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# Superconductivity and Magnetism in BiOCuS 

I.I. Mazin Code 6393, Naval Research Laboratory, Washington, D.C. 20375 

(Dated: Printed on November 1, 2018) 

BiOCu 1$_{−}$ y S was recently reported to superconduct at T ≈ 5 . 8 K at y ≈ 0 . 15 . Band structure calculations indicate that the stoichiometric BiOCuS is a band insulator. In this paper, I show that the hole-doped (whether in the virtual crystal approximation or with actual Cu vacancies) BiOCuS is on the verge of a ferromagnetic instability ( cf. Pd metal), and therefore a conventional superconductivity with T$_{c}$ ∼ 6 K is quite unlikely. Presumably, the hole-doped BiOCuS is another example of superconductivity mediated by spin-fluctuations. 

Recently, Ubaldini et al 1 reported that in the quater- nary compound BiOCu$_{1}$$_{−}$$_{y}$S superconductivity at T$_{c}$ = 6 K can be induced by a small Cu deficiency ( y ∼ 15%) , although not by an electron doping (when substituting O with F). Although another similar study has not con- firmed the superconductivity $^{2}$, this result has triggered some interest in the community $^{2,3}$. I will show below that this interest is well justified, for this material is very un- likely to be a conventional s-wave superconductor, and, if superconducting, is probably another example of a spin- fluctuation mediated superconductivity. 

BiOCuS crystallizes in the well known tetragonal Zr- CuSiAs structure $^{2,4}$, recently made famous by some Fe- based high-temperature superconductors (although there is no commonality whatsoever between the latter and the material in question). The lattice parameters reported in Ref. 1 are 3.8726 and 8.5878 ˚ A, in Ref. 4 3.8691 and 8.5602 ˚ A, and in Ref. 2 3.868 and 8.557 ˚ A. The inter- nal coordinates of Bi and S, respectively, are reported as 0.14829 and 0.6710 (Ref. 4) and 0.151 and 0.648 (Ref. 2). 

In my calculations I used the structure reported in Ref. 4. The variation of crystallographic parameters withing these limits does not affect any conclusions of this paper. 

In agreement with the previous first principles calculations 3,4 I found 5 that in the scalar-relativistic ap- proximation the stoichiometric compound is a band in- sulator, with a direct gap underestimated compared to the measured optical gap $^{4}$, as common in the density functional calculations. Adding spin-orbit interaction on Bi reduces the gap even further, from 0.75 to 0.55 eV, by affecting the unoccupied states. The states right be- low the gap are formed predominantly by the Cu xz and yz orbitals, while the empty states above the gap have quasi-free-electron character (the interstitial region, be- ing ≈ 60% in volume, contributes up to 50% of the density of states of these bands). In agreement with, especially, the full potential all-electron calculations of Ref. 3, I found a flat band right at the top of the valence band at the Z point (Fig. 1), essentially not dispersing up to half-way between Z and X $^{6}$. This band results in a small but very sharp density of states (DOS) peak right below the band gap (Fig. 2), with the total weight of slightly less than 0.3 electron per copper, and a maximum DOS 

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of 4.1 states/eV. f.u. at 130 meV below the gap (corre- sponding to 0.10–0.12 hole/Cu). One can see that this peak indeed originates from the flat band along Z − A , by looking at the Fermi surface for E$_{F}$ set at 120 meV below the band gap, where the peak just starts (approx- imately 0.07 hole/Cu doping). This feature appears as cross-shaped Fermi surface pockets emerging from Z (Fig. 3). 

Given that Cu d orbitals have sizeable Hund rule cou- pling this result suggests that Cu ions under 10-20% hole doping should be magnetic. Indeed, ferromagnetic (anti- ferromagnetic calculation, however, were invariably con- verging to zero magnetization $^{7}$) calculations for y = 0 . 2 in the virtual crystal approximation (0.2 hole per Cu) converge to a stable solution with the magnetic moment M ≈ 0 . 12 − 0 . 13 µ$_{B}$ per Cu, but with hardly any energy gain. Correspondingly, fixed spin moment calculations (Fig. 4) show that in the GGA approximation the energy, within the computational accuracy, does not depend on magnetization up to M = y µ$_{B}$ . Note that this magneti- zation would correspond to a half-metallic state similar to that in the Co-doped FeS$_{2}$ 8 (in fact the physics of the formation of a magnetic state is quite similar here). This means that the calculated spin susceptibility is essentially infinite, a situation extremely close to Pd metal, where 

