Angular dependence of the penetration depth in unconventional superconductors

Abstract

We examine the Meissner state nonlinear electrodynamic effects on the field and angular dependence of the low-temperature penetration depth \ensuremath{\lambda} of superconductors in several kinds of unconventional pairing states, with nodes or deep minima (``quasinodes'') in the energy gap. Our calculations are prompted by the fact that, for typical unconventional superconducting material parameters, the predicted size of these effects for \ensuremath{\lambda} exceeds the available experimental precision for this quantity by a much larger factor than for others. We obtain expressions for the nonlinear component of the penetration depth \ensuremath{\Delta}\ensuremath{\lambda} for different two- and three-dimensional nodal or quasinodal structures. Each case has a characteristic signature as to its dependence on the size and orientation of the applied magnetic field. This shows that \ensuremath{\Delta}\ensuremath{\lambda} measurements can be used to elucidate the nodal or quasinodal structure of the energy gap. For nodal lines we find that \ensuremath{\Delta}\ensuremath{\lambda} is linear in the applied field, while the dependence is quadratic for point nodes. For layered materials with ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\mathrm{\ensuremath{\delta}}}$ type anisotropy, our results for the angular dependence of \ensuremath{\Delta}\ensuremath{\lambda} differ greatly from those for tetragonal materials and are in agreement with experiment. For the two- and three-dimensional quasinodal cases, \ensuremath{\Delta}\ensuremath{\lambda} is no longer proportional to a power of the field and the field and angular dependences are not separable, with a suppression of the overall signal as the node is filled in.