Anisotropy of the Magnetic Penetration Depth

In experiments to determine $\lambda$, single crystals are much preferred over polycrystalline samples. This is true for several reasons. Because $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7- \delta}$ }$ is strongly anisotropic, the orientation of the sample in the applied field is significant. With single crystals one has control over this feature. By positioning the single crystal with its c-axis parallel to the applied field, one can readily proceed to determine the magnetic penetration depth in the ab-plane (or Cu-O plane), $\lambda_{ab}$ (or $\lambda_{\parallel}$). In an analogous fashion, suitable orientation of the single crystal in the applied field in principle allows measurement of $\lambda_{c}$ or $\lambda_{\perp}$(i.e. the magnetic penetration depth perpendicular to the Cu-O planes). This is much more difficult to perform experimentally, however, since the crystals grow in such a way that the $\vec{a}$-$\vec{b}$dimensions are much greater than in the $\vec{c}$-direction.

With polycrystalline samples, the principal axis of each grain is randomly oriented with respect to the applied field. Consequently one must average over all possible orientations of the c-axis to simulate the field distribution and then try to extract a value for the magnetic penetration depths $\lambda_{ab}$ and $\lambda_{c}$. This could be difficult, since different combinations of values of $\lambda_{ab}$ and $\lambda_{c}$ may give similar line shapes.

Further considerations attached to the use of powdered samples include the dissimilarity in shape of the individual grains. Consequently, each grain has a different demagnetization factor and thus a slightly different average field. This leads to an additional broadening of the field distribution, which if not properly taken into account will lead to an underestimate of the magnetic penetration depth.

In general, the magnetic penetration depth in an anisotropic superconductor is determined by replacing the effective mass m^* of the superconducting electrons by an effective-mass tensor m^* [67]. Until very recently, the $\vec{a}$-$\vec{b}$anisotropy had been considered negligible, so it has always been assumed that, for the uniaxial high-temperature superconductors, m^* has a degenerate eigenvalue m_ab^*(i.e. $m_{a}^{*} \approx m_{b}^{*} \equiv m_{ab}^{*}$) associated with supercurrents flowing in the ab-planes that screen magnetic fields perpendicular to the planes, and a nondegenerate eigenvalue m_c^*associated with supercurrents flowing along the c-axis, which help screen magnetic fields parallel to the Cu-O planes [68]. Thus for a uniaxial, anisotropic superconductor:

$$\lambda_{ab} = \sqrt{ \frac{m_{ab}^{*} \, c^{2}}{4 \pi e^{2} n_{s}}}$$ (32)

and

$$\lambda_{c} = \sqrt{ \frac{m_{c}^{*} \, c^{2}}{4 \pi e^{2} n_{s}}}$$ (33)

One can define an anisotropic ratio $\gamma$ for uniaxial superconductors, such that:

$$\gamma = \frac{\lambda_{c}}{\lambda_{ab}} = \sqrt{\frac{m_{c} . . . . . . ngle_{\parallel}} { \langle \triangle B^{2} \rangle_{\perp}}}$$ (34)

For $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7}$ }$, $\gamma \approx 5$ [69].