Multiband magnetism and superconductivity in Fe-based compounds

Multiband magnetism and superconductivity in Fe-based compounds

V. Cvetkovic 1 Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218 1    Z. Tesanovic 1 Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218 11 Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218 1

Recent discovery of high T superconductivity in Fe-based compounds may have opened a new pathway to the room temperature superconductivity. The new materials feature FeAs layers instead of the signature CuO planes of much-studied cuprates. A model Hamiltonian describing FeAs layers is introduced, highlighting the crucial role of puckering of As atoms in promoting d-electron itinerancy and warding off large local-moment magnetism of Fe ions, the main enemy of superconductivity. Quantum many-particle effects in charge, spin and multiband channels are explored and a nesting-induced spin density-wave order is found in the parent compund. We argue that this largely itinerant antiferromagnetism and high T itself are essentially tied to the multiband nature of the Fermi surface.


Theories and models of superconducting state Spin-density waves Charge-density-wave systems

Recently, a surprising new path to room-temperature superconductivity might have been discovered. The quaternary compound LaOFeP was already known to become superconducting below 7K [1], when its doped sibling LaOFFeAs () turned out to have unexpectedly high of 26K [2]. Even higher ’s were found by replacing La with other rare-earths (RE), reaching the current record of 55K [3]. These are the first non-cuprate superconductors exhibiting such high ’s.

The surprise here is that the most prominent characteristic of iron is its natural magnetism. By conventional wisdom, the high superconductivity in RE-OFeAs compounds is unexpected, all the more so since the superconductivity apparently resides in FeAs layers. Following standard ionic accounting, rare-earths are 3, giving away three electrons, while As and O are 3 and 2, respectively. One then expects Fe to be in its 2 configuration, two of its 4s electrons given away to fill As and O p-orbitals, with assistance from a single rare-earth atom. The remaining six d-electrons fill atomic orbitals of Fe in the overall tetragonal As/O environment of Fig. 1; the lower three orbitals should be filled while the upper two orbitals should be empty. However, the Coulomb interactions intervene via the Hund’s rule: the total energy can be reduced by making the spin part of the atomic wavefunction most symmetric and consequently the orbital part of it as antisymmetric as possible, reducing thereby the cost of Coulomb repulsion. The simplest realization of this is to occupy a low orbital with one spin-up and one spin-down electron while storing the remaining four electrons into the spin-up states. The result is a total spin of Fe, with the associated local magnetic moment and likely magnetism in the parent compounds. This is the situation similar to manganese, the Fe’s nearest relative, whose five d-electrons feel the full brunt of the Hund’s rule and typically line up into a large spin state, and very different from copper, where d-orbitals are either fully occupied or contain only a single d-hole, as in the parent state of cuprate superconductors. All told, the circumstances are hardly hospitable to any superconductivity, let alone a high temperature one.

a)     b) c)

Figure 1: (Colour on-line) (a) The three dimensional structure of the parent compound, with FeAs layer (Fe – red, As – green) on top of a REO layers (RE – yellow, O – blue); The blue square in the FeAs plane corresponds to the ‘planar’ unit cell (b). We denote two Fe atoms with A and B, while the two As atoms that are displaced up and down with respect to the layer are presented by dotted and crossed circles respectively. We give our choice of axes in the corner (note that some papers use a coordinate system rotated by 45 degrees). (c) The evolution of d-orbital energy levels from the tetragonal to tetrahedral crystal field environment. The puckering of FeAs planes results in the situation which is “in between”, placing all d-orbitals near the Fermi level.

In this paper, we first argue that the above Hund’s rule route to large local moment magnetism is derailed by significant banding effects, promoting enhanced itinerancy for most Fe d-electrons. We show how such itinerancy arises from the combination of two factors: a sizeable overlap among Fe and As atomic orbitals and the distortion of the overall tetragonal structure into a locally near-tetrahedral environment for Fe ions, both generated by the crucial “puckering” of As atoms out of the FeAs planes (Fig. 1). The puckering rearranges the and crystal-field levels so that – the situation “in between” the purely tetragonal () and the purely tetrahedral () – thus bringing all d-orbitals into a close proximity of the Fermi level , and maximizes direct overlap between Fe d- and As p-orbitals. The end result are numerous bands crossing and a multiply connected Fermi surface, containing both electron and hole sections. We introduce a two-dimensional tight-binding model which captures the relevant features of this multiband problem. Next, we argue that large number of broad bands and the absence of large local Fe moments betrays not only the failure of the atomic Hund’s rule but, via the enhanced metallic screening, the absence of strong local correlations in general. This implies the key role for the nesting properties and we present an analytic calculation of various responses for circular and elliptical bands relevant to this multiband problem. These responses allow us to account for the observed weak antiferromagnetic ordering in parent materials and provide strong clues about the superconducting mechanism itself. In this sense, the Fe-based high superconductors differ from the hole-doped cuprates and are likely more closely related to either the electron-doped cuprates or other weakly to moderately correlated superconductors.

The parent compound of the Fe-based superconductors has a ZrCuSiAs type structure [4], with eight atoms per unit cell, depicted in Fig. 1. The Fe atoms lie in a plane, same as O atoms precisely above them, in the adjacent REO layer. In contrast, the RE and As atoms (also located above each other) are puckered out of plane in a checkerboard fashion. The amount of puckering is significant, turning the in-plane tetragonal structure in the physically relevant FeAs layer into the nearly-tetrahedral one (the angle of the FeAs bond with respect to the vertical is 58.8 as compared to 54.7 for a tetrahedron [5]). As stated above, this has important consequences for promoting banding and rich orbital content near .

There available electronic structure calculations of LaOFeP [6], and of LaOFeAs, doped and undoped [7, 8], consistently convey the key information that all five Fe 3d bands of are located at the Fermi level, in stark contrast with the cuprates. These bands are hybridized with 4p orbitals of As/P located far below the Fermi level, centered between 6 and 2eV below . There are five sections of the Fermi surface: two hole concentric, near-circular quasi-2D ones around the point, two electron elliptical ones, centered around the point, and a 3D hole band with a spherical Fermi surface around the point. Given the fact that the last one vanishes upon doping [8], and that the relevant physics appear to be two-dimensional, we will ignore this Fermi surface and neglect the interlayer couplings altogether. This idea is used in other works which aim to recreate the band-structure, either with all ten bands [9, 10], or with some simpler minimal model [11, 12].

To illustrate the key role of the puckering of As atoms on the electronic structure of FeAs, let us first consider the hypothetical situation in which all As atoms are planar (Fig. 1). The tetragonal crystal field splitting pushes 3d orbitals (, , and ) below the orbitals. In this arrangement, the overlap of Fe orbitals with the neighbouring As p-orbitals either vanishes by symmetry or is very small, the only source of broadening for these bands being the direct overlap of two d-orbitals on neighbouring Fe. The bands, on the other hand, do directly couple to the 4p-orbitals of As, but, since these p-orbitals are deep below the Fermi level, this coupling only pushes the bands further up, increasing the crystal field gap. The consequence is that such a material should likely be a band insulator, turning into a local moment magnet once the Coulomb effects and the Hund’s rule are turned on. A sizeable puckering changes the situation dramatically: first, the Fe crystal field environment turns to near-tetrahedral instead of the tetragonal. In the purely tetrahedral case, the orbitals (, , and ) reverse their position and are above the levels ( and ). In the nearly-tetrahedral case of real FeAs layers, the orbitals are slightly above , and the overlap due to the band formation makes all five bands important. This banding is the other crucial consequence of the puckering: since the orbitals are not entirely in the Fe plane, the overlap between these orbitals and , and d-orbitals becomes significant, and it actually provides the dominant contribution to electron/hole hopping. At the same time, the hopping between and , or and orbitals is only slightly reduced.

Based on the above arguments, we construct a tight-binding model which incorporates the hoppings to the nearest neighbours and includes the relevant overlap integrals. This model which reflects the key qualitative features of the electronic structure calculations [7, 8], and which can serve as the realistic platform for further analytic calculations. As shown below, even such a simplified model must include all five bands and is defined by the tight-binding Hamiltonian


where describes local 3d and 4p orbitals, and accounts for Fe-As, Fe-Fe, and As-As hopping, in that order. is the interaction term and will be discussed shortly. The operator annihilates an electron in orbital on Fe site , and analogously, annihilates an electron on site in orbital . The summation over takes into account all five Fe 3d orbitals, but due to the doubling of the unit cell, there are actually ten of those bands, and the summation over involves all three bands , , and .

The symmetry provides important guidance in understanding the band structure of (1). For example, at point there are two doubly degenerate bands. One of these must be a symmetric combination – relative to A and B sites – of , and orbitals weakly hybridized with the As 4p bands, while the other is the antisymmetric combination. The splitting between these two dublets originates both from the direct spread of the bands (), and from the bands spread (). At the point, these orbitals again two degenerate dublets, albeit in a different linear combination, which are split only by the amount proportional to . From such analyses, we find the orbital energies and hoppings (all in eV’s)

-0.85 -1.4 -1.1 -1.15
-0.55 -0.5 -1.6 -0.55
0.65 -1.4 1.5 3.2
2.1 1.25 0.7
-4.0 -4.0
-0.8 -0.45

The interband couplings are not tabulated and their values are , , and .

The levels reflect our previous discussion of the crystal field splitting: on the scale of ’s. The hoppings reveal that the puckering of As atoms promoted bands to the physically most relevant ones, their coupling to the 4p orbitals being the strongest. These bands provide dominant content of the electronic states at , where they get mixed with the other states, chiefly , to finally form the two hole Fermi surfaces. Clearly, these effects are difficult to reproduce within a simple two-band model. In addition, significant mixing of different d-orbitals with opposite parity relative to the FeAs planes further reinforces the need to include the full d-orbital manifold into the basic description.

a) b)

Figure 2: (Colour on-line) The band structure (a), and the Fermi surfaces (b) following from the tight-binding Hamiltonian (1), and using the parameters of the tight binding fit.

Fig. 2 shows the band structure and the Fermi surface(s) following from (1). The key features of the Fermi surface are faithfully reproduced, with the central hole pockets nearly circular (actually, these are two ellipses which strongly interact and avoid crossing). In the vicinity of the point, the two electron pockets have elliptical shape and do not interact at the crossing points located at the edges of the Brillouin zone.

This brings us to in (1). The picture of puckered As atoms discussed above, promoting the bunching of local d-levels of Fe and their large overlap with As p-orbitals, indicates that the d-bands are near their optimum width, given the restrictions of dealing with 3d electrons and 4p levels far below of . This reduces the importance of the Hund’s rule and points to the d-electron itinerancy, rather than local atomic (ionic) correlations, as the most relevant feature. Indeed, this is consistent with the neutron scattering experiments [13], observing weak antiferromagnetism in the parent compound below 150K instead of the large local moment magnetism expected in the Hund’s rule limit. The AF order is suppressed by doping and ultimately vanishes in the superconducting state. This suggests that should generically be comparable or smaller in magnitude than (1). For example, in the simple single band Hubbard model, with nearly circular (or spherical) Fermi surface, too large on-site repulsion leads to the ferromagnetic Stoner instability, an itinerant prelude to the local moment formation dictated by the Hund’s rule. We thus expect that the main effects of can be understood by a detailed analysis of enhanced spin, charge and interband responses of the non-interacting part of (1).

With this aim in mind, we observe that various pockets of the Fermi surface depicted in Fig. 2 can be viewed as radial and elliptical distortions of the same idealized circle, two of such ideal (hole) pockets located at and two (electron) at M points. As long as such distortions are not too extreme, the responses in different channels can be evaluated analytically, thereby greatly facilitating theoretical analysis. Where comparison is possible, our analytic results appear to agree with numerical calculations in Refs. [8, 11, 12].

We first look at the spin-susceptibility, and analyze how near-nesting of the Fermi surfaces can lead to SDW order. To do this, we have to make some assumptions about the Fermi surfaces and separate the most important contributions. While some nesting takes place within slightly deformed circular Fermi surfaces in Fig. 2, the main contribution to the enhanced response arises from similarly shaped hole and electron pockets, followed by a less important one arising from different hole-hole and electron-electron pockets of the Fermi surface. This is easily appreciated by nothing that, for our idealized circles, the electron-hole nesting leads to a divergent contribution to the electron-hole propagator (i.e., an RPA bubble). So, we concentrate on the spin-susceptibility where one the particle propagators corresponds to the hole band at the point with a circular Fermi surface and Fermi momentum , and the electron band forming a slightly ellipcitally deformed Fermi surface centered around vector, as depicted in Fig. 2. The electronic states at the Fermi level have momentum if parallel to the vector, and if perpendicular. The dispersions are


with being the mass of the electron/hole band. For simplicity we assume that they have the same mass . Wavevector is given relative to the point in the case of the electron band, while is defined with respect to the center of the Brilouin zone. Parameters and are tied to the eccentricity of the Fermi ellipse as , and to the ratio of states enclosed by the two Fermi surfaces . Below, we evaluate the particle-hole bubble due to the near nesting of only one hole and one electron band. Our results are universal, generally applicable to any situation involving elliptical Fermi surfaces, and particularly relevant for FeAs, where one has to sum contributions due to nesting of each individual hole and electron band.

a) b)

Figure 3: (Colour on-line) The arrangement of Fermi surfaces with elliptical bands at the corners of the Brillouin Zone show in Fig. 2 (a), and the regularization of the singular part of the susceptibility due to the elliptical distortion of the electronic Fermi surface (b). For , the hole and the electron Fermi surfaces become identical and the susceptibility diverges.

If the eccentricity were zero, and the two bands had identical Fermi momenta (), the real part of the susceptibility is would have simply been given by


where is the UV band cut-off. A logarithmic singularity occurrs in Eq. (6) when due to the perfect nesting of two hole and electron Fermi pockets. The nesting in FeAs is not perfect due to small distortions in Fig. 2, and this singularity is cut off. Still, it appears nevertheless that this particular response at is the strongest incipient instability of our system. If is overall repulsive and not extremely weak, say modelled as a Hubbard repulsion on Fe sites, , this instability will produce the spin density-wave (SDW) ground state at the commensurate wavevector . It seems natural to associate this Fermi surface instability with the observed weak AFM order of the parent compound [13].

To appreciate how the deformation of the electron Fermi surface cuts off the singularity, we now find the explicit expression for this more general situation. There are two different cases, depending on whether the two Fermi surfaces intersect after one has been moved by (so that their centers coincide). If they do not intersect (equivalent to ), the susceptibility is set by


where and .

Clearly, it is the last two term which cause the nesting divergence in the limit when the ellipse transforms to a circle (). When the Fermi surfaces do intersect (), the last two logarithms in Eq. (7) should be replaced by . This term is responsible for the singularity in this case. The divergent behavior of the real part of the susceptibility is shown in Fig. 3.

Our analysis of the divergent part of the susceptibility was centered on the case when , and the question remains whether that is the global maximum. The derivatives of the susceptibility with respect to are well defined due to the regularization by finite or . It is trivial to demonstrate that the first derivative at vanishes in all directions, which can alternatively be argued based on symmetry. Therefore, one has to look for the sign of the second derivative in both and direction in order to determine whether the susceptibility has a maximum, a minimum or a saddle point at . Even if it turns out that the susceptibility has a maximum, it may be a local maximum, not the global one. While the general treatment of the problem will be presented elsewhere [14], we illustrate the situation by two circular Fermi surfaces with slightly different radii, , and . The susceptibility due to the nesting of these Fermi surfaces is compared for the cases when , and , with being a unit vector pointing in an arbitrary direction. The former corresponds to concentric Fermi surfaces, the latter to two surfaces touching each other. The susceptibility in the former case follows as a special case of Eq. (7)


and the result for touching circles is obtained by replacing the argument under the square root by 2. This is always slightly larger than the susceptibility following from Eq. (8), regardless the value of . Such a result implies a different ordering vector , albeit only in a continuum theory. Our system is on a lattice, and is commensurate with the reciprocal lattice, hence any instability at that wave-vector will be enhanced by Umklapp processes, whereas this is not true for other incommensurate wave-vectors such as . Furthermore, we may argue that two Fermi surfaces touching should not produce any unexpected divergences in the particle-hole channel, by simply observing the analytic expression Eq. (7) when or .

Eq. (7) can be applied to all the possible pairs of hole/electron bands found in the band structure of FeAs. There are two circular hole surfaces of different radii paired with two electron surfaces which are the same, except that they are rotated by 90 – this just exchanges and (the unmarked Fermi surface in Fig. 3a). For the UV cut off we choose the inverse lattice spacing. We now compare the relative values for the doped and parent systems, with the help from the band structure calculations. For the undoped parent system, we estimate [6] , , , and , which yields at . Doping moves upwards, increasing the electronic, and shrinking the hole Fermi surfaces. The corresponding surfaces are now further apart, so the susceptibility is expected to be smaller. Using Ref. [8], we estimate , , , and , giving at . Similar estimate is obtained by using our tight-binding band structure of Fig. 2. This is quite a bit smaller than the undoped value, and suggests rapid suppression of our SDW upon doping, as observed experimentally [13].

The SDW/AF order at discussed above and observed in experiments, could in principle also be interpreted in the local spin picture, as arising from the direct and indirect superexchange between Fe atoms. The direct superexchange is generated by the overlap of 3d orbitals of neighbouring atoms, i.e., overlap between A and B atoms in Fig. 1b. Two A(B) atoms, in contrast, have an insignificant direct overlap. However, from our band structure we know that bands and hybridize with 4p orbitals of As. Let us for example take one A atom in a unit cell in Fig. 1, and consider its overlap with its next neighbour A to the right. Both of these atoms have their orbitals hybridized with the orbital of the As atom standing between. The new hybridized bands both carry a significant fraction of the As orbital, so a hopping between these two atoms is enabled via the intermediate As atom. This hopping gives rise to the indirect superexchange coupling . Similar argument was presented in Ref. [15]. By the same mechanism, the indirect exchange between orbitals of neighboring iron atoms, due to their overlap with p orbitals of As, yields ferromagnetic nearest neighbor contribution to [16, 17]. Our earlier analysis suggests that the total is significantly smaller than (). At such a high ratio of frustrating AF couplings, the AF ordering takes place individually on A and B sublattices [18] irrespective of sign of . At the mean-field level, the relative order on the two sublattices does not affect the ground state energy since each B site interacts with four neighbouring A sites, two of these spins pointing in the direction opposite to the other two; consequently, there is no overall interaction. This implies that, on classical level, the ground state would have been degenerate with its ground state energy independent of the angle between two order parameters. Thus, we include excitations – spin-waves – and investigate how their interaction affects the ground state. For this, we use the standard Holstein-Primakoff bosonization. Assuming that the angle between two order parameters on lattices A and B is , and introducing HP bosons , and on two respective lattices, the following Hamiltonian is obtained

The Bogoliubov transformation of the Hamiltonian (Multiband magnetism and superconductivity in Fe-based compounds) yields two new excitations whose dispersions are given by


We numerically determine zero point energy, and plot it in Fig. 4(a). The energy of the system is minimized when the spins on two sublattices are collinear in agreement with the experiments [19].

a)       b)

Figure 4: (Colour on-line) (a) The zero point energy in units of as a function of angle between two staggered magnetizations, with [20]. The energy is the lowest for . (b) The staggered magnetization in units of per unit cell is plotted for three different spin values , and (from bottom to top, respectively). The dashed line corresponds to the experimentally observed value of [13].

The staggered magnetization must be evaluated numerically for arbitrary . Fig. 4(b) shows the magnetization (per iron site) due to quantum fluctuations as a function of for different values of spin . The fluctuations are spin independent, but the resulting magnetization is not. Unrealistically low spin and large are required in order to explain what is observed experimentally [13], thus hinting at significant itinerancy in the AF state, in line with our previous arguments.

We now turn to the superconductivity itself, clearly the most difficult problem. It is naturally tempting to use the above propensity for SDW at in the parent compund to generate pairing interaction once the system is doped away from the AF order [21]. Following the example of electron-doped cuprates and various organic superconductors, this would imply an ordering of a nodeless kind, with electron and hole pockets in Fig. 2 fully gapped but with gap functions of different relative sign (see, for example, Ref. [8]). It is important to stress here the crucial role played by the Josephson-like interband scattering between hole and electron Fermi surfaces in bringing about this form of superconductivity (see further below, and also [14]). Since hole and electron pockets are not identical, the gap magnitudes would not be either, but the difference could be quite small. Observing such a relative sign difference in the otherwise fully gapped state (an or state) would clearly be strong boost for this picture of superconductivity generated entirely by antiferromagnetic (SDW) spin fluctuations. However, the above change in sign implies sensitivity to ordinary (non-magnetic) impurity disorder which could severely suppress and the gap. This effect, while still present on general grounds, appears less consequential in hole-doped cuprates, due to their strongly correlated, almost local nature. In Fe-based superconductors the correlations are not that strong, as we have just argued above, and this impurity sensitivity becomes an important issue. Finally, in order to generate the (or ) superconducting state, the interband interaction – enhanced by the proximity to the SDW – must overcome the intraband repulsion (see below), not an easy task [14].

In light of this, one should not out of hand dismiss the possibility that Fe-based superconductors are of entirely different kind from even the electron-doped cuprates and similar superconductors, where the purely repulsive interactions suffice to generate pairing near a magnetic instability. Their multiband nature could instead be a realization of the exciton-assisted superconductivity. The phonon interaction appears to be too weak to push to 55 K by itself [8]. However, large number of highly polarizable bands around the Fermi surface leads to strong metallic screening and a possibility of a dynamical overscreening, which turns , the familiar Coulomb pseudopotential of the Eliashberg theory, attractive in the certain finite wavevector and frequency regions. This would pave the way for the exciton-assisted superconductivity, long-anticipated but never unambiguously observed [22]. The basic idea is that the dynamical Coulomb interaction:


() turns attractive at some finite and relatively low . Fe-based superconductors appear to have all the ingredients: their highly polarizable multiband Fermi surface produces neutral plasmon modes corresponding to electron and hole densities oscillating in phase (neutral plasmons). Such modes act as “phonons”, particularly if and are sufficiently different. Furthermore, the nesting features lead to enhanced density response near and this could turn the effective interaction attractive at relatively low . Finally, the interband pairing [23] could still be essential, and acts to further boost irrespective of its sign:


where and are the e-e (h-h) and the interband coupling constants, respectively, and is the characteristic frequency of the exciton-assisted pairing. generated by this mechanism is notoriously difficult to estimate, both due to the competition from structural and covalent instabilities in the particle-hole channel and the need to consider local-field contributions to [24]. Nevertheless, this “hybrid” option – in which phonons help make intraband interactions attractive, or at least only weakly repulsive, and enable the magnetically-enhanced repulsive interband interactions provide the crucial boost to – should be kept in mind as the experimental and theoretical investigations of Fe-based superconductivity continue in earnest.

In summary, Fe-based superconductors appear to offer a glimpse of a new road toward room-temperature superconductivity. We have constructed here a simplified tight-binding model which qualitatively describes the physics of FeAs layers where the superconductivity apparently resides. Analytical results were given for the elementary particle-hole response in charge, spin and multiband channels and used to discuss various features of the SDW/AF order and superconductivity. We stress the importance of puckering of As atoms in promoting d-electron itinerancy and argue that high of Fe-based superconductors might be essentially tied to the multiband character of their Fermi surface, favorable to the (or ) superconducting state. It is tempting to speculate that different ’s obtained for different rare-earth substitutions might be related to the different amount of puckering in FeAs layers and regulating this amount might be the key to even higher .

1 Acknowledgments

The authors are grateful to I. Mazin, C. Broholm and C. L. Chien for useful discussions. This work was supported in part by the NSF grant DMR-0531159 and by the DOE grant DE-FG02-08ER46544.

2 Additional remark

Since this manuscript was originally posted on the arXiv in April 2008 ( there were numerous significant experimental and theoretical developments in this exceptionally fast-paced field. Those most relevant to this work include the observation of the superconducting gap in PCAR [25] and ARPES experiments [26, 27, 28] and various theoretical approaches exemplified by Refs. [29, 30, 31, 32, 33, 34, 35]. Especially notable among these are further theoretical explorations of the (or ) superconducting state in Refs. [36, 14, 37, 38].


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