The magnetic-field-dependence critical current I-c(H-a) is measured for the Mo/Au bilayer transition-edge sensors. A model based on a surface barrier delaying the first penetration of magnetic flux into a flat superconducting strip, including both the geometrical and Bean-Livingston barrier, in combination with two Josephson junction models with different characteristic lengths, is found to ubiquitously fit the I-c(H-a) curves of devices with various geometry designs. The bulk penetration depth, London depairing current density, Josephson junction critical current, self-field coupling coefficient, and conditions of the first vortex entry are investigated throughout the superconducting transition. The evidences of Josephson effect are observed in both Fraunhofer interference pattern and Shapiro steps.
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