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 T_c, 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/k_F 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

$$\rho \sim \frac{T}{T_c} \xi_{BCS}$$ (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 Nd_1.85Ce_0.15CuO_x 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] ( $T_c = 7.0\,\mathrm{K}$), YBaCu₃O_6.95 [7] ( $T_c = 93.2\,\mathrm{K}$) and YBaCu₃O_6.60 ( $T_c = 59\,\mathrm{K}$). 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 YBaCu₃O_6.95, and even more so in YBaCu₃O_6.60. The apparent absence of significant core shrinking in YBaCu₃O_6.60and YBaCu₃O_6.95 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.