Time-Reversal Symmetry Breaking States in High-Temperature Superconductors

The dx2−y2-wave symmetry of the order parameter leads to various physical phenomena of high-temperature superconductors which are not present in standard superconductors. A particularly interesting new aspect occurs in connection with the behavior of the order parameter in the vicinity of interfaces (Josephson junctions) between superconductors and close to surfaces. In this review we discuss the appearance of localized states with broken time-reversal symmetry in these regions, using various different schemes of analysis: the tunneling formalism, the Bogolyubov-de Gennes equation, and the generalized Ginzburg-Landau theory. We demonstrate that this kind of state is connected with the opening of a gap in the quasiparticle density of states which lowers the local free energy density. A direct consequence of time-reversal symmetry breaking can be seen in the presence of spontaneous supercurrents. Other features, such as the possibility of vortices with fractional flux quanta and unusual phase slip effects in Josephson junctions are examined as possible experimental indications for a time-reversal symmetry breaking state. Furthermore, quasiparticle tunneling has been recently discussed and applied to probe the surface of such a superconductor for the violation of time-reversal symmetry. Concerning the magnetic properties surfaces supporting such a state can yield a paramagnetic contribution to the magnetic response of the superconductor. Finally, some examples of superconductors which break time-reversal symmetry in the bulk are discussed. Clear signals in the zero-field relaxation rate of muons leave little doubt that such superconducting states are realized in UPt3, U1−xThxBe13 and Sr2RuO4.