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Nano-scale superconducting quantum interference devices (nano-SQUIDS) wherethe weak-links are made from nano-bridges --- i.e., nano-bridge--SQUIDs (NBSs)--- are one of the most sensitive magnetometers for nano-scale magnetometry.Because of very strong non-linearity in the nano-bridge--electrode joints, theapplied magnetic flux ($\Phi_{a}$) -- critical current ($I_{c}$)characteristics of NBSs differ very significantly from conventionaltunnel-junction-SQUIDs, especially when nano-bridges are long and/or thescreening parameter is large. However, in most of the theoretical descriptions,NBSs have been treated like conventional tunnel-junction-SQUIDs, which arebased on d.c. Josephson effect. Here, I present a model demonstrating that forlong nano-bridges and/or large screening parameter the $I_{c}(\Phi_{a})$ of aNBS can be explained by merely considering the fluxoid quantization in the NBSloop and the energy of the NBS; it is not necessary to take the Josephsoneffect into consideration. I also demonstrate that using the model, we canderive useful expressions like modulation depth and transfer function. I alsodiscuss the role of kinetic inductance fraction ($\kappa$) in determining$I_{c}(\Phi_{a})$.

Nano-bridge Superconducting Quantum Interference Devices: beyond the Josephson limit