3.2.3 Extensions

The temperature dependence of (3.9) discussed above can be modified through several mechanisms which we will consider in turn: anisotropy of $\Delta$, finite lifetime of quasi-particle excitations, and magnetic effects.

The consequences of anisotropy on the ratio R_S/R_N are found by including an angular integral in (3.9), and they can be most easily explained by a comparison between the angular average DOS, g_A(E), and g_S(E) of (3.8).

where P(a) is the distribution of the anisotropy a of the gap around the Fermi surface. Even a small anisotropy, such as that for aluminum[104], transforms the BCS singularity in g_S(E) into a mild van Hove singularity at some average $\Delta_P$,and g_A is still perfectly gapped with g_A(E)=0 for $\vert E\vert \leq \Delta_G$ (see Fig. 3.12a). The effect of anisotropy is thus to reduce the size of the coherence peak in R_S/R_N and to modify the Arrhenius slope relative to the isotropic case. Extreme anisotropy, such as that for non-zero angular momentum pairing states, is similar except that g_A is no longer gapped as there are nodes in $\Delta$.For example, for a d-wave order parameter[108], $g_A(E) \propto E$ as $E \rightarrow 0$. Although g_A is still peaked in this situation, the coherence peak in R_S/R_N may be completely eliminated[109], and the exponential temperature dependence is replaced by a power law $R_S/R_N \propto T^p$, where p = 2 for d-wave, and other values of p are obtained[110] for different nodal structures of $\Delta$.This kind of behaviour has been observed[111,112] in YBa₂Cu₃O_6.95, for which there is strong evidence of a d-wave $\Delta$.A p-wave $\Delta$ may be the source of similar temperature dependence in some Heavy Fermion superconductors[113,114], while one dimensionality may cause it in some organic superconductors[115].

  

Figure 3.12: a) Models of the superconducting DOS: BCS is g_S of Eq.(3.8), Aniso is g_A (Eq.(3.11)) with a gate function distribution P(a) of width 0.1$\Delta$, $\Gamma$ is g_D (Eq.(3.12)) with $\Gamma = 0.1 \Delta$, and $\Delta_2$ is g_SC (Eq.(3.13)) with $\Delta_2 = 0.1 \Delta$.b) The value of the Hebel-Slichter integral for the BCS and lifetime (Eq. (3.12)) broadened g_S(E). The magnetic inelasticity parameter ($\alpha_m$) is appropriate for Mu@C₆₀ in Rb₃C₆₀. The BCS temperature dependence $\Delta(T)$was used.

Finite lifetime ($\tau$) of the quasiparticle excitations of a superconductor due, for example, to electron-phonon, electron-electron or impurity scattering can also modify R_S/R_N. This possibility was suggested by Hebel and Slichter in their original work[95] to explain the small size of the coherence peak they observed in Al. They calculated a DOS which was a version of Eq. (3.8) smeared by convolution with a gate function of width $\tau^{-1}$.A detailed analysis of the temperature dependence of R_S/R_N resulting from this approximation is given by Hebel[116]. A different Ansatz for the DOS was used by Dynes et al.[117] to describe tunneling measurements:

where $\Gamma \sim \tau^{-1}$. However, Allen and Rainer[118] point out that for a lifetime due to electron-phonon scattering, one must resort to the Eliashberg theory of strongly coupled superconductors [52,119] in which the order parameter becomes complex, and the DOS is[120]

where[121] $\Delta_2=\Im{\Delta} \sim \tau^{-1}$($\Im$ is the imaginary part), and $\Delta = \Delta(E,T)$ is determined by the Eliashberg theory and the coupling constant-phonon spectrum product $\alpha^2F(E)$ for the particular material. Fibich[122] first treated the problem of calculating R_S/R_N using (3.13) by neglecting the energy dependence of $\Delta$, and simply using $\Delta$ evaluated at the energy which is most important for the integral (3.9), i.e. $\Delta(E=\Delta_1(T),T)$.The temperature dependence for the imaginary part $\Delta_2(T)$ due to phonon scattering[122,123] and scattering from other quasiparticles[121,124] has been calculated in the low temperature limit. For the temperature dependence of the real part $\Delta_1(T)$ (and for the parameter $\Delta$ in either of the preceding models) it is reasonable[52] to assume that the temperature dependence of the real part of the order parameter is approximately that of the BCS $\Delta$.Recently, it has become feasible[125,118] to calculate R_S/R_N using the full strong-coupling $\Delta(E,T)$,thus avoiding these approximations. As input to such a calculation, one would ideally first obtain a reasonable form for $\alpha^2F(E)$. However, according to Akis[125], the details of $\alpha^2F(E)$ are not important, and the most significant information in determining R_S/R_N(T) is summarized in the ratio $T_{c}/E_{\log}$, where $E_{\log}$ is the logarithmic moment of $\alpha^2F(E)$ (Eq. (1.8)). Note that the effect of impurities in the Eliashberg theory has recently been revisited[126]. These authors find that ``vertex corrections'' from impurity scattering can increase the size of the coherence peak as the mean free path is reduced.

Magnetism may also influence R_S/R_N(T), for instance, in the classical example[127] of gaplessness in a superconductor due to the presence of magnetic impurities, the coherence peak can be reduced or eliminated and the exponential fall-off strongly modified[128,129]. Superconductors that are intrinsically magnetic exhibit similar strong deviations[113,130]. In regard to high-T_c superconductors, antiferromagnetic correlations between quasiparticles have also been shown[109,125] to damp the coherence peak. However, it should be noted that no anomalous behaviour connected with magnetism has been reported yet in A₃C₆₀. Recently[131], the closely related NH₃K₃C₆₀ material which is superconducting under high pressure has been found to exhibit a metal-insulator transition to a magnetic state[132] at about 40K. The small bandwidth and large coulomb interactions between electrons also cause important correlation effects, for example the magnetism in o-A₁C₆₀. The proximity to a similar magnetic phase may be enhanced by the analogous polymerization[133] of the C₆₀ anions in the Pa$\bar{3}$ materials.