5 tuations. Of course, if the level offset Δ is large enough, the two band case should tend to the single band case, as have been shown in ref. $^{26}$, but in the present case, the magnitude of Δ is small enough to retain the nature of the Δ = 0 situation. Apart from the T$_{c}$ , another important issue is the form of the superconducting gap. By symmetry, there are two degenerate d -wave gaps, and the most probable form of the gap below T$_{c}$ is the form d + id $^{′}$, where the two d - wave gaps mix with a phase shift of π/ 2. 32–34 Since the two d -wave gaps have nodal lines at different positions, this kind of mixture leads to a gap that has a finite abso- lute value on the entire Fermi surface. The d + id ′ form of the gap can be constructed from the d -wave gap ob- tained by solving the linearized Eliashberg equation. In the upper panels of Fig.7, we show the contour plot of the d + id ′ gap for the case of n = 1 . 06 and n = 1 . 16 for the model for β -MNCl. To clearly see the anisotropy on the Fermi surface, we plot the magnitude of the gap (normal- ized by the maximum value) in the lower panel of Fig.7. It can be seen that the d + id ′ gap is nearly isotropic for low doping concentration, while it becomes more and more anisotropic for higher doping. The d + id ′ pairing scenario is consistent with the experimental findings that the superconductivity occurs in the spin singlet $^{13}$, s -wave like channel, 7,14,15 while the coherence peak in the spin lattice relaxation rate is absent. 17 It is also consistent with the experimental finding that the gap anisotropy increases upon doping. 15,16 Since the d + id ′ state breaks time reversal symmetry, it is interesting to experimen- tally investigate such a possibility. Finally, we consider the effect of dimensionality by adopting a three dimensional model where the two band model on the HC lattice is connected by vertical (M-M and N-N) hopping integrals t$_{z}$ (inset of Fig.8). Here, we take up to 32 × 32 × 32 k -point meshes and up to 8192 Matsubara frequencies. In Fig.8, we plot T$_{c}$ as a function of t$_{z}$ for the band filling of n = 1 . 08. As can be seen, the three dimensionality suppresses T$_{c}$ , which is in agreement at least qualitatively with the experimental finding that the intercalation of THF molecules between the layers enhances T$_{c}$ . 18 In the present study, we have studied two kinds of model for the the doped β -MNCl. We have found that the single band model does not suffice as a minimal ef- fective model of the material. On the other hand, from the study on the two band model, we have proposed a possibility of spin fluctuation mediated d + id $^{′}$-wave su- perconductivity in the doped β -MNCl. We have shown that the relatively high T$_{c}$ is obtained as a characteristic feature of the spin fluctuation mediated superconductiv- In fact, the suppression of T$_{c}$ with increasing three di- mensionality is a general trend for spin fluctuation me- diated pairing as has been studied in ref. $^{30,31}$. Namely, since the pairing interaction is large around a certain wave vector in spin fluctuation mediated pairing, the fraction of the volume where the pairing interaction is large in the Brillouin zone becomes smaller as the three dimensionality increases. V. CONCLUSION AND FUTURE PROBLEMS C. Effect of dimensionality B. Gap function FIG. 7: (color online) Upper two panels : the contour plots of the d + id ′ gap for n = 1 . 06 (top) and n = 1 . 16 (middle). The model for β -MNCl is used. The yellow (light) solid line (closed around the Brillouin zone edge) represents the Fermi surface. Lower panel : the normalized d + id ′ gap along the Fermi surface. θ is defined in the top panel.