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Minimum model and its theoretical analysis for superconducting materials with BiS$_2$ layers

We perform first principles band calculation of the newly discovered superconductor LaO$_{1-x}$F$_x$BiS$_2$, and study the lattice structure and the fluorine doping dependence of the gap between the valence and conduction bands. We find that the distance between La and S as well as the fluorine doping significantly affects the band gap. On the other hand, the four orbital model of the BiS$_2$ layer shows that the lattice structure does not affect this portion of the band. Still, the band gap can affect the carrier concentration in the case of light electron doping, which in turn should affect the transport properties.