4.1 $\mu^+$ Sites in A₃C₆₀

When muons stop in typical metals, they take up one or a few well-defined crystallographic sites, usually interstitial, and remain diamagnetic; futhermore, they may in exceptional cases diffuse between adjacent equivalent or inequivalent sites, often, because of their small mass, exhibiting interesting quantum effects in their motion.[165] In insulators and semiconductors, $\mu^+$ often captures an electron to form paramagnetic muonium which also occupies a specific interstitial site; however, the paramagnetic states almost never occur in metallic environments either because the spin-flip rate due to collisions with the conduction electrons is sufficiently fast to average the muon-bound electron interaction to zero or because screening of the electrostatic potential precludes a bound state entirely. Although the details of the particular sites adopted by the muon are not critical to the analysis we report in the following sections, a brief discussion is included here for completeness.

In C₆₀ based solids, the available interstitial voids are much larger than in conventional metals, and there are many potential sites for the muon, though some are occupied by alkali ions in the alkali fullerides, i.e. the octahedral (O) and tetrahedral (T) interstitial sites. In pure C₆₀, $\sim$80% of the implanted muons form an exohedrally bonded muonium radical (C₆₀Mu) which has been studied extensively.[102,166,167] In the ionic insulating fullerides K₄ and K₆[14], and the conductors Rb₁[168] and Rb₃[83], a similar fraction of the muons exhibit diamagnetic behaviour. Thus in both metallic and insulating environments, the exohedral C₆₀Mu radical does not survive the charging of the C₆₀. The large fraction of diamagnetic muons in FCC A₃C₆₀ are certainly interstitial in the lattice of C₆₀^-3 ions, but their precise position(s) are currently a matter of speculation. There are, however, some likely candidates which we will now discuss briefly. In the simplest scenario, the muon remains positive in the metal and its site(s) are determined essentially by the minimum electrostatic potential due to the surrounding ions. In such a situation, the muon would adopt a highly symmetric site, of which there are three obvious candidates: midway between two neighbouring O-sites; between T-sites; between an O and a T-site. Of course, polaronic lattice relaxation may complicate this scenario somewhat. Another possibility is that, due to its high electron affinity, the muon forms a complex with one of the interstitial alkali ions analogous to an alkali-hydride molecule. Which of these scenarios is realized will depend delicately on the energies involved. One would expect that, if the nuclear dipolar fields were responsible for the room temperature TF linewidths in the alkali fullerides, that these widths might scale between systems with the average nuclear moment (possibly weighted by the inverse cube of the alkali-halide bond-length, e.g. Table ). We have, at present, no evidence for such a systematic variation either within the A₃C₆₀ superconductors or between different phases. However, the small room temperature linewidths in some cases may have background contributions which may mask these effects. Measurements on single crystals, which have not yet been available in sufficient size for $\mu$SR, might be able to determine the muon site(s).[74,73]

 

In both pure C₆₀ and the insulating alkali fullerides, a small fraction of implanted muons ($\sim$10-20%) form a muonium atom characterized by a large isotropic hyperfine parameter which has been interpreted[169,102,14] to be trapped endohedral muonium (Mu@C₆₀ pictured in Fig. 4.18a). Fig. 4.14 shows the clear signature for this state, i.e. TF precession at frequencies determined by the hyperfine levels of Fig. 3.10. In the metallic systems, however, these precession signals are expected to be unobservable due to relaxation broadening either via the Korringa mechanism or by inhomogeneous broadening in the vortex state of the superconductor or in the low temperature magnetic phases of the A₁ metals. One possibility for observation of the oscillation, however, is at the lowest temperature in zero field in the Meissner state of an A₃C₆₀ superconductor, where a high frequency ``heartbeat'' oscillation ($\approx 4.4$ GHz) might be observed because the Korringa and inhomogeneity broadening mechanisms will be negligible. Such high frequency measurements are technically quite difficult[170] and have not yet been attempted.

  

Figure 4.15: a) The field dependence of the LF relaxation rates at 35K. The fits to the spin-exchange model are discussed in the text. b) The field dependence of the normal state value of (T₁T)^-1 fit to a similar model.

Even in the absence of measurable precession the presence of Mu@C₆₀ can be confirmed in LF experiments. Specifically, provided the T₁ relaxation of the $\mu^+$ in the Mu@C₆₀ atom is in the muon time-range, the relaxation rate will have the characteristic magnetic field dependence of (3.4). We have observed this relaxation in the A₃C₆₀ superconductors and measured its field dependence at T=35K (well into the normal state). The relaxation is single exponential, and the rates as a function of field are plotted in Fig. 4.15a. The fits shown are to Eq. (3.4) with a small additional field-independent rate, R₀. Because the parameters $A_\mu$ and $\nu_{SE}$ are strongly correlated, they could not be determined independently, and we fixed $A_\mu = 4340$ MHz from the precession measurements in the insulators (Fig.4.14). This value is also consistent with measurements of $A_\mu$ in Rb₃C₆₀ discussed below. The resulting parameters are given in Table 4.4 along with the ratios

From the Korringa Law (Eq. 3.7), this ratio is a measure of the ratio of the density of states at the Fermi Surface g_N(0) relative to its value in Rb₃. Additionally, it is reasonable to assume $A_\mu$ is not strongly temperature dependent at low temperature, as is evidenced by Korringa temperature dependence of the relaxation rate in the normal state which is entirely the temperature dependence of $\nu_{SE}$. Hence we can also fit the more accurately determined average value of (T₁T)^-1 in the normal state to (3.4). These average values over several temperatures above T_c are shown in Fig. 4.15b. Fitting the Rb₃C₆₀ results to (3.4) with $A_\mu = 4340$MHz yields $\nu_{SE}/T = 20.2(4)$ MHz/K with $R_0/T = 4.4(7)\cdot10^{-3}\mu$s^-1/K. Using this fit and the points for K₃C₆₀ and Na₂Cs, we calculate the ratios,

which are also given in Table 4.4.

The low values of $\nu_{SE}$ (relative to $A_\mu$)confirm that the slow spin-exchange limit form (3.4) is justified at this temperature and below (for all fields). In fact, $\nu_{SE}$ is remarkably slow compared to spin-exchange rates of Mu in semiconductors such as Si. In that case the spin exchange rate is usually[92] modelled as

where one has extracted the majority of the temperature dependence of the Golden Rule expression (Eq. 3.6) into the carrier concentration, n, and the mean thermal velocity v, leaving the nearly temperature independent sum over matrix elements in the appropriately defined cross-section $\sigma$. For interstitial Mu in Si, the observed $\nu_{SE}$ implies that $\sigma$ is of the order of a typical atomic cross-section (10^-15 cm²). In the metallic case the situation is different since Fermi-blocking prevents all but the electrons within kT of the Fermi surface from participating in the spin exchange collisions. Thus $v \approx v_F$, the Fermi velocity and is nearly T independent, and $n \approx n_0kT/E_F$. Using values[31] for Rb₃C₆₀ ($v_F \approx 10^{-7}$cm/s, $E_F \approx 0.5$ eV, $n_0 \approx 4 \times 10^{21}$ cm^-3), $\sigma$ for Mu@C₆₀ is 10^-18-10^-19 cm² at 35K. The small size of $\sigma$ is attributed to the combination of two factors. First, the conduction band states are made up of C₆₀ molecular orbitals in which the electrons are confined near the hollow carbon cage. The spatial distribution of the conduction electrons is thus quite inhomogeneous on the scale of the unit cell, and Mu@C₆₀ is located in a site of low conduction electron density. Second, possibly for geometric reasons, there is very little hybridization of Mu@C₆₀ with the surrounding C₆₀ orbitals. The sp² carbon orbitals are distorted by curvature of the C₆₀, so that the inner lobes are smaller than the outer ones. Consequently, the tendency for bonding is significantly greater in the exohedral case.[167] The evidence for this is the large vacuum-like isotropic Mu hyperfine interaction. This is also consistent with the apparently very small ¹³C-Mu nuclear hyperfine interaction[14] in pure C₆₀. In contrast, in semiconductors, the hyperfine parameters of muonium are much lower than the vacuum-value.[92] We thus conclude that for Mu@C₆₀ the spin exchange interaction (J in Eq. (3.3)) in A₃C₆₀ is small and the perturbation approach (3.6) is valid.

Comparison of the dependence of the DOS at the Fermi surface g_N(0) on lattice constant a with the dependence of T_c(a) elucidates the exponent of the McMillan equation; furthermore, there is considerable interest in the the role of orientational disorder in determining electronic properties such as g_N(0).[171,172] In the normal state, both $\nu_{SE}$ and (T₁T)^-1 are proportional to the squared density of states at the Fermi surface g_N²(0) (e.g. (3.7)), so the ratios defined above provide a measure of g_N(0) relative to its value in Rb₃. For comparison, similar ratios from NMR[174,175,98,176] yield $\rho^{x} \gt 0.75$ for both x = K₃C₆₀ and Na₂Cs. The low value of our ratios for Na₂CsC₆₀ may indicate that the proportionality constants (hyperfine couplings) between $\nu_{SE}$ (or (T₁T)^-1) and g_N²(0) may vary with structure. This is not unreasonable, since the couplings depend on the detailed structure of the electronic orbitals constituting the conduction band, such as the degree of sp hybridization. It is known that the C-C bond lengths of C₆₀^-3 in cubic Na₂CsC₆₀ differ slightly[43] from those of the neutral C₆₀, but similar measurments on the orientationally disordered systems have not been reported. The exchange coupling for Mu@C₆₀ (J in (3.3)) may be more sensitive to such differences than those of ¹³C NMR because they are determined by the tails of the carbon orbitals protruding into the ball; whereas, the NMR constants are determined by the behaviour of the orbitals at or near the ¹³C nucleus. While the electron-phonon enhancement of g_N (which may differ between the Pa$\bar{3}$ and Fm$\bar{3}$m structures) does not[173] affect T₁, electron-electron interactions can[34], via for example, Stoner enhancement of g_N(0). If there are short-wavelength electronic correlations, it is possible that T₁(T) might vary within the unit cell, causing the Mu@C₆₀ and ¹³C to vary differently. The similarity of the temperature dependence of (T₁T)^-1 for alkali and ¹³C NMR, though, suggests that any electronic correlation contribution to T₁ does not vary significantly with position in the unit cell in either Fm$\bar{3}$m[174] and Pa$\bar{3}$[175] materials. Even if the hyperfine couplings and g_N(0) were identical, it is possible[177] that different levels of disorder could lead to different values of T₁. Thus we conclude that simple comparison of the magnitudes of T₁ (from $\mu$SR or NMR) between non-isostructural A₃C₆₀ superconductors may not represent a comparison of g_N(0). The possibility that C₆₀ polymerization is the source of suppression of the normal state (T₁T)^-1 is discussed in section VII.

  

Figure 4.16: The LF dependence of the Mu@C₆₀ asymmetry (decoupling curve). The field at the inflection point indicates that the Mu hyperfine parameter is large (near its vacuum value). Each point is determined from a common fit to at least 2 different temperatures, ranging from 2.5 to 15 K (e.g. Fig.4.21).

If the T₁ relaxation is too slow to be observed, the amplitude of the signal due to Mu, which is field dependent below $\sim 0.5 T$, can still be used to identify Mu. The field dependence follows:

where x is the reduced field B/B_hyp, and B_hyp is defined as $A_\mu/(\gamma_e+\gamma_\mu)$ (see §7.3.1 of Schenck[73]). Measurement of this field dependence, when the Mu asymmetry is non-relaxing, requires careful accounting of the systematic field dependent shifts in the LF-$\mu$SR baseline. This method has the advantage that it admits the possibility of measuring the hyperfine parameter (although not as accurately as precession) which determines, for example, the inflection point of the decoupling curve (4.4). We have made two measurements of the LF muonium decoupling curve in two different samples of Rb₃. In the first measurement, we acquired pairs of spectra at 35K and 4K at each field in a special apparatus which allowed collection of a reference spectrum in high purity silver in the same conditions of field and temperature simultaneously.[96] Systematic shifts in the baseline could thus be at least partially compensated using the reference data. In the second experiment, we took data in various fields at 2 (or more) temperatures, one where the relaxation was quite fast, and one where it was slow in order to determine the relaxing amplitude, i.e. the muonium asymmetry A_Mu (see Fig. 4.16). The results of this second method are roughly consistent with the first measurement but less scattered. The decoupling in the data is apparently sharper than expected (fit curve). In this data, we did not field cool, but the consequent additional inhomogeneity of the field would not affect the decoupling significantly. Any correlation of the amplitude and relaxation rate, due for example to a non-exponential relaxation, could bias the extracted asymmetry and account for the sharper feature. From the fit to (4.4), we get $A_\mu=4300(400)$MHz. We turn now to briefly discuss the stability of Mu@C₆₀ in Rb₃.

In semiconductors, in which Mu centres undergo spin-exchange with free carriers, it is found that at sufficiently high temperature, the temperature dependent carrier density (which controls the spin and charge exchange rates) becomes high enough to ionize Mu. In doped semiconductors, the relevant temperature scale for this ionization is the bandgap between the impurity and conduction bands, while in intrinsic semiconductors it is the bandgap. The stable Mu state in an n-type system is likely Mu^-, and at high enough carrier density, the metastable neutral Mu state ionizes quickly. We have observed the ionization behaviour (activated increase of the T₁ relaxation rate) of Mu@C₆₀ in Rb₃C₆₀ at 4.2T (see Fig. 4.17). This behaviour indicates the ionization energy is on the order of 600K. The stability of paramagnetic Mu@C₆₀ in Rb₃C₆₀ at low temperature suggests that the Mu impurity level lies in the valence band-conduction band gap (see Fig. 4.18), and that the ionized Mu^- state lies above the Fermi level. This may be due to an enhanced Coulomb repulsion between the two local electrons on the $\mu^+$ from the confinement within the C₆₀ cage. If this picture is correct, it suggests that the 600K energy scale represents the energy of the Mu^- state above the Fermi level. It is not clear at the present time how to reconcile the small size of the ionization energy with the approximate conduction bandwidth of $\sim 0.5$eV.

  

Figure 4.17: Activated increase in (T₁T)^-1 relative to its value at low temperature likely due to Mu@C₆₀ ionizing to endohedral Mu^-. The fit curve is $(1.01(3) + 123(35)\exp{(-588(40)K/T)})$.

  

Figure 4.18: Mu@C60

In conclusion we note that the important features of the muon sites in the A₃C₆₀ superconductors to the analysis included in the following sections are simply that the muons stop randomly on the relevant length scales of the vortex state: the penetration depth $\lambda$ and the vortex spacing; most of the muons remain diamagnetic and sample the field distribution of the vortex state uniformly; and a small fraction of the muons form a paramagnetic muonium centre which is only very weakly coupled to the conduction electrons.