In-field critical current of type-II superconductors caused by strain from nanoscale columnar inclusions

The results of a linear elasticity analysis yields that nanorod inclusions aligned along the $c$ axis of a thin film of ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{7-\delta }$, such as ${\text{BaZrO}}_3$ and ${\text{BaSnO}}_3$, squeeze that matrix by pure shear. The sensitivity of the superconducting critical temperature in that material to the latter implies that the phase boundary separating the nanorod inclusion from the superconductor acts as a collective pinning center for the vortex lattice that appears in external magnetic field. A dominant contribution to the in-field critical current can result. The elasticity analysis also yields that the growth of nanorod inclusions can be weakly metastable when the inclusion is softer than the matrix.

Figure 1

The total elastic energy density Eq. (13), in units of $E_{2\text{D}}^{\left(1\right)}/\pi r_1^2$, versus the density of nanocolumn inclusions, in units of $1/\pi r_1^2$, is plotted. The dashed line above corresponds to the elastic energy of isolated nanocolumn inclusions. The radii in Eq. (13) are set to $r_2^2=\pm 3r_1^2$ for relatively soft and hard nanocolumns, respectively.

Figure 2

A potential-energy landscape for a single vortex line in units of $\left|{\epsilon }_{\text{p}}\right|$ and the coherence length that results from a superposition of 2744 $d$-wave collective pinning centers [Eq. (14)] arranged in a “liquid” fashion is displayed (see Ref. 8).