1.4.1 Fm$\bar{3}$m A₃C₆₀ Superconductors

When all three alkali ions are at least as large as K⁺, the structure of A₃ is FCC (space group Fm$\bar{3}$m). The alkali ions occupy the O and T interstitial sites (see Fig. 1.2). The details of the site may be more complicated as the observation of three (instead of the expected two) ⁸⁷Rb NMR lines[38] indicates. The interstitial voids are large enough to accomodate K⁺, Rb⁺ and Cs⁺ ions, but the FCC structure becomes unstable when the alkali filling reaches Cs₃, which has the A15 structure and is only a superconductor under pressure. Similarly, a related compound with electrically neutral spacer molecules in the interstices of the C₆₀ lattice, NH₃K₃C₆₀, is orthorhombic, and becomes superconducting[39] (T_c = 28K) under pressure (> 10kbar). These two materials show that cubic symmetry is not required for superconductivity in the fullerides. The C₆₀ molecules in these materials are orientationally disordered: specifically, they are merohedrally disordered, i.e. there are two distinct orientation of the 3 orthogonal 2-fold axes of the C₆₀ along the cubic axes, and these orientations are equally populated. The C₆₀ molecules in these phases are not measureably distorted from the isolated C₆₀. The alkali ions act as spacers for the C₆₀, so the average FCC lattice constant increases with the average alkali ionic radius. NMR experiments find that the C₆₀ molecules in K₃ exhibit rotational dynamics down to about 200K (see the review [34] for a discussion).

The superconducting transition temperature in this class of A₃ materials is strongly correlated with the fcc lattice parameter. This can be understood qualitatively in the rigid band picture, as a consequence of the correlation of the conduction bandwidth with the intermolecular overlap which depends strongly on the C₆₀-C₆₀ separation. If the lattice is expanded (but not changed in symmetry), the bandwidth decreases, but the total number of the states (determined by the number of unit cells in the sample) does not, so the density of states per unit energy at the Fermi Surface increases. Throughout this thesis, the electronic density of states is denoted by g(E) with E measured relative to E_F, so the value at the Fermi surface is g(0). The density of states is connected to the transition temperature, in the BCS theory[42] via

where $\Theta_D$ is the Debye temperature, c is a constant of the order unity and V is some mean electron-phonon interaction energy. In the weak-coupling limit, $V g(0) \ll 1$, and using the BCS gap equation, it can be shown that the energy gap is approximately 3.52 times T_c, i.e.

From Eq. 1.2, it can be seen that for a constant $\lambda_{ep}$,an decrease in g(0), will increase T_c exponentially. It is found that hydrostatic pressure as well a effective chemical ``pressure'' (tuning of the FCC lattice constant by alkali size) have the same effect on T_c. T_c varies for this class of A₃ over the range 8K (for K₃ under 2 GPa) through 19.5K for ambient pressure K₃ and 33K for ambient pressure RbCs₂ to a current maximum of 40K[41] for 12 kbar. The FCC lattice constant in the above sequence varies from 13.8Å (K₃, 2GPa) to approximately 14.6Å (RbCs₂, ambient pressure).

These materials are all hard type II superconductors with the magnetic penetration depth $\lambda$ much larger than the superconducting coherence length $\xi$. In table 1.1 these parameters as well as the reduced superconducting energy gap from a number of different techniques are tabulated for the best studied representative of the Fm$\bar{3}$m class of A₃ superconductors: K₃ and Rb₃. All experiments indicate that the Ginzburg-Landau paramter $\kappa_0=\lambda_0/\xi_0$ is on the order of 100. The values of the superconducting energy gap range widely from weak-coupling (Eq. 1.3) to strong ($2\Delta_0/kT_c \gt 3.52$, see below).

Parameter K₃C₆₀ Rb₃C₆₀

T_c [K] 18.5-19.7 27.5-30

$$\lambda_0$$ 2400-8900 2200-8500

$$\xi_0$$ 20-45 20-30

B_c1(0) [mT] 1.2-13.2 1.3-16.2

B_c2(0) [!] 17-50 40-78

$$2\Delta_0/kT_c$$ 3.0-5.3 3.45-5.3

caption

[K₃ and Rb₃ Superconducting Parameter Ranges] Ranges for the reported values of the most important parameters of superconductivity. The ranges are broad because they include values from a wide variety of techniques on samples from different sources. These ranges are considerably narrower when only a subclass of the most ``reliable'' techniques on samples of the highest quality are considered. Detailed references can be found in the reviews: [31,36].

The Fm$\bar{3}$m A₃ materials also possess high low temperature residual resistivity, and consequently small electronic mean free path, l. This may be associated with molecular orientational disorder. Ranges for normal state electronic parameters are given in table 1.2. The other parameters given are the $T \rightarrow 0$ of the resistivity $\rho_0$, the carrier density n, the Fermi Velocity v_F and the parameter r_s which is defined by $n^{-1}=\frac{4\pi}{3}r_s^3$(see Chapter 1 of [20]).

Parameter K₃C₆₀ Rb₃C₆₀

$$\times 10^{21} \times 10^{21}$$ n [cm^-3]

$$\rho_0 \Omega$$ 0.12-2 0.23-0.8

l [Å] 3-70 3-70

$$\times 10^7$$ v_F [cms^-1] -

E_F [eV] 0.05-2.0 0.05-2.0

g(0) [eV^-1] 11-16 19-24

r_s [a₀] 7.2 7.3

caption

[K₃ and Rb₃ Normal Metal Parameters] Representative values and Ranges for the parameters of the normal metallic state of K₃ and Rb₃. The electronic density of states is reported per molecule per spin. a₀ is the Bohr radius.

As mentioned above, the acoustic structure of these materials is characteristic of a molecular solid. The broad frequency spectrum for the various phonon modes is shown in Figure 1.5. By measuring the phonon spectrum (via either Raman or neutron scattering) as a function of alkali doping, it is found that the high frequency intermolecular modes are rather strongly coupled to the conduction electrons, with dimensionless coupling parameter $\lambda_e$ = 0.2-0.64 (for details, see [31]). In addition it is found via inelastic neutron scattering, that the libron mode hardens as the charge of the C₆₀ increases and that the electron-libron coupling is small $\lambda_e < 0.1$.

We now continue with a short description of the other class of A₃ superconductors, those containing Na⁺.