We present exact numerical calculations of supercurrent density, inductance, and impurity-induced flux noise of cylindrical superconducting wires in the non-local Pippard regime, whichoccurs when the Pippard coherence length is larger than the London penetration depth. In this regime the supercurrent density displays a peak away from the surface of the superconductor, signalling a breakdown of the usual approximation of local London electrodynamics with a renormalized penetration depth. Our calculations show that the internal inductance and the bulk flux noise power increases with increasing non-locality. In contrast, the kinetic inductance is reduced and the surface flux noise remains the same. As a result, impurity spins in the bulk may dominate the flux noise in superconducting qubits in the Pippard regime, such as the ones using aluminum superconductors with large electron mean free path.
Superconducting Quantum Interference Devices (SQUIDs) and other superconducting circuits are limited by intrinsic flux noise whose origin is believed to be due to spin impurities. Wepresent a flux vector model for the interaction of spins with thin-film superconducting wires, and show how measurements of flux as a function of the direction of an external magnetic field applied in the plane of the wires can reveal the value of impurity spin quantum number and the nature of its interaction with the circuit. We describe a method to accurately calculate the flux produced by spin impurities in realistic superconducting thin-film wires, and show that the flux produced by each spin is much larger than anticipated by former calculations. Remarkably, flux noise power due to electron spins at the thin edge surface of the wires is found to be of similar magnitude as the one at the wide top surface. In addition, flux noise due to lattice nuclear spins in the bulk of the wires is found to be approximately 5% of the total noise power. We discuss the relative importance of electron and nuclear spin species in determining flux noise, and propose strategies to design low noise SQUIDs.