Optimized positions of Josephson junctions to prevent trapping of magnetic fluxes in cooling under the static environment

We show how to design the position of a Josephson junction (JJ) and holes to prevent trapping of magnetic fluxes on the JJ in the superconducting device when the strength of magnetic field is determined in cooling from normal to superconducting state. The number of magnetic fluxes (NMF) is determined as N=α(B-B0), where B is the external field in cooling and B0 is the threshold of external field that one magnetic flux is trapped, α is the coefficients that might depend on the device size and structure. If the number of holes (NH) is larger than NMF, the magnetic fluxes are trapped into the holes. The holes should be arranged regularly so that all the magnetic fluxes are trapped into the holes. If NMF is larger than NH, the magnetic fluxes occupy the holes initially, and then, the excess magnetic fluxes are trapped around the position of the potential wells arising from the interaction among the previously trapped fluxes. The position of JJ must be designed to avoid the position of a potential well. The cooling speed from normal to superconducting state should be as slow as possible to prevent non-uniformity of temperature distribution that would cause formation of unexpected potential wells outside the designed positions