Experiments on tetragonal Sr2RuO4 and CeIrIn5 indicate the presence of superconductivity with a multi-component superconducting order parameter. Such an order parameter should exhibit an in-plane anisotropy in the upper critical field near the superconducting transition temperature that does not occur for single-component superconductors. Here, this anisotropy is determined from microscopic calculations for arbitrary gap functions. It is shown that this anisotropy is generally not small and, in

more »

... some cases, independent of impurity scattering. Furthermore, this anisotropy is calculated for many detailed microscopic models of Sr2RuO4. For these models this anisotropy is found to be large, which is in sharp contrast to the small anisotropy observed experimentally. However, an accidental cancellation of the anisotropy for gaps on different Fermi surface sheets can lead to a result that is consistent with experiment.