We study theoretically the transport through a finite-size sawtooth lattice coupled to two fermionic reservoirs kept in the superfluid state. We focus on the DC Josephson effect and find that the flat-band states of the sawtooth lattice can support larger critical current and a higher temperature than the dispersive band states. However, for this to occur the boundary states of the finite-size lattice need to be tuned at resonance with the bulk flat-band states by means of additional boundary potentials. We show that transport in a two-terminal configuration can reveal the salient features of the geometric contribution of flat-band superconductivity, namely the linear dependence of key quantities, such as the critical current and critical temperature, on the interaction. Our results are based on parameters of a realistic experimental lattice potential, and we discuss the conditions one needs to reach to observe the predicted effects experimentally.
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