3.2.4 Influence of the Vortex State

Measurements of T₁ in type-II superconductors ($\lambda \gt \sqrt{2} \xi$) are typically done with the applied field above the lower critical field, H_c1, i.e. in the vortex state. The magnetic field interacts with the electronic system in two ways: i) via the Lorentz force embodied in the canonical momentum ${\bf p} + q {\bf A}$ and ii) via the Zeeman interaction of the electronic spins , consequently modifying the quasiparticle excitation spectrum and hence the temperature dependence of R_S/R_N relative to (3.9).

The Lorentz force, together with the magnetic field energy, yields the vortex structure of the mixed state wherein the field and the order parameter are inhomogeneous, for example, the Abrikosov flux lattice (see the review [134]) or more disordered phases [135]. Near H_c2, where the order parameter is small, the effect of the inhomogeneity $\Delta({\bf r})$ has been treated theoretically in the dirty[136] and clean[137] limits and, subsequently, for arbitrary[138] mean free path l. In the dirty limit the effect of the DOS[136] is, in close analogy with the Abrikosov-Gor'kov theory of gapless superconductivity[139,140] in the presence of magnetic impurities,

where $\eta$ is the pair-breaking perturbation parameter[139,136], $\eta \propto \tau B$, where $\tau$ is the collision time. The DOS exhibits no gap. In the clean limit[137] the DOS is highly anisotropic with singular BCS (eq. (3.8)) behaviour along the magnetic field and gapless behaviour perpendicular to the field. The latter property has been confirmed and exploited in measurements of de Haas-van Alphen oscillations in the mixed state at high fields[141]. The effect of finite mean free path in this case is to wash out the anisotropy (and with it the BCS singularity) and to make the density of states a rigorously local property[136].

Away from H_c2, the expansions assuming small $\Delta$ are not applicable. The more general theories are very complex, but some approximate results have been obtained. There are two main approaches used to calculate properties of the vortex state in the regime $H_{c1} \ll H \ll H_{c2}$: the Green's function approach of Gor'kov[142] and the effective Hamiltonian approach of Bogoliubov and deGennes[136]. Using the second approach, the presence of bound, nearly gapless excitations in the vortex cores was predicted[143]. Far from the vortex cores, the excitation spectrum is modified only by the ``Doppler shift'' of the quasiparticle energies due to the circulating supercurrents. Neglecting the core states, Cyrot[144] calculated an explicitly field dependent DOS. These calculations show that the peaks of the zero field DOS are broadened significantly as the vortex spacing decreases below about 10$\xi$.Motivated by STM measurments of the detailed spatial structure of the vortex state, this method has been revisited recently, e.g. see references[145,146]. In contrast to the local nature of the dirty-limit, where the core contributions can be modelled simply as normal electrons, in the clean-limit, the interplay between the core states and the surrounding superconductor may be very important[146]. The Green's Function approach has been employed generally using a linearized version of Gor'kov's equations. For the dirty limit, the field-dependent local and spatially averaged DOS have been calculated numerically[147]. The spatially averaged DOS peaks in this calculation also exhibit field-dependent broadening. Clean-limit calculations have also been performed[148]. Recent use of this technique[150,149] has concentrated on the structure of the vortex core.

It has long been recognized[151] that the Zeeman interaction acts to break the Cooper pairs of a conventional superconductor because it acts in the opposite sense on each member of the pair. The magnetic field at which the Zeeman interaction will destroy superconductivity can be approximated by equating the gain in energy in going to the (spin-polarized) normal state with the condensation energy of the superconductor, thus defining[152] the Pauli limiting field B_P:

where B_c(t) is the thermodynamic critical field and $\chi_i$ is the spin susceptibility in each of the phases. The modification of the upper critical field H_c2 due to these considerations has been calculated[153,152], and its effect on the spectrum of excitations has been predicted to be negligible[139] unless there is some mechanism for mixing quasiparticle states of opposite spin, e.g. spin-orbit scattering. In the case of strong spin-orbit scattering, the quasiparticle spectrum is again of the form (3.14) with the pair-breaking parameter $\eta \propto \tau_{so} B^2$, where $\tau_{so}$ is the time between spin-orbit scattering events[139].

  

Figure 3.13: A generic phase diagram for the behaviour of (T₁T)^-1 in a conventional type II superconductor as described in the text.

Despite these complications in the vortex state, many of the general features of the zero-field (Meissner state) behaviour of R_S/R_N as discussed in section IV.A are observed experimentally. Fig. 3.13 shows the phase diagram of the vortex state, for a hard (H_c1 coincides with the horizontal axis on this scale) type-II superconductor showing regimes of different behaviour for T₁ in the most conventional case. To the lower right is the region of the Hebel-Slichter coherence peak. To the left is the ``Arrhenius region'' where the relaxation rate falls exponentially. Above about 0.7H_c2, the theories based on the gaplessness due to a small inhomogeneous order parameter $\Delta({\bf r})$ are applicable. In particular $R_S/R_N(T,l\rightarrow 0)$ has been calculated[154] with the result, that the peak is reduced but still present in region (ii), and completely eliminated in region (i). The high field damping and elimination of the coherence peak predicted by this theory has been clearly verified experimentally[155] in the A15 superconductor V₃Sn. The calculation of R_S/R_N(T,l) can be found elsewhere[156].

Departures from the exponential fall-off of the relaxation rate with temperature are typically seen at low temperature (region 2C). In this region, the much more weakly temperature dependent relaxation from the electronic excitations in the vortex cores can be significant[157]. One can model the relaxation of (muon or nuclear) spin polarization in such an inhomogeneous case using a local relaxation rate. The average relaxation function is

where the inhomogeneous T₁ rate $R(B({\bf r}),\Delta({\bf r}),T)$is distributed as P(R), and z indicates LF (T₁) relaxation. For $B \ll B_{c2}$, the vortices may be treated (in the dirty limit) approximately[143] as cylinders of normal state material of radius $\xi$.The distribution P(R) in this approximation is bimodal with peaks at R_N and R_S. The relaxation P_Z(t) will not be single exponential, but the average relaxation rate is[159]

where, for ($B \ll B_{c2}$), $f_N \sim \xi^2(B/\Phi_0)$, i.e. the weight in the distribution P(R) at R_N scales linearly with field ($\Phi_0$ is the magnetic flux quantum). This linear field dependence allows the deviation from exponential temperature dependence of the relaxation rate due to the vortex cores to be distinguished from the other mechanisms, such as impurity scattering, (region O) which at low temperature, and especially in low field, may limit the electronic relaxation. From the field dependence, one can thus use this model to extract a rough estimate[158,159] of $\xi$ deep in the superconducting state. Such estimates could be refined by including a more sophisticated local DOS[150,145] in the model for P(R), but dynamic effects might also require consideration. Motion of the vortices on the timescale T₁ would smear the distribution P(R), effacing the bimodal structure of the static vortex state; however, vortex dynamics usually occur on timescales much shorter than a typical T₁[160], though exceptions have been proposed.[161] Another type of dynamics that has been considered in this context is that of spin-diffusion of the nuclear magnetization to the quickly relaxing regions of the cores which effectively allows the relaxation to ``leak'' out of the vortex cores into the surrounding superfluid. This mechanism, though, may be thermodynamically quenched in the inhomogeneous vortex state.[162] Furthermore, such a mechanism is not important in the experiments described here because we are always dealing with a single muon in the sample at any time, and Mu@C₆₀ is static (at least on the length scale of the vortex lattice and on the timescale of the muon).

Deviations from the behaviour summarized in figure (3.13) are expected and observed in many cases: Reduction of the peak region (towards (t,h) = (1,0)) may be the result of any of the mechanisms discussed in the previous section. More detailed reviews of NMR in type-II superconductors can be found elsewhere.[104,163,164]