The Delicate Electronic and Magnetic Structure of the LaOFePn System (Pn = pnictogen)

The occurrence of high temperature superconductivity, and the competition with magnetism, in stoichiometric and doped LaOFeAs and isostructural iron-oxypnictides is raising many fundamental questions about the electronic structure and magnetic interactions in this class of materials. There are now sufficient experimental data that it may be possible to identify the important issues whose resolution will lead to the understanding of this system. In this paper we address a number of the important issues. One important characteristic is the Fe-As distance (or more abstractly the pnictogen (Pn) height $z$(Pn)); we present results for the effect of $z$(Pn) on the electronic structure, energetics, and Fe magnetic moment. We also study LaOFeAs under pressure, and investigate the effects of both electron and hole doping within the virtual crystal approximation. The electric field gradients for all atoms in the LaOFeAs compound are presented (undoped and doped) and compared with available data. The observed $(\pi,\pi,\pi)$ magnetic order is studied and compared with the computationally simpler $(\pi,\pi,0)$ order which is probably a very good model in most respects. We investigate the crucial role of the pnictogen atom in this class, and predict the structures and properties of the N and Sb counterparts that have not yet been reported experimentally. At a certain volume a gap opens at the Fermi level in LaOFeN, separating bonding from antibonding bands and suggesting directions for a better simple understanding of the seemingly intricate electronic structure of this system. Finally, we address briefly on the possible effects of post-lanthanum rare earths, which have been observed to enhance the superconducting critical temperature substantially.

field gradients for all atoms in the LaOFeAs compound are presented (undoped and doped) and compared with available data. The observed (π, π, π) magnetic order is studied and compared with the computationally simpler (π, π, 0) order which is probably a very good model in most respects.
We investigate the crucial role of the pnictogen atom in this class, and predict the structures and properties of the N and Sb counterparts that have not yet been reported experimentally. At a certain volume a gap opens at the Fermi level in LaOFeN, separating bonding from antibonding bands and suggesting directions for a better simple understanding of the seemingly intricate electronic structure of this system. Finally, we address briefly on the possible effects of post-lanthanum rare earths, which have been observed to enhance the superconducting critical temperature substantially. PACS numbers:

I. BACKGROUND AND MOTIVATION
Isostructural and isovalent LaOFeP and LaOFeAs are layered conductors, the first being superconducting at T c =2.5 K 1 while the second becomes antiferromagnetically ordered at T N ≈ 140 K 2,3 and is not superconducting. The discovery of superconductivity at 26 K in carrier-doped LaOFeAs 4 , followed by rapid improvement now up to T c =55 K 5 in this class, makes these superconductors second only to the cuprates in critical temperature. Several dozen preprints appeared within the two months after the original publication, and many hundred since, making this the most active field of new materials study in recent years (since the discovery in MgB 2 , at least).
A host of models and ideas about the "new physics" that must be operating in this class of compounds is appearing, pointing out the need to establish a clear underpinning of the basic electronic (and magnetic) structure of the system. The materials are strongly layered, quasi-two-dimensional in their electronic structure, by consensus. The electronic structure of LaOFeP was described by Lebègue,6 with the electronic structure and its neighboring magnetic instabilities of LaOFeAs being provided by Singh and Du 7 . Several illuminating papers have appeared since, outlining various aspects of the electronic and magnetic structure of LaOFeAs.
The extant electronic structure work has provided a great deal of necessary information, but still leaves many questions unanswered, and indeed some important questions are unaddressed so far. In this paper we address some of these questions more specifically. Stoichiometric LaOFeAs is AFM; then ∼0.05 carriers/Fe doping of either sign destroys magnetic order and impressive superconductivity arises, with T c seemingly depending little on the carrier concentration. Another question is: with the nonmagnetic electronic structure of LaOFeP and LaOFeAs being so similar, why is the former superconducting while the latter is (antiferro)magnetic? Surely this difference must be understood and built into bare-bones models, or else such models risk explaining nothing, or explaining anything. Another question is the effect of the structure. Unusual sensitivity to the As height z(As) has been noted 8 ; T c is reported to increase with applied pressure 9,10 (reduction in volume) for low values of doping (up to x = 0.11 in LaO 1−x F x FeAs, which is reported as the amount of F for optimal doping); there are increases in T c due to replacement of La with other rare earth ions, and the variation in size of the rare earth is often a dominant factor in the observed trends in their compounds. Very important also is the magnetism in these materials, as magnetism is a central feature in the cuprate superconductors and in correlated electron superconductors.
Another important question is: what can be expected if other pnictide atoms can be incorporated into this system: Sb (or even Bi) on the large atom side, or N on the small atom end. In this paper we address these questions.

II. CRYSTAL STRUCTURE
The members of the family of the new Fe-based superconductors crystallize in the ZrCu-SiAs type structure 11,12 (space group P4/nmm, Z = 2). For instance, LaOFeAs is made of alternating LaO and FeAs layers, as presented in Fig. 1 the Q M AFM order, or equivalently as (π, π, 0), while the Q 0 AFM order corresponds to an antiferromagnetic order of the original cell (dashed lines in Fig. 2) with two Fe atoms. Also, FM will refer to a ferromagnetic arrangement of the spins, while NM means non-magnetic.

III. CALCULATION METHOD
To calculate the relevant quantities, we have used density functional theory (DFT) 13

IV. STUDY OF LAOFEAS IN THE TETRAGONAL STRUCTURE
LaOFeAs has a tetragonal structure (as described in Sect. II) at room temperature 4 .
Although it undergoes a structural phase transition at lower temperature 2,3 (see Section V ), the doped (and superconducting) material LaO 1−x F x FeAs remains in this structure down to low temperature, so the study of LaOFeAs in the high symmetry structure is a necessary step towards the understanding of the electronic structure of the whole family of compounds. functional gives reasonable c/a and z(La) in good agreement with experiment, but it predicted z(As) ∼ 0.139, which is 0.011 off the experimental value, about 0.1Å in length.
However, Wien2K with PBE(GGA) XC functional gives an optimized z(As) ∼ 0.149, which agrees well with experimental z(As). Similar results are found in the XFe 2 As 2 family (X=Ba, Sr, Ca) too. It suggests that, GGA (PBE) XC functional optimizes the FeAs-based system much better than LDA (PW92) XC functional. And GGA should have better performance in dealing with the structure (including c/a, equilibrium volume and z(As)) under pressure of this FeAs family. This is probably due to the layered structure of the FeAs family which results in large density gradient between layers, thus GGA has better description of the potential. But in the meantime, GGA (PBE) further overestimates the magnetic moment of Fe, which is already overestimated by LDA (PW92).
B. Effect of z(As) on the electronic structure of LaOFeAs Then we studied how the electronic structure of LaOFeAs depends on the value of z(As). Table II shows the difference between the experimental z(As)(∼ 0.150), the optimized z(As) (∼ 0.139) and a middle value of 0.145 when using FPLO7 with PW92 XC functional: decreasing z(As) (reducing the Fe-As distance) rapidly reduces the differences in energy between  the different magnetic orderings. At z(As) = 0.145, the magnetic moments of the Q M and Q 0 states are reduced significantly in comparison with z(As) = 0.150, and the difference in energy has changed by around 20%, indicating important changes in the electronic structure upon moving the As atom. For z(As) = 0.139, the Q 0 AFM state has lost its moment (become the NM state), while the magnetic moment of the Q M state has decreased even more, with a changing rate of 6.8 µ B /Å , indicating strong magnetophonon coupling. 8 Therefore, using the experimental or optimized value for the internal coordinate of As gives quite different results and might explain several of the discrepancies seen in the previously published works. In Figures 3 and 4, we present the corresponding band structures, total densities of states, and partial densities of states calculated for different values of z(As). Surprisingly, the band structure near E F referred to the common Fermi level barely changes when z(As) decreases. Somewhat away from E F , the bands below the Fermi level are pushed up in en- There is only a weak dependence of the calculated Fe magnetic moment on the electron doping level: 0.1 e − /Fe doping enhances it from 2.12 µ B to 2.16 µ B (see Table I site energy with respect to that of As 4p states is minor.
Notably, the virtual crystal approximation continues to give strong magnetic states, whereas doping is observed to degrade and finally kill magnetism and promote superconductivity. Thus the destruction of magnetism requires some large effect not considered here, such as strong dynamical spin fluctuations.

D. Electric field gradients
We have calculated the electric field gradients (EFG) of each atom in LaOFeAs, studying both the effects of doping and of magnetic order. As shown in Table III and Table IV Applying pressure is often used as a way to probe how the resulting effect on the electronic structure impacts the superconducting critical temperature and other properties. A strong pressure effect was shown experimentally for the members of the LaOFeAs family 9,10,33 , since for example T c = 43 K could be reached under pressure for LaO 1−x F x FeAs, in case of optimal doping 9 . To begin to understand such observations, it is necessary to determine how the electronic structure of the parent compound LaOFeAs is changed by pressure.
In Fig. 6, the magnetic moment of Fe in the Q M AFM phase versus Fe-As distance is presented. Two different behaviours of the magnetic moment are observed. When z(As) is varied at constant volume (zero pressure),the decrease of the magnetic moment of Fe is parabolic. When pressure is applied and all internal positions are optimized (hence z(As) changes) the change is linear until the magnetic moment drops to zero. This linear behavior is followed also when the As height z(As) is shifted by 0.011 to compensate for the PW92  The effect of pressure on the band structure is shown in Fig. 8. While the bands change positions under pressure, in the corresponding DOS (right panel of Fig. 8), the first peak above E F is moved towards the Fermi level when pressure is applied, but the DOS from -0.1 eV to E F is left almost unchanged by pressure. Therefore pressure should induces important changes in the superconducting properties of electron-doped LaOFeAs, while they should be      Fig. 9. The first sheet is an almost perfect cylinder along the Γ − Z line, while the second sheet is made of two ellipsoidal cylinders with some k z bending. They appear to be very similar to the FS computed at ambient pressure 8 . The pressure has almost no effect on the first sheet, but it enhances the distortion of the second sheet. The structural transformation 2,3 changes the √ 2 × √ 2 cell (with four iron atoms; full lines in Fig. 2) from tetragonal (space group P 4/nmm) to orthorhombic (space group Cmma) or equivalently for the primitive cell (with two iron atoms; dashed lines in Fig. 2) from tetragonal (space group P 4/nmm) to monoclinic (space group P 112/n). To simplify our study, the cell doubling along the c axis due to magnetic ordering is neglected for this study, i.e. we consider only the (ππ0) order. We have performed a relaxation (shape of the cell as More important is the dependence of the magnetic moment on the volume (middle plot of Fig. 10). This dependence has two origins: the first one is the usual dependence of the magnetic moments on the volume change, but in LaOFeAs, the magnetic moment on Fe our case, such a state is higher in energy by about 140 meV per Fe atom for the fully relaxed structure, and therefore can safely be ruled out as being the true ground-state of LaOFeAs.
The differences in calculated values that we have noted reflect an unusual sensitivity to details (structure, method, XC functional).

B. (π, π, π) magnetic order
We turn now to the investigation of LaOFeAs taking into account both the true (π, π, π) magnetic order and the structural distortion. In this case, we have used the experimental structural data provided by de la Cruz et al. 2 . As in the case of the (π, π, 0) order, there are two possible magnetically ordered states. Only one gives the (π, π, π) order to be the ground state versus the (π, π, 0) order, and by only few meV per Fe atom. This small energy difference is near the limit of precision of our calculations, but appears to confirm the sign The total and partial densities of states are very similar to the ones in the case of a (ππ0) magnetic order and won't be shown here; but we notice that the rough electron/hole symmetry in view of the study of doped (superconducting) materials is preserved. Also, our calculated Fermi surface (not shown here), made of four sheets, is very similar to the one presented previously 8 for the (π, π, 0) order and folded back along k z : it has two sheets along the Γ − Z direction which are almost perfectly cylindrical, while the two other sheets are more distorted, but still showing a strong two-dimensional character.
As mentioned at the beginning of Section I of this paper, LaOFeAs and LaOFeP are isostructural and isovalent, but they have quite different properties: LaOFeAs is Q M AFM ordered below T N =150 K and not superconducting, while LaOFeP is a T c =2.5 K superconductor 1 without magnetic order. Also, they have completely different response to doping: either electron or hole doping will destroy the Q M AFM ordering in LaOFeAs and make it superconducting with T c over 26 K 4 (43 K under pressure 9 ), while in LaOFeP, doping changes the critical temperature less significantly to only 9 K 1 . A deeper understanding of the differences of the electronic structure of these two compounds can provide insight into the competition between magnetic ordering and superconductivity. For similar reasons, the related compounds LaOFeN and LaOFeSb (although not studied experimentally yet) are potentially of high interest, so we also provide predictions for their electronic structure.   Table VII. Apart from LaOFeP, all the members of the LaOFePn family studied here have a large Fe magnetic moment in the Q M AFM state, the corresponding total energy being significantly lower than the ones corresponding to FM/NM state.

A. LaOFeP
LaOFeP was the first member of the iron-oxypnictide family to be reported to be superconducting 1 . The corresponding electronic structure was studied by Lebègue using ab-    initio calculations 6 , but considering only a non-magnetic ground-state. Since then LaOFeP has been studied using various experimental tools: by using photoemission 37,38,39 , it was shown that the Fe 3d electrons are itinerant, and that there is no pseudogap in LaOFeP.
Also, magnetic measurements revealed 40,41 that LaOFeP is a paramagnet, while electron-loss spectroscopy 42 implied a significant La-P hybridization. The absence of long-range order in LaOFeP was confirmed by Mössbauer spectroscopy 43 and it was proposed that LaOFeP and doped LaOFeAs could have different mechanisms to drive the superconductivity in these compounds. Also, further theoretical studies were performed 39,40,42 but without studying all the possible magnetic states.
In our calculations, we find that for FM order Fe has a weak magnetic moment of about 0.09 µ B , with a total energy very close to the NM one; this result is much like what is found in LaOFeAs. A remarkable difference is that the Q 0 AFM state cannot be obtained.
However, we found the Q M AFM state to be the lowest in energy, but only by about 1.6 meV/Fe, which is about two orders of magnitude less than in LaOFeAs. LaOFeP, therefore,   NM LaOFeP presented earlier 6 .
Therefore, while they are isostructural and significantly covalent, LaOFeP and LaOFeAs present quite important differences in their respective electronic structures. These differences must form the underpinning of any explanation of why LaOFeP is superconducting with a T c which is almost electron-doping independent, while pure LaOFeAs is not superconducting and becomes so only upon doping.

B. LaOFeSb
Since the experimental crystal structure of LaOFeSb is not reported yet, we conducted calculations to obtain the structure. The procedure we used is the following: starting from the experimental volume V 0 of LaOFeAs (but with As replaced by Sb), we first optimized c/a, z(La) and z(Sb). Then we chose a higher volume and again optimized the parameters, finally finding the volume that has the lowest total energy. Using this scheme, the optimized volume is 1.046 V 0 while for LaOFeAs the equilibrium volume is about 0.919 V 0 . Assuming that PW92 overbinds equally for LaOFeSb as for LaOFeAs, the experimental equilibrium volume for LaOFeSb should be 1.046/0.919=1.138 V 0 . Therefore, we performed calculations for a range of volume from V = V 0 to V = 1.150 V 0 , the corresponding structural parameters being presented in Table VIII. Since for LaOFeAs in the Q M AFM phase PW92 underestimated z(As) by 0.011 at its experimental volume, we corrected z(Sb) by adding 0.011 to the optimized z(Sb) (we refer to this position at the ""shifted z(Sb)"). Both for the NM and Q M AFM case, there are very small differences near E F between the optimized z(As) and shifted z(As) in the band structure and DOS, as seen in Fig 14. However, shifting z(Sb) induces important changes in the energy differences between NM and Q M AFM states, as shown in Table IX Thus from these results we expect that doped LaOFeSb should have similar properties (viz, value of T c ) as LaOFeAs.

C. LaOFeN
The structure of LaOFeN is also not reported experimentally. In order to obtain it, the same procedure as for LaOFeSb was used. The lowest total energy is at 0.762 V 0 ' (here V 0 ' is the experimental volume of LaOFeP.). Again assuming PW92 makes a similar error as it makes in LaOFeAs, we estimate its equilibrium volume to be close to 0.825 V 0 '. At 0.825  V 0 ' and for larger volume, the total energy of the Q M AFM state is well below that of the FM/NM state (see Table X). Therefore, LaOFeN, if it exists, should be in the Q M AFM ordered state at low temperature, which is similar to LaOFeAs and LaOFeSb.
Compared to the other LaOFePn compounds, LaOFeN is even closer to being a semimetal when the volume is equal to 0.825 V 0 ', and it becomes a small gap insulator at 0.850 V 0 ' and a higher carrier density metal at 0.800 V 0 ' (see Fig. 15). The DOS for 0.825 V 0 ' shows a pseudogap around E F , but the DOS is somewhat less flat than it is for LaOFeAs.
When LaOFeN is calculated to be insulating (for volumes larger than 0.825 V 0 '), the gap can be taken to define a distinction between bonding (occupied) and antibonding (unoccupied) states.   is no way to ascribe the small FSs to simple overlapping valence and conduction bands: in LaOFeAs and LaOFeSb, the bonding and antibonding bands are never completely separated from each other. In LaOFeN this separation finally becomes apparent, as an actual bandgap does appear.

VII. ROLE OF THE RARE EARTH ATOM IN REOFEAS
After LaOFeAs was discovered, after appropriate variation of the carrier concentration, to be superconducting at 26 K, much substitution on the rare earth (R) site has been done, with impressive increases in the critical temperature. Since all are evidently trivalent and donate three valence electrons to the FeAs layer, it becomes important to uncover the influence of the R atom: is it some aspect of the chemistry, which does differ among the rare earths? is it an effect of size? or can there be some other subtle effect? Table XI is a collection of the lattice constants a and c, volume V of the primitive cell, T c onset of ROFeAs reported from experiment. 44,45,46,47 Both lattice constants, hence the volume, decrease monotonically as the atomic number increases, but T c increases only from La to Gd, whereupon drops for heavier rare earths. Since we have found that small details affect the electronic and magnetic structure -especially z(As) -it is reasonable to assess the size effect. We have performed calculations on Ce, Nd and Gd, using LSDA+U with U=7.0 eV and J=1 eV applied to the R atom to occupy the 4f shell appropriately and keep the

VIII. SUMMARY
We have investigated in some detail the electronic structure and magnetic properties of the LaOFeAs class of novel superconductors using ab-initio methods. The effects of the Fe-As distance, of doping, and of pressure, as well as calculations of the EFGs have been reported. It was found that (approximate) electron-hole symmetry versus doping, and strong magnetophonon coupling are primary characteristics of the LaOFeAs system, and are two of the ingredients that need to be understood to proceed toward the discovery the mechanism of superconducting pairing. We studied effects of the structural distortion and of the (π, π, π) magnetic order, finding that experiments can be reproduced fairly well by our calculations.
Finally, the related materials LaOFeP, LaOFeSb, and LaOFeN were investigated and their properties compared to those of LaOFeAs. From these comparisons, it appears that LaOFeP is significantly different from the other materials studied here; this difference might explain why, at stoichiometry, LaOFeP is superconducting while LaOFeAs is antiferromagnetic. Also, in view of their similarities with LaOFeAs, either pure or doped LaOFeSb and LaOFeN are potential candidates as superconductors.