Tunneling conductance and local density of states in time-reversal symmetry breaking superconductors under the influence of an external magnetic field

We consider different effects that arise when time-reversal symmetry breaking superconductors are subjected to an external magnetic field, thus rendering the superconductor to be in the mixed state. We focus in particular on two time-reversal symmetry breaking order parameters which are believed to be realized in actual materials: $p+\i p'$-wave and $d+\i s$- or $d+\i d'$-wave. The first order parameter is relevant for Sr$_2$RuO$_4$, while the latter order parameters have been suggested to exist near surfaces in some of the high-$T_c$ cuprates. We investigate the interplay between surface states and vortex states in the presence of an external magnetic field and their influence on both the tunneling conductance and the local density of states. Our findings may be helpful to experimentally identify the symmetry of unconventional time-reversal symmetry breaking superconducting states.