Momentum dependence of the half-flux quantum effect in YBCO-Nb rings

We have studied the half-flux quantum effect\footnote{C.C. Tsuei {\it et al.} Phys. Rev. Lett. {\bf 73} 593(1994)} in YBa$_2$Cu$_3$O$_{7-\delta}$(YBCO)-Nb rings with ramp-type junctions,\footnote{H.J.H. Smilde {\it et al.} Appl. Phys. Lett. {\bf 80}, 4579(2002)} in a test for a time reversal symmetry breaking order parameter.\footnote{T.K. Ng and C. Varma, cond-mat/0403379} The angle of one junction normal relative to the YBCO $a$-axis was held fixed, while the other was varied in 5 degree intervals from ring to ring. The epitaxial YBCO thin film rings, with inside and outside diameters of 30$\mu$m and 130$\mu$m, were cooled in various fields and imaged at 4.2K with a SQUID microscope. The spontaneously generated flux $\Phi$ in the rings, when cooled in zero field, alternated systematically with the second junction angle from nearly zero flux ($N=0$) to nearly $\Phi=\Phi_0/2=h/4e$ ($N=1/2$) in a manner consistent with a predominantly $d_{x^2-y^2}$ Cooper pairing symmetry. The transition between the $N=0$ and $N=1/2$ states occured at angles slightly different from multiples of 45$^o$, consistent with a small $a-b$ plane anisotropy in the gap. Deviations from $\Phi_0/2$ flux were small, indicating little, if any, imaginary component to the order parameter, independent of Cooper pair momentum.