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This order analysis of the correction to the WKBJ approximation is based on the approximation that the Fermi surface has a circular symmetry in the (Formula presented) plane. If one goes beyond this approximation, and allows the Fermi surface to only have a square or lower symmetry (for high (Formula presented) materials with a tetragonal or lower symmetry, respectively), then the correction can be nominally of the order of (Formula presented), but is still small, if the deviation from a circular Fermi surface in the (Formula presented) plane is small. [cf., M. Matsumoto and H. Shiba J. Phys. Soc. Jpn. 64, 1703 (1995).] To understand this point, we note that the WKBJ approximation is valid only if the Hamiltonian does not have a discontinuity of the order of (Formula presented) (but it can have a discontinuity of the order of (Formula presented). For surface states on a non-(Formula presented) surface, when the Fermi surface is not circular, such a large discontinuity becomes obvious, if one views the surface state with the generalized image method discussed in Ref. 30. In fact, shift of the surface state energy away from midgap by an amount of the order of (Formula presented) can even occur on the (Formula presented) surface, if the Fermi surface has lower than square symmetry in the (Formula presented) plane.
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Actual interfaces most likely have (Formula presented), but I can assume (Formula presented) here, for the only goal of this calculation is to prove in principle that MIS’s can give rise to ZBCP’s in quasiparticle tunneling along any axis. I have shown already that the area density of MIS’s, with energies showing no shift away from midgap in the WKBJ approximation, is finite for all (Formula presented), albeit lower than that for (Formula presented) by a finite fraction. Thus their difference is important only for a quantitative account of the heights of the observed ZBCP’s, which is very difficult anyway, since such heights would depend on the total interface area in the sample, as well as the precise orientations of all the grains. Y. Tanaka and S. Kashiwaya, in a theory of the Josephson effect in (Formula presented)-wave superconductors [ Phys. Rev. B 53, R11 957 (1996)] have calculated an (Formula presented) function at imaginary frequencies for arbitrary (Formula presented). This (Formula presented) function, if analytically continued to real frequency, would contain all information about the MIS’s. Whereas they did realize the importance of the signs of the products (Formula presented), (Formula presented), and (Formula presented) in giving rise to MIS’s, they obtained the precise combinations of these sign conditions for the existence of the MIS’s only for (Formula presented), corresponding to (Formula presented) in our notation, which is not satisfied in most grain boundaries. If all three products have to be negative for the existence of the MIS’s for general values of (Formula presented) and (Formula presented), most grain boundaries would not have MIS’s. So it is not apparent from their work how easily MIS’s can be obtained, which is crucial for the present argument.
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