Effect of midgap states in d-wave superconductors on the flux quantization in tricrystal rings

To determine whether a tricrystal ring [J. R. Kirtley et al., Phys. Rev. Lett. 76, 1336 (1996)] is a 0 ring or a pi ring, the following expression has been used for the critical current between two d-wave superconductors with misorientation angles theta(1) and theta(2): I-c=I(0)cos2 theta(1)cos2 theta(2). We show that the recently predicted midgap surface states lead to an additional component: I-c(midgap)=-I(MG)sin2 theta(1)sin2 theta(2) (I-MG>0). The midgap component may get enhanced at low temperatures resulting in the conversion of a 0 ring to a pi ring (or vice versa) when the temperature is reduced. The observability of this effect in real structures is discussed.