Electronic and magnetic properties of superconducting LnO$_{1-x}$F$_{x}$BiS$_{2}$ (Ln = La, Ce, Pr, and Nd) from first principles

A density functional theory study of the BiS$_{2}$ superconductors containing rare-earths: \textit{Ln}O$_{1-x}$F$_{x}$BiS$_{2}$ (\textit{Ln} = La, Ce, Pr, and Nd) is presented. We find that CeO$_{0.5}$F$_{0.5}$BiS$_{2}$ has competing ferromagnetic and weak antiferromagnetic tendencies, the first one corresponding to experimental results. We show that PrO$_{0.5}$F$_{0.5}$BiS$_{2}$ has a strong tendency for magnetic order, which can be ferromagnetic or antiferromagnetic depending on subtle differences in 4$f$ orbital occupations. We demonstrate that NdO$_{0.5}$F$_{0.5}$BiS$_{2}$ has a stable magnetic ground state with weak tendency to order. Finally, we show that the change of rare earth does not affect the Fermi surface, and predict that CeOBiS$_{2}$ should display a pressure induced phase transition to a metallic, if not superconducting, phase under pressure.


INTRODUCTION
The excitement generated by the recent finding of two new superconductors, Bi 3 O 2 S 3 and LaO 1−x F x BiS 2 [1,2], led to the unveiling of a whole class of superconductors containing BiS 2 bilayers. These compounds share a common two-dimensional structure with alernating BiS 2 bilayers and spacer layers. It is the BiS planes in the BiS 2 bilayers which are thought to be responsible for superconductivity in these compounds. Soon after the discovery of LaO 1−x F x BiS 2 , chemical substitution to tune the properties was attempted. The first research axis was to replace the lanthanum atom by other lanthanides: cerium, praseodymium, neodymium and ytterbium [3][4][5][6][7]. Hole and electron doping of the parent compound has also been studied in Sr x La 1−x FBiS 2 [8] and La 1−x M x OBiS 2 (M = Ti, Zr, Hf, Th) [9] respectively.
The electronic structure of the two first compounds to be found, Bi 3 O 2 S 3 [10] and La(O,F)BiS 2 [11,12], has been calculated. It has been shown that the superconducting electrons relate to Bismuth 6p and Sulphur 3p bands in the BiS planes, the Fermi surface of LaOBiS 2 displaying an interesting nesting.
In this paper we present our findings from electronic structure calculations for the stoichiometric and doped series of materials LnO 1−x F x BiS 2 with Ln = La, Ce, Pr and Nd, and x = 0, 0.5. By studying these materials, with the exception of the Ce doped compound, we limit our study to those where the crystal structure has been experimentally determined. We have been able to provide a structure through relaxation methods for the Ce compound, since the availability of the La, Pr and Nd compounds provides a reasonable reliability on the starting point.
We first show that the change of rare earth does * Electronic address: cm712@cam.ac.uk † Electronic address: ea245@cam.ac.uk ‡ Electronic address: sss21@cam.ac.uk FIG. 1: Crystal structure of LaOBiS2, represented in Vesta [13]. All the compounds studied share this same structure, with replacement of lanthanum by other lanthanides, or of oxygen by fluorine.
not affect the Fermi surface. We then demonstrate CeO 0.5 F 0.5 BiS 2 has a ferromagnetic tendency while PrO 0.5 F 0.5 BiS 2 tends towards antiferromagnetism. It has been shown that ferromagnetism coexists with superconductivity in CeO 0.5 F 0.5 BiS 2 [14], and this gives a unique strength to our prediction of coexistence of superconductivity and antiferromagnetism in PrO 0.5 F 0.5 BiS 2 . Finally, we predict that applying pressure to CeOBiS 2 would make it metallic, and possibly superconducting.

METHODS
Band structures and densities of states were calculated using the SIESTA method [15,16], implementing the generalized gradient approximation in the shape of the Perdew, Burke, and Ernzerhof functional [17]. It uses arXiv:1312.2615v1 [cond-mat.supr-con] 9 Dec 2013 non-conserving pseudopotentials to replace the core electrons, while the valence electrons are described using atomic-like orbitals as basis states at the double zeta polarized level. Details on the basis set parameters used are available in the supplementary material.
Calculations for x=0.5 have been performed for all the compounds. All these structures share the same space group: P4/nmm ( Figure 1). The calculations for the lanthanum compound were performed without using spin polarisation. However for Ln = La, the presence of unpaired 4f electrons demands the use of spin polarisation in the calculations.
Calculations for the La compound were performed using experimental data for the geometry obtained after high pressure treatment of the sample, for which the maximal superconducting T c (10.6 K) was found [1] The case of the Ce compound is slightly more complicated: as no atomic coordinates have been reported yet, we obtained a geometry from DFT, by relaxing the structure (both atoms and cell), starting from the structure known for the La compound.
We also performed calculations on two parent phases: LaOBiS 2 and CeOBiS 2 . Their space group is also P4/nmm. We used experimental data for both [18,19]: lattice parameters a = 4.008Å c = 13. All the occupancies are equal to one. There are no experimental data available for the other two rareearths to our knowledge.
We performed GGA+U calculations [20] for all the compounds except the lanthanum ones, because of the known strong correlation of the 4f electrons in cerium, praseodymium and neodymium. Given the intrinsic difficulty in the ab-initio determination of a value of U [21], we have explored the behaviour of the system for varying U . U values have been taken between 0 and 9 eV. The influence of U on relaxations has been investigated and proved to be negligible.
All the doped phases calculated have been reported as being superconducting. All the parent phases have been reported as being non-superconducting.

RESULTS
We observe many similarities between the band structures of the different compounds, due to their proximity in terms of composition and crystal structure (Figure 2). More precisely, we find the same four bands (or eight bands with spin polarisation) close or crossing the Fermi level near the R and X points in all the compounds, and similar to the Bi-O-S compounds [10]. We also find the two (or four) bands near the middle of the A-Z segment. The density of states calculations indicate that these bands are formed with Bi and S states. It has already been shown in Bi 3 O 2 S 3 and in the La compound that the main contribution to these bands comes from Bi 6p states, along with some S 3p states. It is confirmed in these systems. These bands cross the Fermi level for all compounds with non-zero x. The position of the Fermi level with respect to the bottom of these bands is very similar in all the compounds with x=0.5.
To explore the character of the bands just below the Fermi level, we plotted the local density of states for the x=0.5 compounds for energies between -0.5 eV and 0 eV (Figure 3). The results confirm what had been found in other BiS 2 -based compound: all these electrons accumulate around the BiS planes [10].
The lanthanide bands are about an electron-volt under and a quarter of electron-volt above the Fermi level. No lanthanide band is crossing it, being repelled by the BiS bands. The evolution of the band structure with U is very similar in all the cases: the lanthanides bands are expelled further from the Fermi level as U grows, sometimes hybridising other bands. The Fermi surface is robust to lanthanide substitution and to the change of U , except for some small pockets arising close to the Gamma and Z points in the praseodymium and neodymium compounds for U very close to zero (U ≤ 0.5 eV and U ≤ 2 for the Pr and Nd compound respectively). The small value of U renders it of little likelihood to be of significance.
Bulk cerium, praseodymium and neodymium have respectively one, three and four 4f electrons. The jump between Ce and Pr is due to the fact that bulk cerium has one electron in a 5d band, whereas bulk praseodymium and neodymium do not. However when we place these elements in the compounds considered here, each atom of these species has one 5d electron, thereby restoring the sequence. As expected the 4f electrons are polarised and the 5d are not. The spin polarisation is therefore 1, 2 and 3 for the Ce, Pr and Nd compounds, respectively. The 5d electron appears unpolarised and strongly hybridised with other bands.
We plotted the energy per unit cell of the three corresponding compounds for a ferromagnetic and an antiferromagnetic ground state and their difference, depending on U ( Figure 5). The antiferromagnetic ground state considered here is the only one that we can obtain in one unit cell: as there are two lanthanide atoms in one unit cell, we did put one with spin up and one with spin down (Figure 4). In the neodymium compound, the difference oscillates around zero, taking values of the order of 10 −3 eV, with no clear tendency. In the praseodymium compound, the energy difference is an order of magnitude larger, negative, and almost linear with U . The ground state has a clear antiferromagnetic tendency for U values within a reasonable range for these systems. For the cerium compound, the difference is as large as in the Pr case but positive. It is almost linear but on two separate segments, as if two different ground states were competing. Regardless of this, however, the ground state has a clear ferromagnetic tendency.
Moreover, for the Ce compound, we calculated the difference in energy between when two consecutive layers have opposite spin, and when their spins are identical, as portrayed in Figure 4. The first case has been simulated FIG. 4: Spin arrangements in LnO0.5F0.5BiS2 considered in this work, plotted with Vesta [13]. On the left is the antiferromagnetic phase considered, and on the right is the arrangement used to test the coupling between layers.
through the doubling of the unit cell in the c direction: in this double unit cell we did put one lanthanide layer with spin up, and the other with spin down. We obtained a raise in energy of 0.067 eV per unit cell, showing that the layers are coupled ferromagnetically.
The magnetism arises from the 4f electrons of the lanthanide atoms, these being the only unpaired electrons. They are in bands quite far from the Fermi level, from -1 eV for U = 0 to much lower when U is higher.
Finally, we consider the effect of pressure on CeOBiS 2 . The band structure of CeOBiS 2 has been calculated with respect to P and U ( Figure 6). Two situations appear to be possible at zero pressure, depending on U : either the BiS bands are crossing partially the Fermi level if U is smaller than 2.5, or not crossing it at all, if U is larger than 2.5. But more interestingly, if we look at the evolution of these bands with pressure, we can see that the bands cross the Fermi level for U larger and larger. It reaches a point where, at 20 GPa, the bands are crossing fully at U = 3.5 and partially at U = 7.5. The bands crossing the Fermi level are not only the BiS bands: other bands are crossing. These are cerium bands (for low U ), or hybridised bands, depending on U : when U gets higher, the cerium bands get too far from the Fermi level to play a part in this.

DISCUSSION
The first results of this study show a number of features that can be generalized in the BiS 2 -based family Difference of total energy per unit cell of LnO0.5F0.5BiS2 (Ln=Ce, Pr, Nd) between a ferromagnetic ground state and an antiferromagnetic ground state, for U varying between 0 and 9. The top plot is for the cerium compound, the one in the middle for neodymium, and the bottom one for praseodymium of superconductors. First, the Bismuth 6p and Sulphur 3p bands are crossing or close to the Fermi level. This crossing seems to be correlated with the appearance of superconductivity. Secondly, the electrons crossing the Fermi level are localised in the BiS planes in all the compounds. This raises the question of the existence of superconductors that have BiS planes without BiS 2 layers. The slight puckering of the BiS planes is apparent in the figure, inducing a clear distortion (along z) of the LDOS, noticeable in the lowest plane depicted, in which only S atoms are captured. It involves some p z participation.
The study of the x = 0.5 phases of the four compounds offers insight into the effect of the doping through rare earths are, and the reasons for changes in T c . The main effect expected from this doping is the increase in the density of states at the Fermi level, that would cause T c to rise according to the BCS equations. But this is not happening at all: for U non zero, there is no difference in the density of states at the Fermi energy between compounds, because the lanthanides' bands are repelled by the BiS bands. The reason why the change of lanthanide has almost no effect whereas the fluorine doping is quite effective is because the extra electrons remain in the bands of the dopant and away from the Fermi level. Also, the Fermi surface is almost identical in the four cases, and does not seem to be responsible for the substantial change observed in T c , nor does the band structure as a whole. Given that it only arises at very low U , the pocket at the Gamma point is very unlikely to have any importance. Indeed, such a small U is very unlikely to be found in any compound with lanthanides.
The evolution of the difference between the energy of the antiferromagnetic and ferromagnetic ground states reflects a rich phenomenology. First, for the Nd compound, the difference is small and variable with U , indicative that there is no clear magnetic tendency, there- fore that the ground state has a very low magnetic critical temperature. Second, for the Pr compound, the difference is larger than the U dependance, showing a clear antiferromagnetic tendency. Similarly for the Ce compound, but here with a clear ferromagnetic tendency. To get a quantitative estimate, we applied a simple 2D Ising model to each Ln(O,F) layer. Each lanthanide atom has 8 lanthanide neighbours. If we take ∆E = 0.03 eV, Onsager's solution for the 2D Ising model gives us T c = 4.32 * 10 − 3 eV = 50 K. Therefore, for a reasonable U , all the spins in one Ln(O,F) layer are aligned approximately under 50 K. This being largely above the superconductivity temperature, it suggests possible coexistence of magnetism with superconductivity. Moreover, the layers are also coupled in the Ce compound. The coexistence of ferromagnetism and superconductivity has been confirmed experimentally in the Ce compound during the making of this study [14]. Given the similarity of energy differences between the Ce and Pr cases, this makes us think the coexistence of antiferro-magnetism and superconductivity in the Pr compound is also likely.
The phase diagram P − U we obtained for CeOBiS 2 is very interesting. It exhibits two main possibilities: either U is larger than 2.5 eV, which is consistent with the absence of superconductivity, or U is smaller than 2.5 eV, which is consistent with its bad metal behaviour observed experimentally. In the latter case, it means that we are likely to observe superconductivity at a low temperature. This is not contradictory with experiment as the magnetization measurements have only been conducted down to 2K. Moreover, the transition temperature measured for the corresponding doped phase is lower than 3K. The other case is even more exciting. If U is larger than 2.5 eV, there should be a insulator-metal transition under pressure, which also could be superconducting at low temperature, if we admit the BiS bands are responsible for superconductivity. The pressure obtained has rather large uncertainties, but we can still conclude that it seems likely the transition temperature could be reachable by commonly known techniques. Of course, this pressure also depends heavily on the U parameter of the compound. The type of doping involved is different from the fluorine type. Indeed, here pressure brings the valence and the conduction bands together until they cross. It would therefore be very interesting to see how this influences superconductivity, in order to favour one type of doping or the other in the future.