Anisotropy of the Energy Gap

In any superconductor with a non-spherical Fermi surface, one expects anisotropy in the superconducting gap function $\Delta (\vec{k})$. In general, the gap function will vary according to the angle with respect to the crystalline axes [70]. That is, the energy required to break a Cooper pair will depend on the direction of $\vec{k}$.

In $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7}$ }$, one expects the size of the energy gap to show some variation in the ab-plane. However this does not necessarily imply nodes in the gap. If the Fermi surface in the ab-plane is not a perfect circle, then the gap will certainly be anisotropic; but as is the case for an anisotropic s-wave pairing state, the gap may remain finite over the entire Fermi surface. Furthermore, if we are to think of the energy gap in the superconducting state of $\mbox{YBa$_{2}$ Cu$_{3}$ O$_{7- \delta}$ }$ as having d_x²-y² symmetry, $\vec{a}$-$\vec{b}$ anisotropy will produce nodes in the gap which are not precisely along $\vert \hat{k}_{x} \vert = \vert \hat{k}_{y} \vert$.

As mentioned above, conventional superconductors also have some anisotropy in the gap function. However, in most of these materials the mean free path is such that $l \ll \xi$ and also $\lambda \ll \xi$, so that the anisotropy is negligible when interpreting the experimental results [71]. In the high-T_c superconductors where $\xi \ll l \ll \lambda$, the anisotropy in the gap might play a significant role.