Theoretical model for parallel SQUID arrays with fluxoid focusing

We have developed a comprehensive theoretical model for predicting the magnetic field response of a parallel superconducting quantum interference device (SQUID) array in the voltage state. The model predictions are compared with our experimental data from a parallel SQUID array made of a yttrium barium copper oxide thin film patterned into wide tracks, busbars, and leads, with 11 step-edge Josephson junctions. Our theoretical model uses the Josephson equations for resistively shunted junctions as well as the second Ginzburg-Landau equation to derive a system of coupled first-order nonlinear differential equations to describe the time evolution of the Josephson junction phase differences which includes Johnson noise. Employing the second London equation and Biot-Savart's law, the supercurrent density distribution is calculated, using the stream function approach, which leads to a two-dimensional second-order linear Fredholm integro-differential equation for the stream function with time-dependent boundary conditions. The model calculates the stream function everywhere in the thin-film structure to determine during the time evolution the fluxoids for each SQUID array hole. Our numerical model calculations are compared with our experimental data and predict the bias-current-versus-voltage and the voltage-versus-magnetic-field response with accuracy. The model elucidates the importance of fully taking Meissner shielding and current crowding into account in order to properly describe fluxoid focusing and bias-current injection. Furthermore, our model illustrates the failure of the simple lumped-element approach to describe a parallel SQUID array with a wide thin-film structure.