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The Vortex Core

In the simple view, a vortex core is a cylinder of normal material whose radius is the Ginzburg-Landau coherence length $\xi(T)$. Over this length scale the order parameter  $\psi(\mathbf{r})$ and the supercurrent density  J(r) fall monotonically to zero at the core centre. Although the Ginzburg-Landau formalism is only truly valid near the transition temperature Tc, the core radius is also commonly defined as the temperature dependent coherence length $\xi(T)$at low temperatures, where it becomes approximately a BCS coherence length $\xi_{BCS}$ [3]. Employing this assumption Caroli, de Gennes and Matricon [20] investigated the quasiparticle excitations of energy  $\varepsilon \ll \Delta_\infty$ localised near an isolated vortex line in a clean ( $\xi_{BCS} \ll l$) type II superconductor, where $\Delta_\infty$is the bulk value of the BCS energy gap. They determined these quasiparticles to have at least an energy $\varepsilon_{min} \approx
\Delta_\infty^2/E_F$, and above this a density of states like that of a cylindrical normal region of radius $\xi(T)$. This traditional vortex core picture implies that at low temperatures the core radius, being roughly a BCS coherence length $\xi_{BCS}$, is essentially temperature independent.

The Kramer-Pesch effect, predicted for isolated flux lines in clean s-wave superconductors [3][4][5], refers to the rapid contraction of the vortex core radius $\rho $ to around a Fermi wavelength 1/kF upon cooling at low temperatures. This flux line narrowing stems from the thermal depopulation of the quasiparticle bound states. The bound state energy levels $E_\mu$ asymptotically approach the BCS energy gap  $\Delta_\infty$ as their corresponding angular momenta $\mu $ become infinite, and the low energy radial wavefunctions are greatest at a distance  $r \simeq \mu / k_F$ from the core centre [4]. The reduction in core radius terminates at $\rho \sim
1/k_F$ in the quantum limit  $T \lesssim T_0 = T_c / (k_F \xi_{BCS})$ [5]. Here only the lowest energy bound state remains occupied [21]. From temperature $T \ll T_c$ down to near the quantum limit temperature T0, the core radius $\rho $ shrinks linearly as

\begin{displaymath}\rho \sim \frac{T}{T_c} \xi_{BCS}
\end{displaymath} (4.10)

Over this distance $\rho $ away from the core centre the supercurrent density  J(r) climbs to its greatest value and the pair potential  $\Delta(\mathbf{r})$ rises very steeply. However the pair potential still attains its asymptotic value over a length scale comparable to the BCS coherence length $\xi_{BCS}$ [4][21]. At the core centre the maximum internal field increases linearly as the temperature drops [22][4]. Experimental evidence for such dramatic vortex core shrinking would contradict the common assumption that, at low temperatures, the value of the core radius $\rho $ remains constant at around a BCS coherence length $\xi_{BCS}$.

Experimental observations reveal the shrinking of the vortex cores upon cooling to be more limited than expected from the predicted Kramer-Pesch effect. Indirect evidence supporting the proposed Kramer-Pesch effect comes from the logarithmic singularity in the current-voltage characteristic for Nd1.85Ce0.15CuOx films [23]. Muon spin rotation ($\mu $SR) measurements of the core radius $\rho $ as a function of temperature show a surprisingly weak Kramer-Pesch effect in NbSe2 [6] ( $T_c = 7.0\,\mathrm{K}$), YBaCu3O6.95 [7] ( $T_c = 93.2\,\mathrm{K}$) and YBaCu3O6.60 ( $T_c = 59\,\mathrm{K}$). The core radius $\rho $ in NbSe2 saturates at $\rho \approx 72\,\textrm{\AA}$, many times larger than the anticipated low temperature radius $\rho $ of around  $10\,\textrm{\AA}$. The temperature dependence of the vortex size is weaker in YBaCu3O6.95, and even more so in YBaCu3O6.60. The apparent absence of significant core shrinking in YBaCu3O6.60and YBaCu3O6.95 possibly reflects the attainment of the quantum limit [5][24][25]. The quantum limit temperature T0 should be much higher in these materials than in NbSe2, since they have a considerably smaller BCS coherence length $\xi_{BCS}$. The substantially larger-than-predicted core radii found at very low temperatures in NbSe2 are attributed to interactions between the vortices, and to their possible zero point motion. To date, all theoretical works concerning the Kramer-Pesch effect suppose isolated vortices, an assumption which likely fails for the transverse field $\mu $SR experiments mentioned here. Also, in quasi two-dimensional (2D) superconductors such as NbSe2 and YBaCu3O $_{7-\delta}$, longitudinal disorder of vortices potentially inflates the value of the core radius $\rho $ determined with $\mu $SR, since a flux line in such materials consists of a column of 2D pancake vortices which could wobble [7]. Flux lines should be stiffer in three-dimensional (3D) superconductors, leading to a simpler dependence of the core radius $\rho $ on temperature. This makes the clean 3D type II s-wave superconductor LuNi2B2C an ideal candidate for observation of the predicted Kramer-Pesch effect with $\mu $SR. The next chapter describes the characteristics of this material.

next up previous contents
Next: Material Properties of LuNiBC Up: Superconductivity Previous: Ginzburg-Landau Theory
Jess H. Brewer