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 π ring, the following expression has been used for the critical current between two d-wave superconductors with misorientation angles θ₁ and θ₂: I_c=I₀cos 2θ₁cos 2θ₂. We show that the recently predicted midgap surface states lead to an additional component: I_c(midgap)=-I_MGsin 2θ₁sin 2θ₂ (I_MG>0). The midgap component may get enhanced at low temperatures resulting in the conversion of a 0 ring to a π ring (or vice versa) when the temperature is reduced. The observability of this effect in real structures is discussed.