Role of Interlayer Electron Hopping for Spin Density Wave State in the Zero-Gap Organic Conductor

We investigate the formation of density waves in the zero-gap state (ZGS) which has been found in the quasi-two-dimensional organic conductor $\alpha$-(BEDT-TTF)${}_2$I${}_3$ salt under the hydrostatic pressure. The ZGS exhibits the cone-like dispersion for both the conduction band and the valence band which degenerate each other at the two-dimensional wave vectors, $\pm \text{mib}{k}_0$ forming a zero gap. By using the extended Hubbard model with repulsive interaction we calculate the onset temperature of the spin density wave (SDW) as a function of the interlayer electron hopping, which by itself does not break the cone-like dispersion. It is shown that the SDW with wave number $2\text{mib}{k}_0$ is induced by the combined effect of the interaction, the inter-band excitation across the zero-gap and the interlayer hopping.