Introduction

Spin-mediated superconductivity is one of the plausible models explaining the formation of Cooper pairs in iron-based superconductors. To investigate this hypothesis, magnetism has been intensively studied. The neutron scattering technique is a powerful method of examining spin fluctuations, because it can clarify both the energy and momentum dependences over the entire Brillouin zone. Revealing the overall spectrum of spin fluctuation is essential for understanding their magnetism and the mechanism of spin-mediated superconductivity.

The magnetism dependence of superconductivity should be demonstrated to prove that the superconductivity is due to spin fluctuations. AFe2As2 (A = Ba, Sr or Ca) is one of ideal systems for this purpose, because its superconductivity can be induced in several ways: electron doping1,2, hole doping3, chemical pressure4 and external pressure5. Systematic and comparative studies among those samples can solve the problem.

The antiferromagnetic (AF) long-range ordering commonly observed in the parent compounds disappears and Tc increases by applying pressure or upon doping in the underdoped region5,6. Apparently, AF long-range ordering competes against superconductivity. In the overdoped region, on the other hand, it is unclear why Tc decreases and the superconductivity disappears with increasing doping. Inelastic neutron scattering (INS) studies on electron-doped Ba(Fe,Ni)2As2 have clarified that the spin fluctuations at high-energy region (>100 meV) are insensitive to doping7 whereas those at low-energy region (<50 meV) disappear as Tc decreases in the overdoped region8. It has been demonstrated that low-energy spin fluctuations are correlated with superconductivity. For hole-doped (Ba,K)Fe2As2, on the other hand, low-energy spin fluctuations remain even in heavily overdoped KFe2As2 below E = 20 meV8,9. Thus, why Tc decreases upon hole doping in the hole-overdoped region remains as an unanswered question.

INS studies on the spin fluctuations of non-doped AFe2As2 (A = Ba, Sr or Ca) over the entire Brillouin zone have clarified that the spin dispersion can be well described by the J1-J2 model. However, it is unclear whether the model, which is based on a localized spin picture, is valid because carriers show an itinerant character. In fact, many calculations based on itinerant models have been proposed to explain the spin spectra10,11,12. Some models, for example based on the combination of density functional theory and dynamical mean-field theory, attempt to involve both itinerant and localized characters13,14. To establish a definitive model of magnetism, further examination of spin fluctuations is required.

Although systematic studies on spin fluctuations of electron-doped Ba(Fe,Ni)2As2 have been reported7,15, the spin fluctuations of hole-overdoped samples over the entire Brillouin zone have not yet been established. In previous INS experiments on KFe2As2, spin fluctuations were observed up to E = 20 meV, which is halfway to the zone boundary8,9. In optimum-doped (Ba,K)Fe2As2, a conflict between INS and resonant x-ray inelastic scattering (RIXS) has been found. The spin dispersion is robust upon doping according to the results of INS8 whereas softening has been observed in RIXS experiments16. To solve this problem, further study of the hole doping dependence is essential. We, thus, report the overall spectra of spin fluctuations in overdoped Ba1-xKxFe2As2 obtained by the INS technique.

Results

Figure 1(a–e) show the observed two-dimensional constant-energy images of spin excitations in the case of Ba0.5K0.5Fe2As2. Figure 1(f–j) show the results of calculations with peak positions derived from the Gaussian fitting of constant-energy spectra. We describe the (H, K) plane with orthorhombic notation, even though superconducting Ba1-xKxFe2As2 has a tetragonal crystal structure to facilitate comparison with non-doped BaFe2As2. Clear incommensurate peaks appeared around the (±1, 0) and (0, ±1) splitting along the longitudinal direction with a wave vector of (±2δ, 0) and (0, ±2δ), respectively, where δ = 0.06 at E = 13 meV. The (1 ± 2δ, 0) position corresponds to [π(1 ± 2δ, 0)] in the ab plane and (0.5 ± δ, 0.5 ± δ) in tetragonal notation. As the energy increases, spin excitations start to split along the transverse direction corresponding to (±1 ± 2δ, K) or (H, ±1 ± 2δ) and reach the magnetic zone boundary with merging signals from next zone boundaries. In contrast, magnetic excitations along the longitudinal direction corresponding to (±1 ± 2δ + H, 0) or (0, ±1 ± 2δ + K) are strongly damped, consistent with previous observations for the BaFe2As2 system17. Figure 1(k,l) show the dispersion cuts along the transverse direction (1, K). A clear spin wave-like dispersion was observed up to E = 200 meV, where it reaches the zone boundary.

Figure 1
figure 1

Constant-energy images of spin excitations of Ba0.5K0.5Fe2As2 at T = 6 K at (a) E = 13 ± 0.5 meV obtained with incident neutron energy of Ei = 31meV; (b) E = 40 ± 1 meV with Ei = 65 meV; (c) E = 80 ± 3 meV with Ei = 202 meV; (d) E = 110 ± 5 meV with Ei = 202 meV; (e) E = 200 ± 10 meV with Ei = 409 meV. The color bars represent the vanadium-normalized absolute intensity (d2σ/(dΩdE)) in units of mbarn sr−1 meV−1 f.u.−1. (f–j) Simulated pictures of net intensity derived from obtained fitting parameters (peak intensity, peak position, FWHM and background) corrected by the magnetic form factor. The dashed box indicates the AF zone boundaries. (k,l) Dispersion cuts of Ba0.5K0.5Fe2As2 with Ei = 202 and 409 meV, respectively, along the K direction. The solid lines are a guide to the eye. (m) Constant-energy cuts of spin excitations along the K direction as a function of energy. The blue arrows represent peak positions of magnetic signals. Peaks observed on both sides without the blue arrows are magnetic signals originating from another magnetic Brillouin zone. Solid and dotted lines are the results of Gaussian fitting and the magnetic zone boundaries, respectively.

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Figure 2(a–j) show two-dimensional constant-energy images of spin excitations and the results of calculations for x = 1. Well-defined incommensurate peaks with incommensurability larger than that for x = 0.5 are observed at E = 5meV. With increasing energy, spin excitations split along the transverse direction, similarly to the case of x = 0.5. The dispersion cut along the transverse direction (0.68, K) shows that the spin excitations reach the zone boundary around E = 80 meV (Fig. 2(k)). Nevertheless, magnetic signals exist even considerably above E = 80 meV with a vertical dispersion exhibiting a chimney-like structure (Fig. 2(l,m)). The energy-constant cuts along the K direction clearly show that the magnetic signals extend up to E ~ 200 meV (Fig. 2(m)).

Figure 2
figure 2

Constant-energy images of spin excitations of KFe2As2 at T = 6K at (a) E = 5 ± 0.5 meV obtained with Ei = 30 meV; (b) E = 15 ± 0.5 meV with Ei = 30 meV; (c) E = 38 ± 1 meV with Ei = 75 meV; (d) E = 76 ± 2 meV with Ei = 149 meV; (e) E = 160 ± 20 meV with Ei = 423 meV. (f–j) Simulated pictures of net intensity derived from obtained fitting parameters (peak intensity, peak position, FWHM and background) corrected by the magnetic form factor. The dashed box indicates the AF zone boundaries. (k) Dispersion cut of KFe2As2 with Ei = 149 meV along K direction. The solid lines are guide to the eye. The vertical white dashed lines depict the magnetic zone boundaries. (l) High-energy spin excitation of KFe2As2 with Ei = 423 meV. (m) Variation in the scattered intensity along the K direction for different excitation energies. The blue arrows represent peak positions of magnetic signals. Peaks observed on both sides without the blue arrows are magnetic signals originating from another magnetic Brillouin zone. Solid and dotted lines depict the results of Gaussian fitting and the magnetic zone boundaries, respectively.

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Next, we overview the overall spin dispersion of Ba1-xKxFe2As2. Figure 3(a–c) show the spin excitation dispersions for x = 0.5 and 1 at T = 6 K derived from Gaussian fitting of the constant-energy spectra with those of the non- and underdoped samples. In x = 0.5, a spin wave dispersion is observed up to E = 200 meV, similarly to the cases of x = 0 and 0.33 (Fig. 3(a)). In x = 1, on the other hand, the dispersive spin excitations reach the zone boundary around E = 80 meV, which is considerably lower than the energy for x = 0.5 (Fig. 3(b)). Instead, a vertical dispersion with a chimney-like structure was observed from E = 80 meV up to 200 meV. In x = 0.5, signals of the chimney-like structure can also be found above E = 200 meV, but they are less clear than those in x = 1 (Fig. 1(m)).

Figure 3
figure 3

Dispersions of spin excitations for Ba0.5K0.5Fe2As2 and KFe2As2 along transverse direction.

(a–c) Spin excitation dispersions of Ba0.5K0.5Fe2As2 (red) and KFe2As2 (blue) at T = 6K plotted as filled circles. The horizontal and vertical lengths of the green rectangles represent the full width at half maximum of the Gaussian fits of the constant-energy spectrum and the energy range in the fitting, respectively. Scan directions are depicted by the arrows in the insets. The dashed and dotted lines in (c) are dispersion of BaFe2As2 at T = 7K (ref. 17) and Ba0.67K0.33Fe2As2 at T = 9 K (ref. 8), respectively. The vertical dashed lines depict the magnetic zone boundaries for Ba0.5K0.5Fe2As2. (d) Energy dependence of ∫χ”(q, 3ω)dq for Ba0.5K0.5Fe2As2 (filled red circles) and KFe2As2 (filled blue circles) at T = 6 K. The solid lines are a guide to the eye. The black dashed lines depict ∫χ”(q, ω)dq for BaFe2As2 at T = 5 K (ref. 15), Ba0.67K0.33Fe2As2 at T = 9 K (ref. 8) and BaFe1.82Ni0.18As2 (ref. 7) at T = 5 K, respectively.

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Figure 3(d) shows the energy dependence of the dynamical magnetic susceptibility ∫χ”(q, ω)dq for x = 0.5 and 1 at T = 6 K. ∫χ”(q, ω)dq for x = 0, 0.33 (Tc = 38.5 K) and BaFe2-yNiyAs2 (y = 0.18, Tc = 8 K) reported in7,8,15 are also depicted for comparison. It can be seen that ∫χ”(q, ω)dq for x = 0.5 exhibits essentially equivalent behavior to that for x = 0.33. Compared with the case of x = 0, on the other hand, the signals in the high-energy region are much lower for x = 0.5, while the peak energy remains around E = 150~200 meV. For x = 1, ∫χ”(q, ω)dq above E = 100 meV is further low, with the peak energy decreasing to around E = 30 meV. The large reduction in the high-energy spin fluctuations with hole doping results in suppression of the total fluctuating moment, which has been estimated to be <m2> = 1.45 and 0.65 μB2/Fe, for x = 0.5 and 1, respectively (Fig. 4). In contrast, ∫χ”(q, ω)dq in the low-energy region is almost independent of the doping level except for the sharp peak attributed to the spin resonance. Thus, the suppression of superconductivity in the hole-overdoped region cannot be due to a decrease in low-energy magnetic intensity as for electron-doped Ba(Fe,Ni)2As27,8.

Figure 4
figure 4

Phase diagram of Ba1-xKxFe2As2.

Symbols denote Tc (open circles, ref. 28), The Néel temperature TN (open squares, ref. 6), the total fluctuating magnetic moment <m2> (closed circles), band width of spin wave W (closed squares), and the incommensurability δ (closed diamonds, ref. 24).

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Discussion

The present observations demonstrate that the energy scale of the dispersive spin wave is robust upon hole doping up to x = 0.5, which is followed by a rapid decrease up to x = 1 (Fig. 4). The decrease appears to be related to the appearance of the incommensurate spin structure. In fact, the band width is robust in electron-doped Ba(Fe,Co,Ni)2As2, which exhibits a commensurate spin structure except in the incommensurate AF state, which appears in a narrow doping range and has one-order smaller incommensurability than that of KFe2As27,18. The smaller band width of the spin wave leads to weaker effective magnetic exchange coupling J according to the Heisenberg model. The results, thus, suggest that J is correlated with the periodicity of spin fluctuations. The small value of J in KFe2As2 is consistent with the fact that its electronic interaction strength U is quite large19,20.

The chimney-like structure can originate from particle-hole excitations, which define the itinerant character of spin fluctuations. Note that the chimney-like structure resembles the spin excitations in the itinerant AF metals Cr21, Cr0.95V0.0522 and Mn2.8Fe0.2Si23. The present results show that the band width decreases and the chimney-like dispersion appears with hole doping. This is qualitatively consistent with DFT + DMFT calculations13,14, which also supports the origin of the chimney-like dispersion to be particle-hole excitations.

The present observation of a large reduction in high-energy spin fluctuations upon hole doping is in contrast to the case of electron-doped Ba(Fe,Ni)2As2, where high-energy spin fluctuations are independent of doping7. Because hole-doped (Ba,K)Fe2As2 exhibits a higher maximum Tc of 38 K than electron-doped Ba(Fe,Ni)2As2 (Tc = 20 K)2 even though (Ba,K)Fe2As2 exhibits weaker spin fluctuations in the high-energy region, this reduction of the high-energy spin fluctuations does not appear to suppress Tc. The suppression of Tc in hole-overdoped (Ba,K)Fe2As2 can rather be attributed to the reduction of J, which remains almost constant from non-doped to optimum-doped region and followed by rapid reduction in the overdoped region. The J dependence of the superconductivity has also been suggested in studies on spin resonance24,25,26. Stronger magnetic correlation leads to a larger energy split between the resonance and the superconducting gap energy. In fact, the resonance energy in overdoped (Ba,K)Fe2As2 approaches the superconducting gap energy with doping up to x = 0.7724, which can result from the reduction of J. These results lead to the conclusion that there is a strong relationship between magnetism and superconductivity in Ba1-xKxFe2As2.

Method

Single crystals of Ba0.5K0.5Fe2As2 (Tc = 36 K) and KFe2As2 (Tc = 3.4 K) were grown by a KAs self-flux method27,28. The magnetic susceptibility was measured by a SQUID from Quantum Design.

The INS measurement was performed using the Fermi chopper spectrometer 4SEASONS in J-PARC29. We co-aligned 160 and 300 pieces of single crystal with x = 0.5 (~5 g) and x = 1 (~5 g), respectively. We employed the multi-Ei method30 with incident neutron energies of Ei = 31, 65, 110, 202, 409, 720 meV for Ba0.5K0.5Fe2As2 and Ei = 30, 75, 149, 423 meV for KFe2As2. The incident beam was parallel to the c-axis. We converted signal intensities into absolute units using a vanadium standard. The data were processed by the “Utsusemi” visualization software developed at J-PARC31. Throughout this letter, wave vectors are specified in the orthorhombic reciprocal lattice.

Additional Information

How to cite this article: Horigane, K. et al. Spin excitations in hole-overdoped iron-based superconductors. Sci. Rep. 6, 33303; doi: 10.1038/srep33303 (2016).