Possibility of spin fluctuation mediated d + id ′ pairing in a doped band insulator β -MNCl (M=Hf,Zr) superconductors arXiv:1001.3167v1 [cond-mat.supr-con] 18 Jan 2010 Kazuhiko Kuroki Department of Applied Physics and Chemistry, The University of Electro -Communications, Chofu, Tokyo 182-8585, Japan We study two types of models for the superconducting layered nitride β -MNCl(M=Hf,Zr); a single band model on a triangular lattice, and a two band model on a honeycomb lattice. We find that the former model does not suffice as an effecitive model, while the latter one can be a good candidate. We propose from the study on the two band model a possibility of spin fluctuation mediated d + id $^{′}$- wave superconductivity in the doped β -MNCl. We show that the relatively high T c obtained in the doped band insulator is a characteristic feature of the spin fluctuation mediated superconductivity on a honeycomb lattice. We also find that the gap anisotropy on the Fermi surface strongly increases upon increasing the doping concentration, and the inter layer hopping suppresses superconductivity. These results are in qualitative agreement with the experimental findings. Layered nitride β - M NCl 1 ( M =Hf, Zr) doped with carriers is one of the most interesting group of supercon- ductors. The mother compound β - M NCl is composed of alternate stacking of honeycomb (HC) M N bilayer and Cl bilayer. 2 This is a band insulator and becomes a su- perconductor upon doping electrons by Na or Li interca- lation. They exhibit relatively high T$_{c}$ up to ∼ 25K for M =Hf and ∼ 15K for M =Zr. The bilayer HC lattice structure consisting of M and N is considered to be play- ing the main role in the occurrence of superconductivity, and the two dimensional nature of the superconductivity has been revealed by nuclear magnetic resonance (NMR) 3 and muon spin relaxation ( µ SR) studies. 4,5 Despite the relatively high T$_{c}$ , experimental as well as theoretical studies indicate extremely low density of states (DOS) at the Fermi level. 6–8 In fact, they have the highest T$_{c}$ among materials with the specific heat coeffi- cient γ as small as ∼ 1mJ/molK $^{2}$. The electron phonon coupling is also estimated to be weak, 6,8–10 and the iso- tope effect is found to be small. 11,12 These experiments suggest that some kind of unconventional pairing mech- anism may be at work, but on the other hand, NMR knight shift measurement suggests spin singlet pairing, 13 and the tunneling spectroscopy 14 and specific heat 7 ex- periments find a fully open, seemingly s -wave like gap. However, regarding this fully open gap, recent experi- ments show that the anisotropy of the gap increases with doping. Namely, when the doping concentration is small, the specific heat coefficient increases linearly as a func- tion of the magnetic field, suggesting an isotropic gap, 15 while a steep increase of the coefficient at low magnetic field is observed for higher doping. 7,15 Also, 2Δ / ( k$_{B}$T$_{c}$ ) determined from the specific heat measurement ranges from ∼ 5 in the lightly doped regime to less than ∼ 3 in the heavily doped regime, 15 which may be an indica- tion that the maximum and minimum values of the gap have a substantial difference when heavily doped, and the specific heat is mainly governed by the minimum value. Moreover, a recent superfluid density measurement by µ SR shows that the gap is nearly isotropic when the dop- ing concentration is small, while the anisotropy increases for large doping. 16 Very recently, absence of coherence peak in the spin lattice relaxation rate has been found in an NMR experiment, 17 again suggesting unconventional pairing, most probably with some kind of sign change in the superconducting gap. Furthermore, for Li$_{x}$HfNCl, an intercalation of organic molecules tetrahydrofuran (THF) between the layers is found to enhance T$_{c}$ . 18 Given these experimental circumstances, here we study two types of models for the doped β -MNCl, a two band model on a HC lattice, and a single band model where the nitrogen site is effectively integrated out. While the latter model does not seem to suffice for explaining the relatively high T$_{c}$ in this series of material, the former one provides a possibility of spin fluctuation mediated d -wave superconductivity in the doped β -MNCl. 19 Spin fluctuation mediated pairing in a doped band insulator may sound surprising and odd at first sight since one cannot expect strong spin fluctuations. Nevertheless, we show that the relatively high T$_{c}$ despite the low DOS and the temperature independent spin susceptibility is a characteristic feature of the spin fluctuation mediated superconductivity of a doped band insulator on the HC lattice. The most probable pairing state below T$_{c}$ is the d + id ′ state, whose gap anisotropy on the Fermi surface is found to be strongly enhanced upon increasing the car- rier concentration. We also study the effect of the three dimensionality, where the inter layer hopping suppresses superconductivity. We first obtain the effective model for β -MNCl. Al- though the material consists of bilayer HC lattice, here we consider models on a single layer lattice, and try to reproduce the bands that lie close to the Fermi level in the first principles band calculations. The simplest one is a single band model on a triangular lattice, in which the nitrogen site degrees of freedom is effectively integrated out, so that there is only one site per unit cell. This is a model that corresponds to the single band Hubbard II. FORMULATION I. INTRODUCTION