Ginzburg-Landau theory of the superheating field anisotropy of layered superconductors

We investigate the effects of material anisotropy on the superheating field of layered superconductors. We provide an intuitive argument both for the existence of a superheating field, and its dependence on anisotropy, for $\kappa =\lambda /\xi$ (the ratio of magnetic to superconducting healing lengths) both large and small. On the one hand, the combination of our estimates with published results using a two-gap model for ${\mathrm{MgB}}_2$ suggests high anisotropy of the superheating field near zero temperature. On the other hand, within Ginzburg-Landau theory for a single gap, we see that the superheating field shows significant anisotropy only when the crystal anisotropy is large and the Ginzburg-Landau parameter $\kappa$ is small. We then conclude that only small anisotropies in the superheating field are expected for typical unconventional superconductors near the critical temperature. Using a generalized form of Ginzburg Landau theory, we do a quantitative calculation for the anisotropic superheating field by mapping the problem to the isotropic case, and present a phase diagram in terms of anisotropy and $\kappa$, showing type I, type II, or mixed behavior (within Ginzburg-Landau theory), and regions where each asymptotic solution is expected. We estimate anisotropies for a number of different materials, and discuss the importance of these results for radio-frequency cavities for particle accelerators.

Figure 1

Illustrating vortex nucleation in a type-II superconductor occupying the half space $x>0$, and subject to a magnetic field parallel to the vacuum-superconductor interface.

Figure 2

(a) Illustration of the penetration of a vortex core into a type-II anisotropic superconductor with anisotropy axis $\mathit{c}\parallel z$ (perpendicular to the plane of the figure). (b) Vortex and vortex core acquire an ellipsoidal shape when $\mathit{c}$ lies in the $xy$ plane. Here the superconductor surface lies horizontally and vertically, for $\mathit{c}$ parallel to $x$ and $y$, respectively. The magnetic field is parallel to $z$ for both (a) and (b). We can estimate the superheating field from the calculation of the work necessary to push a vortex core into the superconductor, thus destroying the Meissner state. For anisotropic vortices, the superheating field turns out to be proportional to the area of the green (black) boxes for the superconductor boundary surface parallel (perpendicular) to the $\mathit{c}$ axis. These estimates simplify the calculations of Bean and Levingston [16], which consider the vanishing of the surface energy barrier felt by a single penetrating vortex in a type-II superconductor. More generally, the Ginzburg-Landau approach takes into account the cooperative effects due to the penetration of multiple vortices [2].

Figure 3

Showing regions in ${\kappa }_{\parallel }\times \gamma$ space where the crystal is always type I (left region to blue solid lines), always type II (right region to red solid lines), or might be of either type (region between red and blue lines), depending on the orientation of the crystal. The shaded blue and orange regions correspond to regions where the Ginzburg-Landau superheating field anisotropy can be approximated by ${\gamma }^{1/2}$ and 1, respectively, within $10\%$ of accuracy.

Figure 4

Anisotropy of the Ginzburg-Landau superheating field as a function of mass anisotropy for several values of ${\kappa }_{\parallel }$. The dotted lines are the limiting solutions given by Eq. (13).

Figure 5

Ginzburg-Landau superheating field $H_{\text{sh}}/\left(\sqrt{2}{H}_c\right)$ as a function of $\kappa$. The points correspond to the solutions using ${\kappa }_{\parallel }$ and ${\kappa }_{\perp }$ for ${\mathrm{Ag}}_5{\mathrm{Pb}}_2{\mathrm{O}}_6$ (blue), ${\mathrm{C}}_8\mathrm{K}$ (purple), ${\mathrm{MgB}}_2$ (red), and BSCCO (dark red).

Figure 6

Illustrating a vortex and vortex core (blue and red regions) of ${\mathrm{MgB}}_2$ near $T=0$ (we increased ${\xi }_a$ by a factor of 30 with respect to ${\lambda }_a$, so that the core features become discernible; the small black region in the center corresponds to the actual scale). Near zero temperature, the field penetration region is calculated to be nearly isotropic (${\lambda }_a\approx {\lambda }_c$), whereas the core shape anisotropy is predicted to reach a maximum (${\xi }_a\approx 6{\xi }_c$) (Ref. [23]).