A theoretical model for parallel SQUID arrays with fluxoid focussing

We have developed a comprehensive theoretical model for predicting the magnetic field response of a parallel SQUID array in the voltage state. The model predictions are compared with our experimental data from a parallel SQUID array made of a YBCO thin-film patterned into wide tracks, busbars and leads, with eleven 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 2D second-order linear Fredholm integro-differential equation for the stream function with time-dependent boundary conditions. The novelty of the model is that it 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 unprecedented accuracy. The model elucidates the importance to fully take Meissner shielding and current crowding into account in order to properly describe fluxoid focussing 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.