Josephson (001) tilt grain boundary junctions of high-temperature superconductors

Abstract

We calculate the Josephson critical current I_c across in-plane (001) tilt grain boundary junctions of high-temperature superconductors. We solve for the electronic states corresponding to the electron-doped cuprates, two slightly different hole-doped cuprates, and an extremely underdoped hole-doped cuprate in each half-space, and weakly connect the two half-spaces by either specular or random Josephson tunnelling. We treat symmetric, straight, and fully asymmetric junctions with s-, extended-s, or d _{x ²−y ²} -wave order parameters. For symmetric junctions with random grain boundary tunnelling, our results are generally in agreement with the Sigrist–Rice form for ideal junctions that has been used to interpret ‘phase-sensitive’ experiments consisting of such in-plane grain boundary junctions. For specular grain boundary tunnelling across symmetric junctions, our results depend upon the Fermi surface topology, but are usually rather consistent with the random facet model of Tsuei et al. Our results for asymmetric junctions of electron-doped cuprates are in agreement with the Sigrist–Rice form. However, our results for asymmetric junctions of hole-doped cuprates show that the details of the Fermi surface topology and of the tunnelling processes are both very important, so that the ‘phase-sensitive’ experiments based upon in-plane Josephson junctions are less definitive than has generally been thought.

Acknowledgments

The authors would like to thank T. Claeson, S. E. Shafranjuk, and A. Yurgens for useful discussions.

Notes

†In their equation (15), Shirai et al. 33 defined their quasiparticle dispersion to be ε _q = −(t + ζ ₁)η _q −ζ ₂ γ _q −(μ +8Wn), with η _q = 2(cosq_x + cosq_y ), their equation (17), in units of the lattice constant a = 1. Following their equation (21), the definitions t = W, ζ ₁=−0.19t, ζ ₂=0.0t, μ =−0.2833t, and the hole density were given, where the summation is only over the two spin states. Putting these numbers into the above dispersion, one obtains ε _q = −1.606t(cosq_x +cosq_y +4.06), which never vanishes, and hence does not exhibit a Fermi surface. However, the q -independent part of this expression is inconsistent with their equation (4) for the effective chemical potential, , where the summation is over the four near-neighbour in-plane sites denoted by ρ and the two spin states. Hence, the 8 in their equation (15) should be replaced with the number 4 to avoid overcounting the two spin states. With this modification, the correct dispersion of Shirai et al. becomes ε _q } = −1.606t(cos q_x +cos q_y +1.94), which has the Fermi surface at small pockets in the corners of the first Brillouin zone shown in figure 2. Although Shirai et al. may have intended to include this factor of 4 in their definition of n, which would have led to a Fermi surface resembling FS2 in figure 2, from the equations in their manuscript, they very likely used the above dispersion corresponding in our notation to J _| = 201 meV, ν = 0, and μ = −1.94, where J _| can be determined from their value of Δ =0.0799t, which we set equal to 10 meV. Hence, their strange results for the symmetric (mirror) and especially the straight (parallel) 18.4○ junctions do not appear to correspond to any results by other authors for any FS studied.