m

0.4

0.6

0.8

t.O

r/r~

Fig. 15.12. The temperature dependence of the normalized superconducting energy gap for K3C60 (dark squares) and Rb3C60 (open triangles) plotted as a function of reduced temperature T/Tc. The solid line corresponds to the temperature dependence of A(T)/A(O) calculated by BCS theory [15.54].

Two independent determinations of the superconducting energy gap were made using the NMR technique. In the first method, the temperature dependence of the spin-lattice relaxation rate (1/7"1) in the superconducting state was measured [15.55, 56], and the results were interpreted using the relation [15.77] 1 ~ e x p ( - ~-).

T1

(15.17)

The interpretation of these experiments is further discussed in w Early work yielded values for 2A(O)/kBTc of 4.0 and 3.1 for K3C60 and Rb3C60 , respectively, but a later determination yielded (2A/kBTc) = 3.5 for both compounds [15.55, 78], in excellent agreement with conventional BCS theory. Closely related to the NMR spin relaxation experiments are the muon spin relaxation studies (see w on Rb3C60 powder samples with Tc = 29.4 K [15.57] into which muonium was introduced endohedrally into the C60 anions, denoted by Mu@C60 [15.79]. The conduction electrons associated with the C60 anions screen the positive charge on the muon, so that no local dipole moment is present on the muon. The spin relaxation time 7'1 of the muon due to the conduction electrons in the normal and superconducting states was measured as a function of temperature for various values of magnetic field, the field being necessary for observation of the muon spin resonance phenomenon. By making measurements as a function of magnetic field, the effect of the magnetic field on the superconducting en-

636

15.

Superconductivity

ergy gap could be taken into account by an extrapolation procedure to the small-field limit. The coupling between the muon spin and the conduction electrons arises because of the small but nonzero value of the ~r-electron wave functions for the conduction electrons at the center of the C60 anion. The muon spin relaxation results show a good fit of the temperature dependence of (1/T1) to the theory of Hebel and Slichter for spin relaxation in the superconducting state (see Fig. 15.13), yielding a superconducting energy gap value of A/k a = 53 + 4 K corresponding to 2A/kaTc -- 3.6+0.3, which is in good agreement with the very closely related NMR spin relaxation experiments [15.55] mentioned above and discussed more fully in w A broadening of the density of states associated with the presence of field inhomogeneities due to trapped magnetic flux was reported in the muon spin resonance experiment [15.79]. Broadening effects in the density of states near Tc were also reported in the scanning tunneling microscopy (STM) studies of the density of states [15.54]. The coherence peak predicted in the Hebel-Slichter theory [15.78] is clearly observed in the muon spin res-

15

-

'T

9

(o)

,,,.. I

10 0 r,...

/

I'-"-.,..

i-9

0 10 ~

10

...... i

20

30 40 t (K)

50

60

. . . . . . . . . . . . . . .

H=O.3T

v LID

Fig. 15.13. The temperature dependence of the muonium spin resonance relaxation time T1 in Rb3C60. In (a) ( T 1T ) -~ vs. T is plotted in a magnetic field of 1.5 T. The solid curve is a fit to the theory of Hebel and Slichter with a broadened density of states, while (b) shows an Arrhenius plot of Tj-1 in a magnetic field of 0.3 T [15.57].

0

,i-,,,.-,

(

x

0.2 0.05

1

9

1

!

l

0.1 1/T (K-')

1,

,

1

t

0.15

15.4. TemperatureDependence of the Superconducting Energy Gap

637

onance experiment as shown in Fig. 15.13 and arises from the divergence of the superconducting density of states at the energy gap edges and from the reduced density of states of excited quasiparticles at low-temperature. The observation of the Hebel-Slichter peak in (1//'1) in the muon spin relaxation studies is consistent with a weak coupling superconductor. The successful observation of a Hebel-Slichter peak in the muon spin resonance (/zSR) experiment was important historically [15.57], because in early NMR work the Hebel-Slichter peak was not observed in the NMR measurements of the temperature dependence of the spin-lattice relaxation time /'1 [15.55]. The successful observation of this peak using the /zSR technique suggested that small magnetic fields should be used to look for the Hebel-Slichter peak with the NMR technique, and this strategy eventually led to its successful observation in NMR spin-lattice relaxation studies [15.80]. The/~SR technique has also been applied to study AL(T), and from these studies values of AL(0) have been found for K3C60 and Rb3C60, as given in Table 15.1. It was also concluded that superconducting fullerenes have an isotropic energy gap and could follow an s-wave BCS model [15.64]. Analysis of the Knight shift of the NMR resonance for three different nuclei in the same sample as the temperature is lowered below Tc provides a determination of the temperature dependence of the spin susceptibility which yields a sensitive measure of 2A/kBTc [15.80,81] (see w Explicit application of this approach provided a second NMR technique for sensitive determination of 2A/kBT~. Measurements of the frequency shift in the NMR spectra for the 13C, 87Rb, and 133Csnuclei in the Rb2CsC60 compound below Tc [15.80, 81] (see w yield a good fit for 2A/kBTc = 3.52 for both solids (see Fig. 16.10) and further show that 2A/kBTc = 4 is outside the error bars for the Rb2CsC60 measurements (see w Analysis of infrared reflectivity spectra above and below Tc also provides information on the superconducting energy gap. From a Kramers-Kronig fit to the frequency-dependent reflectivity measurements, the optical conductivity trl(tO) in the far-infrared region of the spectrum is deduced (see Fig. 13.31). From the frequency at which o-l(to) ~ 0, the superconducting energy gap is determined. Such measurements have also been carried out [15.58] for both K3C60 and Rb3C60 (see w and show good agreement with predictions of the BCS model with values of (2A/kBTc) = 3.6 and 3.0 for K3C60 and Rb3C60, respectively [15.58,59]. The absolute reflectivity measurements [15.59] yielded (2A/kBTc) = 3.6 and 2.98 for K3C60 and Rb3C60, respectively, in good agreement with the previous infrared measurements [15.581. The ratio of the frequency-dependent optical conductivities in the superconducting and normal states (trl,~/trl,~) was also fit to predictions from

638

15. Superconductivity

the BCS theory [15.66] using the response function of the Mattis and Bardeen theory for the high-frequency conductivity in the superconducting state [15.82], which depends parametrically only on 2A. A reasonable fit to the Mattis-Bardeen theory was obtained for the superconducting energy gap values, using parameters determined directly from the KramersKronig analysi s [15.66]. However, using the more complete Eliashberg theory [15.83], which incorporates the excitation spectrum a2eF(o~) that is responsible for the electron pairing, a substantial improvement was obtained in fitting the experimental optical conductivity data [15.66]. These fits employed a Lorentzian function for a2eF(w) peaked at co0 -~ 1200 cm -~ with a full width at half-maximum (FWHM) intensity of--~320 cm -1, which simulates the high-frequency intramolecular mode spectrum of C60. The fits remained good for various positions of ~o0 for this excitation band, as long as h~o0 >> kB Tc. This good agreement was interpreted as providing strong evidence for an effective pairing interaction, mediated by these high-frequency intramolecular vibrational modes [15.66]. Furthermore, the possibility of important contributions to the pairing from low-frequency intermolecular or librational modes [w < 100 cm -1 (see w was considered, and it was concluded that, within the Eliashberg theory, these low-frequency modes led to strong coupling, in sharp contrast to the main experimental result for oq(w) indicating that 2A ,-, 3.5kBTc [15.66]. In summary, NMR and muon spin relaxation measurements, NMR Knight shift measurements,~ and detailed far-infrared reflectivity determinations of the superconducting energy gap are consistent with the predictions of BCS theory for a weak coupling superconductor. The STM measurements of the superconducting energy gap agree in a number of ways with the other energy gap determinations, except for the values of 2A/kBTc, where the STM data suggest that the superconducting fullerenes are strong coupling superconductors. A two-peak model for the Eliashberg spectral function o~2F(w) utilizing both the high-frequency intramolecular vibrations and the low-frequency intermolecular vibrations [15.84] can be made to account for a value of (2A/kBT~) ,-~ 5. Nevertheless, the present consensus supports the lower value of (2A/kBTc) = 3.5, consistent with a weak coupling BCS superconductor. 15.5.

ISOTOPE EFFECT

There have been several measurements of the isotope effect in superconducting M3C60compounds. The main objective of these measurements was to provide insight into whether or not the pairing mechanism responsible for superconductivity in the doped fullerenes involves electron-electron or electron-phonon coupling. In the experiments reported to date, the isotopic

15.5. Isotope Effect

639

enrichment has mainly involved 13C substituted for 12C [15.69, 85,86]. In measurements of the isotope effect a fit is made to the relation Tc o~ M -'~, where M is the isotopic mass and a is the isotope shift exponent. In analyzing experiments on the isotope effect, the observed c~ values are compared with the BCS prediction of a = 0.5. The observation of an isotope shift of Tc for the carbon isotopes but not for the alkali metal isotopes indicates that carbon atom vibrations play an important role in the superconductivity of the alkali metal-doped fullerenes. Magnetization experiments using a SQUID magnetometer on Rb3C60 where up to 75% of the carbon was the 13C isotope gave a = 0.37 :E 0.05 [15.69]. The magnetization results of Chen and Lieber [15.85] on K3C60 prepared from 99% 13C powder are shown in Fig. 15.14 in comparison with the corresponding data for K312C60, from which a value of a = 0.30 4-0.06 is obtained. From these two experiments one could conclude that the measured values imply a somewhat smaller a value than expected for a BCS model superconductor. A comparison with a determinations for other superconductors (see Table 15.3) shows a values close to 0.5 for classical BCS superconductors, but significant departures from a = 0.5 are found for transition metal superconductors and high-Tc materials. It is of interest to note that a large range of values for a have been reported for high-T~ cuprate materials (see Table 15.3). Shown in Table 15.3 are results for a for various stoichiometries of the La2_xSrxCuO 4 and La2_xCaxCuO 4 systems. The results for the La2_xSrxCuO 4 system show relatively high values for a (0.4-0.6) for x < 0.15, but much lower values of a for x > 0.15 [15.88]. Some variations of a in samples with similar nominal stoichiometries were reported [15.88, 89], and these variations are reflected by the range of c~

9

13

o o

N 9

o

....

~ ....

, ....

,.'-'

O

E -0.0~, k..

.

O C

o 0 il r 0

9

-1~-

-0.18 9

0

! ...,1

0

18.0

18.5

'~

" ....

10

19.0

z (K)

i ....

20

19.

IL.

30

20.0

Fig. 15.14. High-resolution temperature-dependent magnetization measurements on K313C60 (filled circles) and K312C60 (open circles) samples, highlighting the depression in Tc for the K313C60 isotopically substituted material. The inset shows a full magnetization curve for a K313C60sample [15.87].

640

15.

Superconductivity

Table 15.3

Isotope effect for various superconductors. Materials Sn Pb Hg Ru Os Lal.ss7Sr0.113CuO4 Lal.ssSr015CuO 4 Lal.925Ca0.075CuO 4 Lal 9Ca0.1CuO4 K 3C60 K3C60 K3C60 .

.

.

.

.

.

.

.

.

?'ca (K)

a

3.7 7.2 4.1 0.51 0.66 29.6 37.8 9.2 19.5 19.3 19.3 19.3

0.47 0.49 0.50 0.0 0.15 0.60--0.64 0.08-0.37 0.81 0.50--0.55 0.30 4- 0.06 0.37 4- 0.05 1.4 4- 0.5

.

.

.

Isotopic abundance

Reference [15.95] [15.95]

[15.95] [15.95] [15.95] 85-90% 180, 15-10% 160 85-90% lsO, 15-10% 160 92%180, 8% 160 17-93% 180, 83-7% 160 1% 12C, 99% 13C 25% 12C, 75% 13C 67% lzC, 33% 13C .

.

.

.

[15.88] [15.881 [15.89] [15.89] [15.851 [15.69] [15.86, 931 ,,

aThe listed Tc value corresponds to 100% abundance of dominant isotope.

values quoted in Table 15.3. These data are significant in showing the wide deviations from a = 0.5 that are observed in high-T~ cuprate materials. These measurements, taken for a wide range of 180 and 160 isotopic abundances, show that a is not sensitive to isotopic oxygen substitutions [15.89]. Similar results have been obtained using the Lal.85Sr0.15Cu~_xNixO4 system as x was varied from x = 0 (a = 0.12) to x = 0.3 (a = 0.45), with a monotonic increase in a observed with increasing x [15.90]. Other pertinent results on the dependence of a on stoichiometry of high-To cuprate materials have been reported for the Y B a 2 C u 3 O T _ 8 system giving a values in the range 0.55-0.10 as Pr was substituted for Y [15.91] and for the Nd-Ce-CuO system a < 0.05 where the ~80:160 ratio was varied [15.92]. Measurements of the isotope effect on doped C60 samples with only a 33% enrichment of 13C gave a much larger value of a = 1.44-0.5 [15.86, 93]. The large range in the values of the exponent a reported by the various groups seems to be associated with the difficulty in determining T~ with sufficient accuracy when the normal-superconducting transition is not sharp. The importance of using samples with high isotopic enrichment can be seen from the following argument. Since Tc = C M -'~, we can differentiate this expression and write

Arc T

= - a (-~---) f

(15.18)

where M is the isotopic mass and f is the fractional isotopic substitution 0 < f <__ 1, assuming that AM is related to the difference between the average elemental mass and that of a particular isotope. Equation (15.18)

15.6. Pressure-DependentEffects

641

thus shows that ATc is measured more accurately when the isotope effect is measured on light mass species and when the fractional abundance f of the isotope under investigation is large. To provide another perspective on the experimental determination of the isotope effect, two distinct substitutions of ~50% 13C were prepared in the Rb3C60 compound [15.94]. In one sample Rb3(13Cl_xl2Cx)60called the mass-averaged sample, each C60 molecule had ~50% of 12C and 13C, while in the other sample Rb3[(13f60)l_x(12C60)x)] called the mass-differentiated sample, approximately half of the C60 molecules were prepared from the ~2C isotopes and the other half from 13C isotopes. The samples thus prepared were characterized by infrared spectroscopy to confirm that the C60 molecules were mass averaged in the first sample and mass differentiated in the second sample. Measurements of the isotope effect in the massaveraged sample confirmed the a = 0.3 previously reported for K3C60 [15.85], while the mass-differentiated sample yielded a value of a = 0.7 [15.94]. No explanation has yet been given for the different values of a obtained for these two kinds of samples. On the basis of the superconducting energy gap equation [Eq. (15.9)], all of the reported values of the exponent a suggest that phonons are involved in the pairing mechanism for superconductivity and that the electron-phonon coupling constant is relatively large [15.69,86]. Future work is needed to clarify the experimental picture of the isotope effect in the M3C60 compounds, and if the large value of a = 1.4 were to be confirmed experimentally, detailed theoretical work is needed to explain the significance of the large a value. 15.6.

PRESSURE-DEPENDENTEFFECTS

Of interest also is the dependence of the superconducting parameters on pressure. Closely related to the high compressibility of C60 [15.96] and M3C60 (M "- K, Rb) [15.17] is the large (approximately linear) decrease in Tc with pressure observed in K3C60 [15.17] and Rb3C60 [15.17] for measurements up to a pressure of 1.9 GPa (see Table 15.1). Figure 15.15 is a plot of the temperature-dependent susceptibility for several values of pressure for a pressed powder sample of K3C60 [15.23]. This figure clearly shows a negative pressure dependence of T~, which is more clearly seen in the pressuredependent plot of T~ in Fig. 15.16. The pressure coefficients of T c for K3C60 and Rb3C60 are similar to those measured for the high-Tc superconductors in the La2_x(Ba,Sr)xCuO 4 system where (aTc/ap) ,~ +0.64 K&bar [15.9799], much greater than those for the A15 superconductors (+0.024 K&bar), but also less than those for BEDT-TTF salt organic superconductors for which ( a T c / d p ) " - ~ - 3 K&bar [15.29].

642

15. Superconductivity

..n g,

s

!.,

6

//

.m

"~

i

4

t~

2I

Fig. 15.15. Plot of the temperature dependence of the susceptibility of a pressed K3C60 sample at the various pressures indicated [15.23].

0 5

0

10

15

20

25

Temperature OK)

20 m

X

9

K3 C60 o\

i

\

A

0

o

10

o~

F--

Fig. 15.16. Pressure dependence of Tc~ (transition onset) and of To2 [the morphology-dependent kink in the x(T) curves] (see Fig. 15.15) for a pressed powder sample of K3C60 [15.23].

9

0

l

5

mm

T~l

o

To2

. ~~

J

10

.i.

z

15

20

Pressure (kbar)

When the pressure dependence of Tr for K3C60and Rb3C60 is expressed as (1/Tc)(~Tc/ep)p_Oin the limit of p = 0, a common value o f - 0 . 3 5 GPa -1 is obtained for both K3C60 and Rb3C60 [15.60, 100]. In this interpretation, the smaller size of the K + ion relative to Rb + (0.186 A) is attributed to an effective relative "chemical pressure" of 1.06 GPa, which is found by considering the compressibilities of the two compounds (see Table 15.1). By displacing the pressure scale of the T~(p) data for K3C60 by 1.06 GPa, the pressure dependence of Tc for K3C60 and Rb3C60 could be made to coincide, and the results for both compounds could be fit to the same functional

15.7. Mechanismfor Superconductivity

643

form Tc(p) = T~(0) exp(-Tp)

(15.19)

with the same value of T = 0.44 4-0.03 GPa -1 [15.17]. A plot of T~ vs. pressure obtained from curves (such as in Fig. 15.15) is presented in Fig. 15.16, showing a nearly linear decrease of T~ with pressure. The result given by Eq. (15.19) can be interpreted in terms of a T~ that depends only on the lattice constant for the M3C60compound and not on the identity of the alkali metal dopant (see w The large effect of pressure on T~ is due to the sensitivity of the intermolecular coupling to the overlap between nearest-neighbor carbon atoms on adjacent molecules, which are separated by 3.18 A in the absence of pressure. When this intermolecular carboncarbon distance becomes comparable to the nearest-neighbor intramolecular C-C distance (-~1.5 A), a transition to another structural phase takes place [15.17].

15.7.

MECHANISM FOR SUPERCONDUCTIVITY

The observation of a 13C isotope effect on T~ indicates that C-related vibrational modes are involved in the pairing mechanism. Several other experimental observations suggest that the role of the alkali metal dopant is simply to transfer electrons to C60-derived states (tlu band) and to expand the lattice. This is consistent with the absence of an isotope effect for Rb in RbaC60 and with the strong correlation of Tc with lattice constant rather than, for example, with the mass of the alkali metal ion. If metal ion displacements were important in the pairing mechanism, then Rb3C60 would be expected to have a lower Tc than K3C60, contrary to observations. Which C60 vibrational modes are most important to Tc is not completely resolved. The dependence of T c not on the specific alkali metal species but rather on the lattice constant a0 of the crystal implies that the coupling mechanism for superconductivity is likely through lattice vibrations and is more closely related to intramolecular than to intermolecular processes. Raman scattering studies (see w of the Hg-derived, intramolecular modes in M3C60show large increases in the Raman linewidths of many of these modes relative to their counterparts in the insulating parent material C60 or in the doped and insulating phase M6C60.This linewidth broadening strongly suggests that an important contribution to the electron-phonon interaction comes from the intramolecular modes. Theoretical support for the broadening of certain intramolecular vibrational modes as a result of alkali metal doping has been provided by considering the effect of the Jahn-Teller mechanism on the line broadening

644

15.

Superconductivity

through both an adiabatic and a nonadiabatic electron-intramolecular vibration coupling process [15.101]. Broadening was found in modes with Hg symmetry, as well as in some modes with Hu and T~u symmetries, and the broadening results from a reduction in symmetry due to a Jahn-Teller distortion. An especially large effect on the highest Hg mode due to doping has been predicted [15.101]. The broadening effect appears to be in agreement with phonon spectra on alkali metal--doped fullerenes as observed in neutron scattering studies (see w While keeping these fundamental characteristics in mind, theorists have been trying to identify the dominant electron pairing mechanism for superconducting fullerenes. A number of approaches have been attempted and are summarized below. In the first, only the electron-phonon interaction is considered and electron-electron interactions are neglected. This main approach has been followed by most workers, with different possible side branches pursued, as outlined below. The second approach considers a negative electron-electron interaction whereby superconductivity is explained by an attractive Hubbard model. Returning to the various branches of the first approach, one branch considers a BCS version of the electron-phonon interaction without vertex corrections, while another branch includes vertex corrections for the enhancement of T~. Within the first approach, much effort has gone into the identification of the phonon modes which are most important in the electron-phonon interaction. Much of the discussion has focused on the use of the Eliashberg equations to yield the frequency-dependent ote2F (to) spectral functions from which the electron-phonon coupling parameter ,h,ep c a n be determined by the relation aep --

2

dto a2eF(t~

(15.20)

to

and the average phonon frequency __ tOph

"--

~ph is then

determined by

2 f0~ _d~ae2F(to, In to }. exp { ~ep

(15.21)

It is readily seen that a given spectral weight a2F(to) at low frequency contributes more to Aep than the corresponding weight at high to because of the to factor in the denominator of Eqs. (15.20) and (15.21). Since T~ depends predominantly on the lattice constant, which in turn depends on the density of states, as discussed in w T~ is most sensitive to molecular properties. Thus, it is generally believed that the intramolecular phonons are dominant in the electron-phonon interaction [15.50]. This conclusion is based on detailed studies of both the temperature dependence of the transport properties [15.50] (see w and the electron-phonon

15.7. Mechanism for Superconductivity

645

coupling constant directly [15.51,52]. Although most authors favor dominance by the high-frequency intramolecular vibrations, some authors have emphasized the low-frequency intramolecular vibrations [15.102] for special reasons, such as an effort to explain possible strong coupling in K3C60 and Rb3C60 as implied by the STM tunneling experiments to determine the superconducting energy gap [15.54]. Thus, it may be said that no firm conclusion about the dominant modes has yet been reached [15.84,103, 104]. If both high- and low-frequency phonons were to contribute strongly to the electron-phonon interaction, then two peaks in the Eliashberg spectral function a2eF(to) would occur in Eqs. (15.20) and (15.21), such that O)ph = O)~1/AePto2A2/~ep' --

(15.22)

where Aep is obtained from Eq. (15.20) by integration of 2c~2F(~0)/~o over all phonon frequencies and where A1 and A2 are related to the contribution from each peak in aZeF(~o) [15.84]. According to this two-peak approach, phonons with energies h~Oph greater than ~rkBT~ are effective at pair breaking, while the low-energy phonons are not. In narrowband systems, such as in M3C60, we expect a Coulomb repulsion to be present between electrons that are added to the C60 molecule in the degenerate fl, (or flu) levels or energy bands. In the limit of a strong Coulomb repulsive interaction, the tlu energy band is split into upper and lower Hubbard bands, and thus the solid becomes a magnetic insulator [15.105,106] or acts as a heavy fermion system. If M 3 C 6 0 is a Mott insulator, then the superconducting phase must have off-stoichiometric compositions M 3 _ 8 C 6 0 , where a value of 8 = 0.001 could account for the normal state conductivity. It has been argued that the success of LDA band calculations for K3C60 and Rb3C60 in predicting a linear dependence of Tc on pressure [15.107] and a variety of experimental efforts to look for anomalous behavior very close to the x = 3 stoichiometry indicates that K3C60 and Rb3C60 are metals and not Mott insulators. In the case of solid C60, the competition between the electron-electron interaction and the transfer energy from molecule to molecule must be considered in treating the superconductivity of ~r electrons in the flu (gu) energy band. Below we also review efforts to treat the pairing of two electrons on the basis of the electron-electron interaction. White and his co-workers discussed the attraction between two electrons localized on a single C60 anion by defining the pair-binding energy, Epair [15.108], E;air -- 2(I)i - -

(1)i_ 1 - -

(I)i+1

(i = 1, 3, 5),

(15.23)

where ~ is the total energy of a molecule when the molecule has i additional electrons. If gpair is positive, it is energetically favorable for two

15. Superconductivity

646

adjacent molecules to have (i + 1) electrons on one molecule and ( i - 1) electrons on the other. This transfer of electrons has been proposed as a possible mechanism for superconductivity in fullerenes [15.109,110]. This situation has been examined by a numerical calculation using the extended Hubbard Hamiltonian, U -- -- y ~ tij(CtitrC.ia --b H.c.)+ ~- ~ nitrni_ o. -+- V ~ nianja, (15.24) (i,j),o"

i,tr

(i,j),tr,tr'

where C~ creates an electron with spin tr on the ith site, and n~ = C~C~ is the density of electrons with spin tr on site i, and t~j denotes the transfer integral for the transfer of charge. U and V represent on-site and offsite electron-electron interactions, the sums in Eq. (15.24) are taken over orbital and spin states, and H.c. denotes Hermitian conjugate. The exact diagonalization of ~f given by Eq. (15.24) has been done for small clusters with N electrons on a one-dimensional ring [15.111], torus (N = 16), cube (N = 8), and truncated tetrahedron (N = 12) [15.108], which are all smaller than the case of N = 60 for C60. The calculated results show that E~r of Eq. (15.23) can be positive or negative, depending on the relative ratios of the parameters t, U, and V. If an electron-electron pairing mechanism were to be important, it would need to explain not only superconductivity arising from the partial filling of the tl~-derived band but also, for the case of the alkaline earth compounds, the partial filling of the 6gderived band, including the strong hybridization of the t~g electronic states with the alkaline earth metal s and d states. Recently, the total energy of the c6n0- ion has been calculated using semiempirical self-consistent field (SCF) and configurational interaction (CI) programs from the library MOPAC [15.112]. The calculated results show that the total energy as a function of n has a positive curvature, which implies that Epair is always negative. In this sense the effective attractive interaction between two electrons in a molecule might not be relevant as a mechanism for superconductivity. When the electron-phonon interaction screens the Coulomb repulsion between two electrons, the effective Coulomb interaction,/Jeff, can be negative. In this case we can apply an attractive Hubbard model to such a system. Starting from an attractive Hubbard model, it has been shown [15.113] that the superconducting state is more stable than the charge density wave (CDW) state in the fcc structure of K3C60, similar to the case in BaBiO3. The effective intramolecular interaction between r electrons, Ueff , c a n be negative for the case in which the optic phonons involving the K § ions couple strongly to the zr electron carriers on C60 molecules of the K3C60 crystal. The absolute value of Ueef caused by two optic modes that are related to octahedral-tetrahedral and octahedral--octahedral sites (estimated

15.7.

M e c h a n i s m for Superconductivity

647

as Ueff "~ -0.65 and -0.27 eV, respectively) may be larger than that of the repulsive interaction caused by the screened intramolecular Coulomb repulsion Ue_e(1.2--1.7 eV) or the negative values caused by a bond dimerization effect (~ -0.026 eV) or the Jahn-Teller interaction (~ -25 meV) [15.114]. Within mean field theory, the superconducting state is more stable than the CDW state for any negative/Jeff. This fact can be understood both by the calculation of Tc in the weak coupling limit of Ueff and by the perturbation expansion in t of the total energy for the two states in the strong coupling limit of IUeff/tl >> 1. A number of proposed mechanisms for superconductivity in doped fullerenes involve the Hubbard model. Therefore criticisms of using a Hubbard model should be mentioned. The validity of the Hubbard model requires that the intra-site screening be much smaller than the intersite screening and that the bare intrasite Coulomb repulsion should be much larger than the bare intersite repulsion. Since the nearest-neighbor C-C distance between adjacent C60 molecules is .v3 3, and of the same order as the diameter of C60 (~7/~), the bare intrasite and intersite repulsions should be of comparable magnitude. The large number of on-site electrons (240) for C60 suggests that the intrasite screening is important. Thus use of the Hubbard model for describing the electronic structure of fullerenes has been questioned. It has also been argued that the energy needed to transport an electron should be the difference Uintr a - Uinte r rather than the intracluster repulsion U~ntra, and it is expected that Uintr a - Uinte r < ( Uintr a [15.107, 115]. Another proposed theory of superconductivity for doped C60 is based 2 on a parity doublet [15.116] formed from an h,10 t~,t~g configuration in the doped C60 material. In this case, the t~lut~gelectron pair forms the parity doublet that triggers the superconducting transition. Also, the possibility of plasmons mediating an attractive interaction between electrons has been suggested as a pairing mechanism for low-carrier-density systems [15.117]. Although many attempts have been made to explain the superconducting state in doped fullerenes as special cases of a Hubbard system with U ranging from negative to positive values, none of the theories are yet conclusive, partly because of their inability to account for many of the published experimental results. Likewise, many first principles calculations elucidate certain fundamental issues but do not make systematic predictions which can be tested. Most experiments suggest that the transfer energy is comparable to the phonon energy and to the electron-electron interaction. Quantitative applications of the theory using realistic parameters are needed. Furthermore, justification is also needed of the methodology that was used to determine reliable values of the parameters that are employed in theoretical calculations. Particular insights into the dominant superconducting mech-

648

15. Superconductivity

anism are expected through gaining a better understanding of the isotope effect and pressure-dependent phenomena. While it is generally believed that superconductivity in doped fullerenes arises from some kind of pairing mechanism between electrons, the detailed nature of the pairing mechanism is not well established. Although the electron-phonon mechanism has been most widely discussed, it appears that electron-electron pairing mechanisms have not yet been ruled out. For those who believe that the electron-phonon interaction is the dominant pairing mechanism, heated discussion currently focuses on which phonons play a dominant role in the coupling. To produce pairing, many authors have invoked a dynamic Jahn-Teller mechanism induced by pertinent intramolecular modes, and further clarification is needed of the role of the Jahn-Teller effect in fullerene superconductivity. Experimental evidence in support of Jahn-Teller distortions for C3o anions comes from both EPR studies (w and optical studies (w A host of other fundamental theoretical issues remain to be clarified. Since the charge distribution in the superconducting fullerenes is mainly on the surface of the icosahedral C3o anions, in contrast to the more distributed charge distribution found in conventional solids, modifications are probably necessary to the traditional interpretation of a variety of measurements, such as the plasma frequency, specific heat, magnetic susceptibility, and temperature- and magnetic-field dependent transport measurements, to mention but a few. While concern about the validity of the Migdal theorem has been raised, since the vibrational frequencies are comparable in magnitude to the Fermi energy and to the bandwidth of the LUMO levels, no clear picture has yet emerged on which aspects of the Migdal theory need modification. While many workers in the field agree that many-body effects are important, no consensus has been reached on how to handle strong correlations between electrons in a system where the diameter of the C60 molecules and the nearest-neighbor distances are of comparable magnitude and the electronic energy bandwidths are very narrow.

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649

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[15.65] [15.66] [15.67] [15.68] [15.69]

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