6 sudden disappearance of the T$_{c}$ and a superconductor- insulator transition is observed. 35 Recently, this increase of T$_{c}$ has been shown to be correlated with the increase of the uniform spin susceptibility, which can be considered as a support for the present spin fluctuation scenario. 15 In fact, although the spin fluctuation is nearly tempera- ture independent in our model (so that the development of the spin fluctuation upon lowering the temperature need not be observed experimentally), it is strongly dop- ing dependent and is enhanced upon lowering the doping rate. 15 Nonetheless, the origin of the sudden drop of T$_{c}$ at a certain doping rate can not be explained within the present two band model (see Fig.4) and remains as an open problem. Also, the almost constant behavior of T$_{c}$ in the heavily doped regime 18 is puzzling since for such a large doping, the Fermi level hits the second band and a large difference in the DOS should be detected (which is not the case experimentally). A rigid band picture may not be valid in the heavily doped regime, as suggested in some experiments 36 and band calculations. 37 The author acknowledges Y. Taguchi, Y. Iwasa, Y. Kasahara, H. Tou, R. Kadono, M. Hiraishi, S. Yamanaka, T. Ekino, A. Sugimoto, H. Fukuyama, T. Oguchi, M. Onoue, and H. Aoki for fruitful discussions. Numerical calculation has been done at the computer center, ISSP, University of Tokyo. This study has been supported by Grants-in Aid from MEXT of Japan and JSPS. ity in a doped band insulator on the HC lattice. We have also found that the gap anisotropy on the Fermi surface strongly increases upon increasing the doping concentra- tion, and the inter layer hopping suppresses superconduc- tivity. These results are in qualitative agreement with the experimental findings. One remaining issue is the curious doping dependence of the superconductivity. For Li$_{x}$ZrNCl, T$_{c}$ shows an increase upon lowering the carrier concentration until a References FIG. 8: T c plotted as a function of t z for the model shown in the inset. t = 1 . 2eV, t $^{′}$/t = 0 . 35, Δ /t = 2 . 7, U/t = 6, n = 1 . 08.  10  15  20  25  30 0  0.04  0.08  0.12 t /t T [K] c z t zGLYPH<13> t zGLYPH<13> tGLYPH<13> t'GLYPH<13> MGLYPH<13> NGLYPH<13> 34 M. Ogata, J. Phys. Soc. Jpn. 72 , 1839 (2003). 33 B. Kumar and B.S. Shastry, Phys. Rev. B 68 , 104508 (2003). 32 G. Baskaran, Phys. Rev. Lett. 91 , 097001 (2003). 31 P. Monthoux and G.G. Lonzarich, Phys. Rev. B 63 , 054529 (2001). 1 S. Yamanaka et al. , Nature 392 , 580 (1992). 20 I. Hase and Y. Nishihara, Phys. Rev. B 60 , 1573 (1999). 21 H. Sugimoto and T. Oguchi, J. Phys. Soc. Jpn. 73 , 2771 (2004). 22 Generally, the on-site repulsion on the M site and N site can be different, but here we assume them to be equal for simplicity. 23 N.E. Bickers, D.J. Scalapino, and S.R. White, Phys. Rev. Lett. 62 , 961 (1989). 24 T. Dahm and L. Tewordt, Phys. Rev. Lett. 74 , 793 (1995). 25 S. Grabowski, M. Langer, J. Schmalian, K.H. Bennemann, Europhys. Lett. 34 , 219 (1996). 26 K. Kuroki and R. Arita, Phys. Rev. B 63 , 174507 (2001). 28 Possibility of high T c superconductivity on the HC lattice has also been proposed in the context of RVB theory of graphene by Pathak et al. , arXiv: 0809.0244. 27 K. Kuroki, in Proc. Int. Workshop on Superconductivity in Diamond and Related Materials, Sci. Technol. Adv. Mater. 9 , 044202 (2008). 29 K. Kuroki and R. Arita, Phys. Rev. B 64 , 024501 (2001). 30 R. Arita, K. Kuroki, and H. Aoki, Phys. Rev. B 60 , 14585 (1999). 19 Theoretically, a possibility of unconventional pairing me- diated by charge fluctuations has been proposed in A. Bill et al. , Phys. Rev. B 66 100501(R) (2002). 18 T. Takano, T. Kishiume, Y. Taguchi, and Y. Iwasa, Phys. Rev. Lett. 100 , 247005 (2008). 17 H. Tou, private communications. 16 M. Hiraishi et al. , arXiv: 0910.5557. 15 Y. Kasahara et al. , Phys. Rev. Lett. 103 , 077004 (2009). 14 T. Ekino et al. , Physica C 388-389 , 573 (2003); Physica B 328 , 23 (2003). 13 H. Tou, Y. Maniwa, and S. Yamanaka, Phys. Rev. B 67 , 100509 (2003). 12 Y. Taguchi et al. , Phys. Rev. B 76 , 064508 (2007). 11 H. Tou et al. , Phys. Rev. B 67 , 100509 (2003). 10 R.Heid and K.P.Bohnen, Phys. Rev. B 72 , 134527 (2005). 9 A. Kitora, Y. Taguchi, and Y. Iwasa, J. Phys. Soc. Jpn. 76 , 023706 (2007). 8 R. Weht et al. , Europhys. Lett. 48 , 320 (1999). 7 Y. Taguchi, M. Hisakabe, and Y. Iwasa, Phys. Rev. Lett. 94 , 217002 (2005). 6 H. Tou et al. , Phys. Rev. Lett. 86 , 5775 (2001). 5 T. Ito et al. , Phys. Rev. B 69 , 134522 (2004). 4 Y. J. Uemura et al. , Physica (Amsterdam) 289-290B , 389 (2000). 3 H. Tou et al. , Phys. Rev. B 63 , 020508 (2000). 2 S. Shamoto et al. , Physica (Amsterdam) 402C , 283 (2004).