Abstract
Since the discovery of the metallic antiferromagnetic (AF) ground state near superconductivity in iron pnictide superconductors^{1,2,3}, a central question has been whether magnetism in these materials arises from weakly correlated electrons^{4,5}, as in the case of spin density wave in pure chromium^{6}, requires strong electron correlations^{7}, or can even be described in terms of localized electrons^{8,9} such as the AF insulating state of copper oxides^{10}. Here we use inelastic neutron scattering to determine the absolute intensity of the magnetic excitations throughout the Brillouin zone in electrondoped superconducting BaFe_{1.9}Ni_{0.1}As_{2} (T_{c}=20 K), which allows us to obtain the size of the fluctuating magnetic moment 〈m^{2}〉, and its energy distribution^{11,12}. We find that superconducting BaFe_{1.9}Ni_{0.1}As_{2} and AF BaFe_{2}As_{2} (ref. 13) both have fluctuating magnetic moments 〈m^{2}〉≈3.2 μ_{B}^{2} per Fe(Ni), which are similar to those found in the AF insulating copper oxides^{14,15}. The common theme in both classes of hightemperature superconductors is that magnetic excitations have partly localized character, thus showing the importance of strong correlations for hightemperature superconductivity^{16}.
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Main
In the undoped state, iron pnictides such as BaFe_{2}As_{2} form a metallic lowtemperature orthorhombic phase with the antiferromagnetic (AF) structure as shown in Fig. 1a (ref. 17). Inelastic neutron scattering measurements have mapped out spin waves throughout the Brillouin zone in the AF orthorhombic and paramagnetic tetragonal phases^{13}. On Co and Nidoping to induce optimal superconductivity via electron doping, the orthorhombic structural distortion and static AF order in BaFe_{2}As_{2} are suppressed and the system becomes tetragonal and paramagnetic at all temperatures^{18}. In previous inelastic neutron scattering experiments on optimally electrondoped Ba(Fe,Co,Ni)_{2}As_{2} superconductors ^{11,12,19,20,21,22}, spin excitations up to ∼120 meV were observed. However, the lack of spin excitation data at higher energies in absolute units precluded a comparison with spin waves in undoped BaFe_{2}As_{2}. Only the absolute intensity measurements in the entire Brillouin zone can reveal the effect of electron doping on the overall spin excitation spectra and allow a direct comparison with the results in the AF insulating copper oxides^{14,15}. For the experiments, we chose to study wellcharacterized electrondoped BaFe_{1.9}Ni_{0.1}As_{2} (refs 20, 22) because large single crystals were available^{23} and their properties are similar to Codoped BaFe_{2}As_{2} (refs 11, 12, 19, 21, 24).
By comparing spin excitations in BaFe_{1.9}Ni_{0.1}As_{2} and BaFe_{2}As_{2} throughout the Brillouin zone, we were able to probe how electron doping and superconductivity affect the overall spin excitation spectra. We demonstrate that whereas the lowenergy spin excitations are affected, the highenergy excitations show a very weak temperature and doping dependence. Comparison of our results with various theories suggests that neither a fully itinerant nor a localized picture explains the magnetic excitation spectrum. However, a combination of density functional theory (DFT) and dynamic mean field theory (DMFT) provides a natural way to improve on both these pictures.
Figure 1c–e summarizes our key findings for the electrondoped iron arsenide superconductor BaFe_{1.9}Ni_{0.1}As_{2} and the comparison with the spin waves in BaFe_{2}As_{2}. The data points in Fig. 1c and d show the dispersion of spin excitations for optimally doped BaFe_{1.9}Ni_{0.1}As_{2} along [1,K] and [H,0]and the solid lines show the fit of BaFe_{2}As_{2} spin waves to an effective Heisenberg J_{1a}−J_{1b}−J_{2} model with (ref. 13). Figure 1e shows the local dynamic susceptibility per formula unit (f.u.), which contains two Fe(Ni) atoms, in absolute units, defined as (ref. 12), where χ^{′′}(q,ω)=(1/3)t r(χ_{α β}^{′′}(q,ω)), at different energies for BaFe_{2}As_{2} and BaFe_{1.9}Ni_{0.1}As_{2}. It is clear that electron doping on BaFe_{2}As_{2} affects only the lowenergy spin excitations by broadening the spin waves below 80 meV, but has no impact on spin waves above 100 meV (see Supplementary Information). The quasiparticles that form within the spin density wave gap are sensitive to the Fermi surface change on doping BaFe_{2}As_{2} and, hence, the resulting lowenergy itinerant spin excitations change substantially, whereas the higher energy spin excitations are hardly affected.
To substantiate the key conclusions from the data and calculations presented in Fig. 1, we show in Fig. 2 the twodimensional constantenergy (E) images of spin excitations of BaFe_{1.9}Ni_{0.1}As_{2} in the (H,K) scattering plane for several Brillouin zones at 5 K. In the undoped phase, spin waves in BaFe_{2}As_{2} exhibit an anisotropy spin gap of Δ=9.8 meV (ref. 25). On doping, the anisotropy spin gap disappears and the spin excitations form transversely elongated ellipses that decrease in intensity with increasing energy^{12,21}. For energy transfers of E=10±3, 33±3, 43±3, 60±10 and 81±10 meV (Fig. 2a–e), spin excitations are peaked at the AF wave vector Q=(1,0) in the centre of the Brillouin zone (shown as a dashed square box). As the energy increases to E=113±10 and 135±10 meV (Fig. 2f,g), spin excitations start to split along the Kdirection and form a ring around the Γ point. Finally, spin excitations near the zone boundary at E=157±10 and 214±10 meVform four blobs centred at Q=(1,1) (Fig. 2h,i).
To determine the dispersion of spin excitations for BaFe_{1.9}Ni_{0.1}As_{2}, we cut through the twodimensional images similar to Fig. 2 along the [1,K] and [H,0] directions. Figure 3a–f shows constantenergy cuts along the [1,K] direction for E=25±5, 55±5, 95±10, 125±10, 150±10, and 210±10 meV. The scattering becomes dispersive for spin excitation energies above 95 meV. Figure 3g–i shows similar constantenergy cuts along the [H,0] direction. The solid lines in the figure show identical spin wave cuts for BaFe_{2}As_{2} (ref. 13). As both measurements were taken in absolute units, we can compare the impact of electron doping on the spin waves in BaFe_{2}As_{2}. At E=25±5 meV, spin excitations in superconducting BaFe_{1.9}Ni_{0.1}As_{2} are considerably broader in momentum space and weaker in intensity than spin waves (Fig. 3a,g). On increasing the excitation energy to 55±5 meV, the dispersive spin waves in BaFe_{2}As_{2} become weaker and broader (Fig. 3b,h). For energies above 95 meV, spin excitations in BaFe_{1.9}Ni_{0.1}As_{2} are almost indistinguishable from spin waves in BaFe_{2}As_{2} in both the linewidth and intensity (Fig. 3c–f,i). Based on these constantenergy cuts, we show in Fig. 1c,d the comparison of spin excitation dispersions of BaFe_{1.9}Ni_{0.1}As_{2} (filled circles and horizontal bars) with those of spin waves in BaFe_{2}As_{2} (solid lines). Inspection of Figs 1–3 reveals that electron doping to BaFe_{2}As_{2} broadens and suppresses only lowenergy spin excitations and has no influence on spin waves above 100 meV (see Supplementary Information).
To demonstrate further the effect of electron doping on the spin waves of BaFe_{2}As_{2}, we show in Fig. 4a–d constant Q cuts at different wave vectors along the [1,K]direction for spin excitations in BaFe_{1.9}Ni_{0.1}As_{2}. Near the Brillouin zone centre at Q=(1,0.05) and (1,0.2), welldefined spin excitations are observed near E=40 and 60 meV, as shown in Fig. 4a and b, respectively. The intensity of the scattering from the spin excitations in BaFe_{1.9}Ni_{0.1}As_{2} is, however, much lower than that of BaFe_{2}As_{2}, shown as solid lines in the figures. On increasing the wave vectors to Q=(1,0.35) and (1,0.5), the magnetic scattering peak near E=100, and 120 meV, and are essentially indistinguishable from spin waves in BaFe_{2}As_{2} as shown in Fig. 4c and d. Furthermore, spin excitations have virtually no temperature dependence between 5 K and 150 K (Fig. 4b).
Finally, in Fig. 4e,f, we show the temperature dependence of spin excitations at energies near the neutron spin resonance E=9 meV (refs 20, 22) and at E=90±5 meV, respectively. Whereas the intensity of the resonance at E=9 meVincreases markedly below T_{c}, consistent with earlier work^{20,22}, spin excitations at 90±5 meV are identical on cooling from 150 K to 5 K. We note that highenergy spin waves in BaFe_{2}As_{2} are also weakly temperature dependent^{13}. Figure 4g shows the energy dependence of the dynamic spin–spin correlation lengths, which are about ξ≈14 Å and independent of excitation energy. For comparison, the dynamic spin–spin correlation length (the solid line in Fig. 4g) in BaFe_{2}As_{2} decreases with increasing energy and becomes similar to that of BaFe_{1.9}Ni_{0.1}As_{2} for excitation energies above 100 meV.
To check if spin excitations in the AF BaFe_{2}As_{2} and superconducting BaFe_{1.9}Ni_{0.1}As_{2} can be understood in an itinerant picture, we calculate the local susceptibility χ^{′′}(ω) using the random phase approximation (RPA) based on realistic Fermi surfaces and band structures ^{26}. Within the RPA, the polarization bubble χ^{0} is computed from the DFT using Kohn–Sham Green’s functions, while the irreducible vertex Γ^{irr} is approximated by the screened Coulomb parameters and . Using and and performing calculations above T_{N} (ref. 26), we find that the RPA estimate of χ^{′′}(ω) for BaFe_{2}As_{2} and BaFe_{1.9}Ni_{0.1}As_{2} (dashed blue and red lines in Fig. 1b) increases approximately linearly with energy and has absolute values about a factor of three smaller than the observation (Fig. 1e). Although the RPA calculation depends on the Coulomb parameters used, we note that the fiveorbital Hubbard model calculation using and produces essentially similar local magnetic spectra ^{27}. Therefore, a pure RPAtype itinerant model underestimates the absolute spectral weight of the magnetic excitations in iron pnictides.
The solid blue and red lines in Fig. 1b show the calculated local susceptibility using a combined DFT and DMFT in the paramagnetic state. Within DFT+DMFT, χ^{′′}(q,ω) is computed by the Bethe–Salpeter equation using the polarization function χ^{0} and the twoparticle local irreducible vertex function Γ^{irr}(ref. 26). χ^{0} is computed from the interacting oneparticle Green’s function determined by the charge selfconsistent full potential DFT+DMFT method and Γ^{irr} is extracted from the twoparticle vertex function of the auxiliary impurity problem. The latter is defined by the DMFT procedure using the projection of all electronic states to the d character within the iron muffin–tin sphere. By comparing DFT+DMFT and RPA calculations in Fig. 1b with data in Fig. 1e, we see that the former is much closer to the observation. Note that the calculation is done in the paramagnetic state, hence the lowenergy modifications of the spectra due to the long range AF order are not captured in this calculation. RPA can describe only the itinerant part of the electron spectra, whereas DFT+DMFT captures the essential aspects of both the quasiparticles and the local moments of iron formed by strong Hund’s coupling (see Supplementary Information for a more detailed discussion). The improved agreement of DFT+DMFT thus suggests that both the quasiparticles and the local moment aspects of the electrons of iron are needed to obtain the correct intensity and energy distribution of neutron scattering spectra^{26}.
One way to quantitatively compare spin excitations in iron pnictides with those in copper oxides is to estimate their total fluctuating moments, defined as (ref. 12). Based on Fig. 1e, we find that 〈m^{2}〉=3.17±0.16 and 3.2±0.16 μ_{B}^{2} per Fe(Ni) for BaFe_{2}As_{2} and BaFe_{1.9}Ni_{0.1}As_{2}, respectively. Using the formula for magnetic moment of a spin 〈m^{2}〉=(g μ_{B})^{2}S(S+1) (where g=2; ref. 28), we find an effective iron spin S of about 1/2, similar to that of CaFe_{2}As_{2} (ref. 29). These results also show that superconductivity in electrondoped system hardly changes the total size of the fluctuating moment. In the fully localized (insulating) case, the formal Fe^{2+} oxidation state in BaFe_{2}As_{2} would give a 3d^{6} electronic configuration. Hund’s rules would yield S=2 and 〈m^{2}〉=24 μ_{B}^{2} per Fe. This is considerable more than the observed values, suggesting significant hybridization of Fe 3d with pnictide p orbitals and among themselves, which leads to a metallic state where the Hund’s coupling is less important than in the atomic limit^{30}. For comparison, we note that 〈m^{2}〉>1.9 μ_{B}^{2} per Cu for the AF insulating La_{2}CuO_{4} measured over a similar energy range ^{14,15}. From Fig. 1e, we see that the large fluctuating moment 〈m^{2}〉 in iron pnictides arises mostly from highenergy spin excitations that are essentially independent of temperature^{13} and electron doping within the errors of our measurements (Fig. 1). As there are currently no highenergy spin excitation data in absolute units for optimally holedoped Ba_{0.67}K_{0.33}Fe_{2}As_{2} (ref. 31), it is unclear how hole doping BaFe_{2}As_{2} modifies the spinwave spectra.
The DFT+DMFT calculation suggests that both the band structure and the local moment aspects (for example Hund’s coupling) of the iron electrons are needed to obtain a good description of the magnetic response in BaFe_{2}As_{2} and BaFe_{1.9}Ni_{0.1}As_{2}. The weak dependence of the fluctuating moment on electron doping is consistent with the Hund’s metal picture, where electron filling associated with the Fe 3d^{6} electrons by Nidoping is not expected to drastically affect the local moments. What is surprising is that the similarities between the local susceptibilities of the iron pnictides studied here and the parents of the cuprate superconductors. The large fluctuating moment, arising from Hund’s rule coupling, and concentrated at higher energy in iron pnictides, nevertheless gives an imprint on the massive and anisotropic lowenergy quasiparticles^{32}, which form Cooper pairs at low energy. This physics is different from the physics of the doped charge transfer insulators appropriate for copper oxides^{10}, hence the electron correlations in iron pnictides and copper oxides have different microscopic origins, although they are important for understanding the magnetism and superconductivity in both materials.
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Acknowledgements
We thank T. A. Maier, J. P. Hu and T. Xiang for helpful discussions. The work at Knoxville is supported by the US National Science Foundation (NSF) DMR1063866 and OISE0968226. Work at the Insitute of Physics is supported by the Ministry of Science and Technology China 973 programs (2012CB821400, 2011CBA00110) and National Natural Science Foundation of China (NSFC11004233). The work at Rutgers is supported by Department of Energy (DOE) BES DEFG0299ER45761 (G.K.), American Chemical Society Petroleum Research Fund 48802 and the Alfred P. Sloan foundation (K.H.).
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P.D. and M.L. planned neutron scattering experiments. M.L., L.W.H., H.L., R.A.E., T.G. and P.D. carried out neutron scattering experiments. Data analysis was done by M.L. with help from L.W.H., R.A.E., T.G. and S.M.H. The samples were grown by H.L. and coaligned by M.L. and M.W. The DFT and DMFT calculations were done by H.P., K.H. and G.K. The paper was written by P.D., K.H. and G.K. with input from S.M.H. and M.L. All coauthors provided comments on the paper.
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Liu, M., Harriger, L., Luo, H. et al. Nature of magnetic excitations in superconducting BaFe_{1.9}Ni_{0.1}As_{2}. Nature Phys 8, 376–381 (2012). https://doi.org/10.1038/nphys2268
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DOI: https://doi.org/10.1038/nphys2268
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