Theory of a mesoscopic superconducting phase gradiometer

We analyze an all-superconducting mesoscopic DC SQUID, comprising a pair of thin-film strips connected by a pair of parallel ultra-narrow wires. We focus on the magnetoresistance of the device in the size range in which the strips are narrower than the penetration depth (i.e.~the mesoscopic regime), and especially on the low-magnetic-field regime, in which no vortices are present in the strips. The resistance originates in dissipative order-parameter fluctuations in the wires, and its magnetic-field dependence comes from the phase-gradient of the order parameter in the strips, associated with screening currents. We present a theory of the magnetoresistance of this mesoscopic DC SQUID, based on the theory of intrinsic resistance (e.g.~of Josephson junctions or superconducting nanowires). We compare this theory with recent experimental data obtained from DNA-templated devices, and discuss why the device can be regarded as a superconducting phase gradiometer.