Anisotropy of incommensurate magnetic excitations in slightly overdoped
  Ba$_{0.5}$K$_{0.5}$Fe$_2$As$_2$ probed by polarized inelastic neutron
  scattering experiments

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Abstract

Polarized neutron scattering experiments on the slightly overdoped superconductor Ba$_{0.5}$K$_{0.5}$Fe$_2$As$_2$ reveal broad magnetic resonance scattering peaking at approximately 15 meV. In spite of doping far beyond the suppression of magnetic order, this compound exhibits dispersive and anisotropic magnetic excitations. At energies below the resonance maximum, magnetic correlations polarized parallel to the layers but perpendicular to the propagation vector are reduced by a factor two compared to those in the two orthogonal directions; in contrast correlations at the peak maximum are isotropic.

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Doping dependence of spin excitations and its correlations with high-temperature superconductivity in iron pnictides

Meng Wang, Chenglin Zhang, Xingye Lu … (2013)

In conventional Bardeen-Cooper-Schrieffer (BCS) superconductors1, superconductivity occurs when electrons form coherent Cooper pairs below the superconducting transition temperature T c. Although the kinetic energy of paired electrons increases in the superconducting state relative to the normal state, the reduction in the ion lattice energy is sufficient to give the superconducting condensation energy (E c=−N(0)Δ2/2 and Δ≈2ħωD , where N(0) is the electron density of states at zero temperature, ħωD is the Debye energy, and V 0 is the strength electron-lattice coupling)1 2 3. For iron pnictide superconductors derived from electron or hole-doping of their antiferromagnetic (AF) parent compounds4 5 6 7 8 9, the microscopic origin for superconductivity is unclear. Although spin excitations arising from quasiparticle excitations between the hole pockets near Γ and electron pockets at M in reciprocal space have been suggested as the microscopic origin for superconductivity10 11, orbital fluctuations may also induce superconductivity in these materials12. Here we use inelastic neutron scattering (INS) to systematically map out energy and wave vector dependence of the spin excitations in electron and hole-doped iron pnictides with different superconducting transition temperatures. By comparing the outcome with previous spin wave measurements on the undoped parent compound BaFe2As2 (ref. 13), we find that high-T c superconductivity only occurs for iron pnictides with low-energy (≤25 meV or ~6.5 k B T c) itinerant electron-spin excitation coupling and high-energy (>100 meV) spin excitations. Since our absolute spin susceptibility measurements for optimally hole-doped iron pnictide reveal that the change in magnetic exchange energy below and above T c 14 15 can account for the superconducting condensation energy, we suggest that the presence of both high-energy spin excitations giving rise to a large magnetic exchange coupling J and low-energy spin excitations coupled to the itinerant electrons are important for high-T c superconductivity in iron pnictides. For BCS superconductors, the superconducting condensation energy E c and T c are controlled by the strength of the Debye energy ħωD and electron-lattice coupling V 0 (refs 1, 2, 3). A material with large ħωD and lattice exchange coupling is a necessary but not a sufficient condition to have high-T c superconductivity. On the other hand, a soft metal with small ħωD (such as lead and mercury) will also not exhibit superconductivity with high-T c. For unconventional superconductors such as iron pnictides, the superconducting phase is derived from hole and electron doping from their AF parent compounds4 5 6 7 8 9. Although the static long-range AF order is gradually suppressed when electrons or holes are doped into the iron pnictide parent compound such as BaFe2As2 (refs 5, 6, 7, 8, 9), short-range spin excitations remain throughout the superconducting phase and are coupled directly with the occurrence of superconductivity16 17 18 19 20 21 22 23 24 25. For spin excitations-mediated superconductors, the superconducting condensation energy should be accounted for by the change in magnetic exchange energy between the normal (N) and superconducting (S) phases at zero temperature. For an isotropic t-J model26, ΔE ex(T)=2J[‹S i+x ·S i ›N−‹S i+x ·S i ›S], where J is the nearest neighbour magnetic exchange coupling and ‹S i+x ·S i › is the dynamic spin susceptibility in absolute units at temperature T 14 15. Since the dominant magnetic exchange couplings are isotropic nearest neighbour exchanges for copper oxide superconductors27 28, the magnetic exchange energy ΔE ex(T) can be directly estimated using the formula through carefully measuring of J and the dynamic spin susceptibility in absolute units between the normal and superconducting states29 30 31. For heavy Fermion superconductor such as CeCu2Si2, one has to modify the formula to include both the nearest neighbour and next nearest neighbour magnetic exchange couplings appropriate for the tetragonal unit cell of CeCu2Si2 to determine ΔE ex(T) (ref. 32). In the case of iron pnictide superconductors5 6 7 8 9, the effective magnetic exchange couplings in their parent compounds are strongly anisotropic along the nearest neighbour a o and b o axis directions of the orthorhombic structure (see inset in Fig. 1a)13 33 34. Although the electron doping induced lattice distortions in iron pnictides35 may affect the effective magnetic exchange couplings36, our INS experiments on optimally electron-doped BaFe1.9Ni0.1As2 indicate that the high-energy spin excitations, which determines the effective magnetic exchange couplings13 33 34, are weakly electron doping-dependent24. Therefore, we can rewrite the relation between the magnetic exchange coupling and the magnetic exchange energy as15 Here the scattering function S(Q,E=ħω) is related to the imaginary part of the dynamic susceptibility χ″(Q,ω) via S(Q,ω)=[1+n(ω,T)]χ″(Q,ω), where [1+n(ω,T)] is the Bose population factor, Q the wave vector, and E=ħω the excitation energy. J 1a is the effective magnetic coupling strength between two nearest sites along the a o direction, while J 1b is that along the b o direction and J 2 is the coupling between the next nearest neighbour sites (see inset in Fig. 1a)13. To determine how high-T c superconductivity in iron pnictides is associated with spin excitations, we consider the phase diagram of electron and hole-doped iron pnictide BaFe2As2 (Fig. 1a)9. In the undoped state, BaFe2As2 forms a metallic low-temperature orthorhombic phase with collinear AF structure as shown in the inset of Fig. 1a. INS measurements have mapped out spin waves throughout the Brillouin zone and determined the effective magnetic exchange couplings13. Upon doping electrons to BaFe2As2 by partially replacing Fe with Ni to induce superconductivity in BaFe2−x Ni x As2 with maximum T c≈20 K at x e=0.1 (ref. 37), the low-energy ( 150 meV), while spin excitations in the hole-doped compound beyond 100 meV are suppressed by shifting the spectral weight to lower energies. This is in qualitative agreement with our absolute intensity measurements (Fig. 1h). The reduction of the high-energy spin spectral weight and its transfer to low energy with hole doping, but not with electron doping, is not naturally explained by the band theory (Fig. 7) and requires models that incorporate both the itinerant quasiparticles and the local moment physics9 42. The hole doping makes the electronic state more correlated, as local moment formation is strongest in the half-filled d 5 shell, and mass enhancement larger thereby reducing the electronic energy scale in the problem. The dashed lines in Fig. 7 show the results of calculated local susceptibility using RPA, which clearly fails to describe the electron- and hole-doping dependence of the local susceptibility. Estimation of the superconductivity-induced magnetic exchange energy Finally, to determine how low-energy spin excitations are coupled to superconductivity in Ba0.67K0.33Fe2As2, we carried out a detailed temperature-dependent study of spin excitation at E=15±1 meV. Comparing with strongly c axis modulated low-energy (E<7 meV) spin excitations18, spin excitations at the resonance energy are essentially 2D without much c axis modulations. In previous work18, we have shown that spin excitations near the neutron spin resonance are longitudinally elongated and change dramatically in intensity across T c. However, these measurements are obtained in arbitrary units and therefore cannot be used to determine the magnetic exchange energy. Figure 8a–d shows the 2D mapping of the resonance at T=25, 38, 40, and 45 K, respectively. While the resonance reveals a clear oval shape at temperatures below T c consistent with earlier work (Fig. 8a,b)18, it changes into an isotropic circular shape abruptly at T c (Fig. 8c,d) as shown by the dashed lines representing full-width-at-half-maximum of the excitations (Supplementary Fig. S7). Temperature dependence of the resonance width along the [H,0] and [1,K] directions in Fig. 8e reveals that the isotropic to anisotropic transition in momentum space occurs at T c. Figure 8f shows temperature dependence of the resonance from 9–40 K, which vanishes at T c. Figure 8g plots temperature dependence of the mode energy together with the sum of the superconducting gaps from the hole and electron pockets40. Figure 8h compares temperature dependence of the superconducting condensation energy43 with superconductivity-induced intensity gain of the resonance. By calculating spin excitations induced changes in magnetic exchange energy using equation (1) (see Methods and Supplementary Fig. S8)15, we find that the difference of magnetic exchange interaction energy between the superconducting and normal state is approximately seven times larger than the superconducting condensation energy43, thus indicating that AF spin excitations can be the major driving force for superconductivity in Ba0.67K0.33Fe2As2. Discussion One way to quantitatively estimate the impact of hole/electron doping and superconductivity to spin waves of BaFe2As2 is to determine the energy dependence of the local moment and total fluctuating moments ‹m 2› (ref. 24). From Fig. 1h, we see that hole-doping suppresses high-energy spin waves of BaFe2As2 and pushes the spectral weight to resonance at lower energies. The total fluctuating moment of Ba0.67K0.33Fe2As2 below 300 meV is ‹m 2›=1.7±0.3 per Fe, somewhat smaller than per Fe for BaFe2As2 and BaFe1.9Ni0.1As2 (ref. 24). For comparison, BaFe1.7Ni0.3As2 and KFe2As2 have ‹m 2›=2.74±0.11 and per Fe, respectively. Therefore, the total magnetic spectral weights for different iron pnictides have no direct correlation with their superconducting T cs. Table 1 summarizes the comparison of effective magnetic exchange couplings, total fluctuating moments, and spin excitation band widths for BaFe2−x Ni x As2 with x e=0,0.1,0.3 and Ba1−x K x Fe2As2 with x h=0.33, 1. From Fig. 1h, we also see that the spectral weight of the resonance and low-energy (<100 meV) magnetic scattering in Ba0.67K0.33Fe2As2 is much larger than that of electron-doped BaFe1.9Ni0.1As2. This is consistent with a large superconducting condensation energy in Ba0.67K0.33Fe2As2 since its effective magnetic exchange coupling J is only ~10% smaller than that of BaFe1.9Ni0.1As2 (Fig. 1h)43 46. For electron-overdoped nonsuperconducting BaFe1.7Ni0.3As2, the lack of superconductivity is correlated with the absence of low-energy spin excitations coupled to the hole and electron Fermi surface nesting even though the effective magnetic exchange couplings remain large40 41. This means that by eliminating [‹S i+x · S i ›N−‹S i+x · S i ›S], there is no magnetic driven superconducting condensation energy, and thus no superconductivity. On the other hand, although the suppression of correlated high-energy spin excitations in KFe2As2 can dramatically reduce the effective magnetic exchange coupling in KFe2As2 (Figs 1e and 6), one can still have superconductivity with reduced T c. If spin excitations are a common thread of the electron pairing interactions for unconventional superconductors15, our results reveal that the large effective magnetic exchange couplings and itinerant electron-spin excitation interactions may both be important ingredients to achieve high-T c superconductivity, much like the large Debye energy and the strength of electron-lattice coupling are necessary for high-T c BCS superconductors. Therefore, our data indicate a possible correlation between the overall magnetic excitation band width, the presence of low-energy spin excitations, and the scale of T c. This suggests that both high-energy spin excitations and low-energy spin excitation itinerant electron coupling are important for high-T c superconductivity. Methods Sample preparation Single crystals of Ba0.67K0.33Fe2As2, KFe2As2, and BaFe1.7Ni0.3As2 are grown using the flux method18 25. The actual crystal compositions were determined using the inductively coupled plasma analysis. We coaligned 19 g of single crystals of Ba0.67K0.33Fe2As2 (with in-plane and out-of-plane mosaic of 4°), 3 g of KFe2As2 (with in-plane and out-of-plane mosaic of ~7.5°), and 40 g of BaFe1.7Ni0.3As2 (with in-plane and out-of-plane mosaic of ~3°). Neutron scattering experiments Our INS experiments were carried out on the MERLIN and MAPS time-of-flight chopper spectrometers at the Rutherford-Appleton Laboratory, UK13 24. Various incident beam energies were used as specified, and mostly with E i parallel to the c axis. To facilitate easy comparison with spin waves in BaFe2As2 (ref. 13), we defined the wave vector Q at (q x , q y , q z ) as (H,K,L)=(q x a o/2π, q y b o/2π, q z c/2π) reciprocal lattice units (rlu) using the orthorhombic unit cell, where a o≈b o=5.57 Å, and c=13.135 Å for Ba0.67K0.33Fe2As2, a o≈b o=5.43 Å, and c=13.8 Å for KFe2As2, and a o=b o=5.6 Å, and c=12.96 Å for BaFe1.7Ni0.3As2. The data are normalized to absolute units using a vanadium standard with 20% errors24 and confirmed by acoustic phonon normalization (see Supplementary Note 1). Supplementary Discussion provides additional data analysis on electron-doped iron pnictides, focusing on the comparison of electron overdoped nonsuperconducting BaFe1.3Ni0.3As2 with optimally electron-doped superconductor BaFe1.9Ni0.1As2 and AF BaFe2As2. DFT+DMFT calculations Our theoretical DFT+DMFT method for computing the magnetic excitation spectrum employs the ab initio full potential implementation of the method, as detailed in ref. 47. The DFT part is based on the code of Wien2k48. The DMFT method requires solution of the generalized quantum impurity problem, which is here solved by the numerically exact continuous-time quantum Monte Carlo method49 50. The Coulomb interaction matrix for electrons on iron atom was determined by the self-consistent GW method in Kutepov et al. 51, giving U=5 eV and J=0.8 eV for the local basis functions within the all electron approach employed in our DFT+DMFT method. The dynamical magnetic susceptibility χ″(Q,E) is computed from the ab initio perspective by solving the Bethe-Salpeter equation, which involves the fully interacting one particle Greens function computed by DFT+DMFT, and the two-particle vertex, also computed within the same method (for details see Park et al. 42). We computed the two-particle irreducible vertex functions of the DMFT impurity model, which coincides with the local two-particle irreducible vertex within the DFT+DMFT method. The latter is assumed to be local in the same basis in which the DMFT self-energy is local, here implemented by projection to the muffin-tin sphere. Calculation of magnetic exchange energy and superconducting condensation energy for Ba0.67K0.33Fe2As2 In a neutron scattering experiment, we measure scattering function S(Q,E=ħω) which is related to the imaginary part of the dynamic susceptibility via S(Q,ω)=[1+n(ω,T)]χ″(Q,ω), where n(ω,T) is the Bose population factor. The magnetic exchange coupling and the imaginary part of spin susceptibility are related via the formula15: where g=2 is the Landé g-factor. Hence, we are able to obtain the change in magnetic exchange energy between the superconducting and normal states by the experimental data of χ″(Q,ω) in both states using equation (1). Strictly speaking, we want to estimate the zero temperature difference of the magnetic exchange energy between the normal and the superconducting states, and use the outcome to compare with the superconducting condensation energy15. Unfortunately, we do not have direct information on the normal state χ″(Q,ω) at zero temperature. Nevertheless, since our neutron scattering measurements at low-energies showed that the χ″(Q,ω) are very similar below and above T c near the AF wave vector Q AF=(1,0,1) and only a very shallow spin gap at Q=(1,0,0) (see Fig. 1f,h in Zhang et al. 18), we assume that there are negligible changes in χ″(Q,ω) above and below T c at zero temperature for energies below 5 meV. For spin excitation energies above 6 meV, the Bose population factors between 7 and 45 K are negligibly small. In previous work on optimally doped YBa2Cu3O6.95 superconductor, we have assumed that spin excitations in the normal state at zero temperature are negligibly small and thus do not contribute to the exchange energy30. The directly measured quantity is the scattering differential cross section where k i and k f are the magnitudes of initial and final neutron momentum and F(Q) is the Fe2+ magnetic form factor, and (γr e )2=0.2905, barn·sr−1. The quantity χ″(Q,E) in both superconducting and normal states can be fitted by a Gaussian for resonance wave vector (1,0) and by cutting the raw data. The outcome is summarized in the Supplementary Table S1, where the unit of E is meV and that of A s(n) is mbarn˙meV−1·sr−1·Fe−1. For the case below 5 meV, we assume that A n decreases to zero linearly with energy and A s=A n (see Fig. 1h in Zhang et al. 18), while the σ keeps the value at 5 meV. The assumption is shown in Supplementary Figure S8, where the resonance is seen at E=15 meV. Because the condensation energy is only defined at zero temperature, we take T=0 in the formula equation (3) and the integral gives: The magnetic exchange coupling constants in an anisotropic model are estimated to be which are 10% smaller than that of BaFe2As2 (ref. 13) and we estimate S to be close to ½ (ref. 24). Hence the exchange energy change is The condensation energy U c for optimally doped Ba0.67K0.33Fe2As2 can be calculated to be from the specific heat data of Popovich et al. 43 Therefore, we have the ratio ΔE ex/U c≈7.4, meaning that the change in the magnetic exchange energy is sufficient to account for the superconducting condensation energy in Ba0.67K0.33Fe2As2. Author contributions This paper contains data from three different neutron scattering experiments in the group of P.D. lead by M.W. (Ba0.67K0.33Fe2As2), C.Z. (KFe2As2), and X.L. (BaFe1.7Ni0.3As2). These authors made equal contributions to the results reported in the paper. For Ba0.67K0.33Fe2As2, M.W., H.L., E.A.G. and P.D. carried out neutron scattering experiments. Data analysis was done by M.W. with help from H.L. and E.A.G. The samples were grown by C.Z., M.W., Y.S., X.L. and coaligned by M.W. and H.L. RPA calculation is carried out by T.A.M. The DFT and DMFT calculations were done by Z.Y., K.H. and G.K. Superconducting condensation energy was estimated by X.Z. For KFe2As2, the samples were grown by C.Z and G.T. Neutron scattering experiments were carried out by C.Z., E.A.G. and P.D. For BaFe1.7Ni0.3As2, the samples were grown by X.L., H.L., and coaligned by. X.Y.L. and M.Y.W. Neutron scattering experiments were carried out by X.L., T.G.P. and P.D. The data are analysed by X.L. The paper was written by P.D., M.W., X.L. and C.L.Z. with input from T.M., K.H. and G.K. All co-authors provided comments on the paper. Additional information How to cite this article: Wang, M. et al. Doping dependence of spin excitations and its correlations with high-temperature superconductivity in iron pnictides. Nat. Commun. 4:2874 doi: 10.1038/ncomms3874 (2013). Supplementary Material Supplementary Information Supplementary Figures S1-S9, Supplementary Tables S1-S2, Supplementary Note 1 and Supplementary Discussion