## Abstract

High-temperature superconductivity in iron pnictides occurs when electrons and holes are doped into their antiferromagnetic parent compounds. Since spin excitations may be responsible for electron pairing and superconductivity, it is important to determine their electron/hole-doping evolution and connection with superconductivity. Here we use inelastic neutron scattering to show that while electron doping to the antiferromagnetic BaFe_{2}As_{2} parent compound modifies the low-energy spin excitations and their correlation with superconductivity (<50 meV) without affecting the high-energy spin excitations (>100 meV), hole-doping suppresses the high-energy spin excitations and shifts the magnetic spectral weight to low-energies. In addition, our absolute spin susceptibility measurements for the optimally hole-doped iron pnictide reveal that the change in magnetic exchange energy below and above *T*_{c} can account for the superconducting condensation energy. These results suggest that high-*T*_{c} superconductivity in iron pnictides is associated with both the presence of high-energy spin excitations and a coupling between low-energy spin excitations and itinerant electrons.

## Introduction

In conventional Bardeen-Cooper-Schrieffer (BCS) superconductors^{1}, 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)Δ^{2}/2 and Δ≈2*ħ*ω_{D}, where *N*(0) is the electron density of states at zero temperature, *ħ*ω_{D} 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 compounds^{4,5,6,7,8,9}, 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 superconductivity^{10,11}, orbital fluctuations may also induce superconductivity in these materials^{12}. 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 BaFe_{2}As_{2} (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 *ħ*ω_{D} and electron-lattice coupling *V*_{0} (refs 1, 2, 3). A material with large *ħ*ω_{D} 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 *ħ*ω_{D} (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 compounds^{4,5,6,7,8,9}. Although the static long-range AF order is gradually suppressed when electrons or holes are doped into the iron pnictide parent compound such as BaFe_{2}As_{2} (refs 5, 6, 7, 8, 9), short-range spin excitations remain throughout the superconducting phase and are coupled directly with the occurrence of superconductivity^{16,17,18,19,20,21,22,23,24,25}. 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* model^{26}, Δ*E*_{ex}(*T*)=2*J*[‹**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 superconductors^{27,28}, the magnetic exchange energy Δ*E*_{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 states^{29,30,31}. For heavy Fermion superconductor such as CeCu_{2}Si_{2}, 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 CeCu_{2}Si_{2} to determine Δ*E*_{ex}(*T*) (ref. 32). In the case of iron pnictide superconductors^{5,6,7,8,9}, 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 pnictides^{35} may affect the effective magnetic exchange couplings^{36}, our INS experiments on optimally electron-doped BaFe_{1.9}Ni_{0.1}As_{2} indicate that the high-energy spin excitations, which determines the effective magnetic exchange couplings^{13,33,34}, are weakly electron doping-dependent^{24}. Therefore, we can rewrite the relation between the magnetic exchange coupling and the magnetic exchange energy as^{15}

Here the scattering function *S*(**Q**,*E*=*ħ*ω) 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*=*ħ*ω 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 BaFe_{2}As_{2} (Fig. 1a)^{9}. In the undoped state, BaFe_{2}As_{2} 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 couplings^{13}. Upon doping electrons to BaFe_{2}As_{2} by partially replacing Fe with Ni to induce superconductivity in BaFe_{2−x}Ni_{x}As_{2} with maximum *T*_{c}≈20 K at *x*_{e}=0.1 (ref. 37), the low-energy (<80 meV) spin waves in the parent compounds are broadened and form a neutron spin resonance coupled to superconductivity^{20,21,22,23}, while high-energy spin excitations are weakly affected^{24}. With further electron doping to *x*_{e}≥0.25, superconductivity is suppressed and the system becomes a paramagnetic metal (Fig. 1a)^{37}. For hole-doped Ba_{1−x}K_{x}Fe_{2}As_{2} (ref. 38), superconductivity with maximum *T*_{c}=38.5 K appears at *x*_{h}≈0.33 (ref. 5) and pure KFe_{2}As_{2} at *x*_{h}=1 is a *T*_{c}=3.1 K superconductor^{6}. In order to determine how spin excitations throughout the Brillouin zone are correlated with superconductivity in iron pnictides, we study optimally hole-doped Ba_{0.67}K_{0.33}Fe_{2}As_{2} (*T*_{c}=38.5 K, Fig. 1c), pure KFe_{2}As_{2} (*T*_{c}=3 K, Fig. 1b), and nonsuperconducting electron-overdoped BaFe_{1.7}Ni_{0.3}As_{2} (Fig. 1d). If spin excitations are responsible for mediating electron pairing and superconductivity, the change in magnetic exchange energy between the normal and superconducting state should be large enough to account for the superconducting condensation energy^{15}.

From density functional theory (DFT) calculations, one finds that Fermi surfaces for the undoped parent compound BaFe_{2}As_{2} consist of hole-like pockets near the Brillouin zone centre and electron pockets near the zone corner (Fig. 2a)^{10}. Figure 2a–d shows the evolution of Fermi surfaces as a function of electron- and hole-doping obtained from the tight-binding model of Graser *et al.*^{39} When electrons are doped into the parent compounds, the hole Fermi surfaces decrease in size while the electron pocket sizes increase (Fig. 2b)^{40}. As a consequence, quasiparticle excitations between the hole and electron Fermi surfaces form transversely elongated spin excitations that increase with increasing electron doping^{23,24,25}. For electron-overdoped iron pnictides, the hole pockets sunk below the Fermi surface (Fig. 2c)^{40}. The absence of interband transition between the hole and electron Fermi surfaces is expected to result in a complete suppression of the low-energy spin excitations at the AF ordering wave vector. This is indeed confirmed by nuclear magnetic resonance experiments on electron overdoped Ba(Fe_{1−x}Co_{x})_{2}As_{2} (ref. 41). For optimally hole-doped iron pnictide Ba_{0.67}K_{0.33}Fe_{2}As_{2} (inset in Fig. 1a), DFT theory based on sign reversed quasiparticle excitations between hole and electron pockets (Fig. 2d) predicts correctly the longitudinally elongated spin excitations from **Q**_{AF}=(1,0)^{23,18}. In addition, INS work on powder^{16,17} and single crystals^{18} of hole-doped Ba_{1−x}K_{x}Fe_{2}As_{2} reveal that the low-energy spin excitations are dominated by a resonance coupled to superconductivity. For pure KFe_{2}As_{2} (*x*_{h}=1), low-energy (<14 meV) spin excitations become longitudinally incommensurate from **Q**_{AF}=(1,0) (inset in Fig. 1a)^{19}. These results, as well as the work on electron-doped iron pnictides^{23,24,25}, have shown that the low-energy spin excitations in iron-based superconductors can be accounted for by itinerant electrons on the hole and electron-nested Fermi surfaces^{9}.

Here we use INS to show that the effect of hole-doping to BaFe_{2}As_{2} is to suppress high-energy spin excitations and transfer the spectral weight to low-energies that couple to the appearance of superconductivity (Fig. 1h). The overall spin excitations spectrum in optimally hole-doped superconducting Ba_{0.67}K_{0.33}Fe_{2}As_{2} is qualitatively consistent with theoretical methods based on DFT and dynamic mean filed theory (DMFT)^{42}. By using the INS measured magnetic exchange couplings and spin susceptibility in absolute units, we calculate the superconductivity-induced lowering of magnetic exchange energy and find it to be about seven times larger than the superconducting condensation energy determined from specific heat measurements for Ba_{0.67}K_{0.33}Fe_{2}As_{2} (ref. 43). These results are consistent with spin excitations-mediated electron pairing mechanism^{15}. For the nonsuperconducting electron-overdoped BaFe_{1.7}Ni_{0.3}As_{2}, we find that while the effective magnetic exchange couplings are similar to that of optimally electron-doped BaFe_{1.9}Ni_{0.1}As_{2} (Fig. 1g)^{24}, the low-energy spin excitations (<50 meV) associated with the hole and electron pocket Fermi surface nesting disappear (Fig. 2c), thus revealing the importance of Fermi surface nesting and itinerant electron-spin excitation coupling to the occurrence of superconductivity (Fig. 1h). Finally, for heavily hole-doped KFe_{2}As_{2} with low-*T*_{c} superconductivity (Fig. 1b), there are only incommensurate spin excitations below ~25 meV possibly due to the mismatched electron-hole Fermi surfaces^{19,44} and the correlated high-energy spin excitations prevalent in electron-doped and optimally hole-doped iron pnictides are strongly suppressed (Fig. 1e), indicating a dramatic softening of effective magnetic exchange coupling (inset Fig. 1h). Therefore, high-*T*_{c} superconductivity is likely associated with two ingredients: a large effective magnetic exchange coupling^{15}, much like the large Debye energy for high-*T*_{c} BCS superconductors, and a strong itinerant electrons-spin excitations coupling from the Fermi surface nesting^{10}, similar to electron-phonon coupling in BCS superconductors.

## Results

## Electron and hole-doping evolution of spin excitations

To substantiate the key conclusions of Fig. 1, we present the two-dimensional (2D) constant-energy images of spin excitations in the (*H*,*K*) plane at different energies for KFe_{2}As_{2} (Fig. 3a–c), Ba_{0.67}K_{0.33}Fe_{2}As_{2} (Fig. 3d–f), and BaFe_{1.7}Ni_{0.3}As_{2} (Fig. 3g–i) above *T*_{c}. In previous INS work on KFe_{2}As_{2}, longitudinal incommensurate spin excitations were found by triple axis spectrometer measurements for energies from 3 to 14 meV in the normal state^{19}. While we confirmed the earlier work using time-of-flight INS for energies below *E*=15±1 meV (Fig. 3a,b), our new data collected at higher excitation energies reveal that incommensurate spin excitations converge into a broad spin excitation near *E*=20 meV and disappear for energies above 25 meV (Fig. 3c; Supplementary Fig. S1). For Ba_{0.67}K_{0.33}Fe_{2}As_{2}, spin excitations at *E*=5±1 meV are longitudinally elongated from **Q**_{AF} as expected from the DFT calculations (Fig. 3d)^{18,23}. At the resonance energy (*E*=15±1 meV)^{16}, spin excitations are isotropic above *T*_{c} (Fig. 3e). On increasing energy further to *E*=50±10 meV, spin excitations change to transversely elongated from **Q**_{AF} similar to spin excitations in optimally electron-doped superconductor BaFe_{1.9}Ni_{0.1}As (Fig. 3f)^{24}. Figure 3g–i summarizes similar 2D constant-energy images of spin excitations for nonsuperconducting BaFe_{1.7}Ni_{0.3}As. At *E*=9±3 (Fig. 3g) and 30±10 meV (Fig. 3h), there are no correlated spin excitations near the **Q**_{AF}. Upon increasing energy to *E*=59±10 meV (Fig. 3i), we see clear spin excitations transversely elongated from **Q**_{AF} (Fig. 3i; Supplementary Fig. S2). To further illustrate the presence of a large spin gap in BaFe_{1.7}Ni_{0.3}As, we compare spin waves in BaFe_{2}As_{2}^{13} and paramagnetic spin excitations in BaFe_{1.7}Ni_{0.3}As. Figure 4a,b shows the background subtracted spin wave scattering of BaFe_{2}As_{2} for the *E*_{i}=250, 450 meV data, respectively, projected in the wave vector (**Q**=[1,*K*]) and energy space at 7 K^{13}. Sharp spin waves are seen to stem from the AF ordering wave vector **Q**_{AF}=(1,0) above the ~15 meV spin gap^{45}. Figure 4c,d shows identical projections for spin excitations of BaFe_{1.7}Ni_{0.3}As at 5 K. A large spin gap of ~50 meV is clearly seen in the data near **Q**_{AF}=(1,0). A detailed comparison of spin excitations in BaFe_{2−x}Ni_{x}As_{2} with *x*_{e}=0, 0.1, 0.3 is made in Supplementary Figs S3 and S4.

Figure 5a–d shows 2D images of spin excitations in BaFe_{1.7}Ni_{0.3}As_{2} at *E*=70±10, 112±10, 157±10, and 214±10 meV, respectively. Figure 5e–h shows wave vector dependence of spin excitations at energies *E*=70±10, 115±10, 155±10 and 195±10 meV, respectively, for Ba_{0.67}K_{0.33}Fe_{2}As_{2}. Similar to spin waves in BaFe_{2}As_{2} (ref. 13), spin excitations in BaFe_{1.7}Ni_{0.3}As_{2} and Ba_{0.67}K_{0.33}Fe_{2}As_{2} split along the *K*-direction for energies above 80 meV and form rings around **Q**=(±1,±1) positions near the zone boundary, albeit at slightly different energies. Comparing spin excitations in Fig. 5a–d for BaFe_{1.7}Ni_{0.3}As_{2} with those in Fig. 5e–h for Ba_{0.67}K_{0.33}Fe_{2}As_{2} in absolute units, we see that spin excitations in BaFe_{1.7}Ni_{0.3}As_{2} extend to slightly higher energies and have larger intensity above 100 meV.

As discussed in the spin wave measurements of BaFe_{2}As_{2} (ref. 13), the magnon band top energy at **Q**=(1,1) governs the effective magnetic exchange couplings *J* (*J*_{1a}, *J*_{1b}, and *J*_{2}). To estimate the change of *J* for hole-doped Ba_{0.67}K_{0.33}Fe_{2}As_{2}, we calculate the energy cut at **Q**=(1,1) by exploring the Heisenberg Hamiltonian of the parent compound. It turns out that *J*_{1a}, *J*_{1b}, and *J*_{2} have comparable effect on the band top. On the basis of the dispersion of Ba_{0.67}K_{0.33}Fe_{2}As_{2}, the effective magnetic exchange *J* is found to be about 10% smaller for Ba_{0.67}K_{0.33}Fe_{2}As_{2} compared with that of BaFe_{2}As_{2} (Fig. 6). For comparison, if we assume the band top for KFe_{2}As_{2} is around *E*=25 meV, the effective magnetic exchange should be about 90% smaller for KFe_{2}As_{2}. Of course, we know this is not an accurate estimation since spin excitations in KFe_{2}As_{2} are incommensurate and have an inverse dispersion. In any case, given the zone boundary energy of *E*≈25 meV (Supplementary Fig. S1), the effective magnetic exchange couplings in KFe_{2}As_{2} must be much smaller than that of BaFe_{2}As_{2}.

## Theoretical calculations of spin excitations

To understand the wave vector dependence of spin excitations in hole-doped Ba_{0.67}K_{0.33}Fe_{2}As_{2}, we have carried out the random phase approximation (RPA) calculation of the dynamic susceptibility in a pure itinerant electron picture using the method described before^{25}. Figure 5i,j shows RPA calculations of spin excitations at *E*=70 and 155 meV, respectively, for Ba_{0.67}K_{0.33}Fe_{2}As_{2} assuming that hole doping induces a rigid band shift^{25}. While a pure RPA type itinerant model can explain longitudinally elongated spin excitations at low-energies^{18}, it clearly fails to describe the transversely elongated spin excitations in hole-doped iron pinctides at high energies (Fig. 5e,g). For comparison with the RPA calculation, we also used a combined DFT and DMFT approach^{24,42} to calculate the imaginary part of the dynamic susceptibility χ^{″}(**Q**,ω) in the paramagnetic state. Figure 5k,i shows calculated spin excitations at *E*=70 and 155 meV, respectively. Although the model still does not agree in detail with the data in Fig. 5e,g, it captures the trend of spectral weight transfer away from **Q**_{AF}=(1,0) on increasing energy and forming a pocket centred at **Q**=(1,1).

## Dispersions of spin excitations and local dynamic susceptibility

By carrying out cuts through the 2D images similar to Figs 3d–f and 5e–h along the [1,*K*] and [*H*,0] directions (Supplementary Figs S5 and S6), we establish the spin excitation dispersion along the two high symmetry directions for Ba_{0.67}K_{0.33}Fe_{2}As_{2} and compare with the dispersion of BaFe_{2}As_{2} (Fig. 1f)^{13}. In contrast to the dispersion of electron-doped BaFe_{1.9}Ni_{0.1}As_{2} (ref. 24), we find clear softening of the zone boundary spin excitations in hole-doped Ba_{0.67}K_{0.33}Fe_{2}As_{2} from spin waves in BaFe_{2}As_{2} (ref. 13). We estimate that the effective magnetic exchange coupling in Ba_{0.67}K_{0.33}Fe_{2}As_{2} is reduced by about 10% from that of BaFe_{2}As_{2} (Fig. 6). Similarly, Fig. 1g shows the dispersion curve of BaFe_{1.7}Ni_{0.3}As_{2} along the [1,*K*] direction plotted together with that of BaFe_{2}As_{2} (ref. 13). For energies below ~50 meV, spin excitations are completely gapped marked in the dashed area probably due to the missing hole-electron Fermi pocket quasiparticle excitations^{40}. On the basis of the 2D spin excitation images similar to Fig. 3a–c; we plot in Fig. 1e the dispersion of incommensurate spin excitations in KFe_{2}As_{2}. The incommensurability of spin excitations is weakly energy dependent below *E*=12 meV but becomes smaller with increasing energy above 12 meV. Correlated spin excitations for energies above 25 meV are suppressed as shown in the shaded area in Fig. 1e.

To quantitatively determine the effect of electron and hole doping on the overall spin excitations spectra, we calculate the local dynamic susceptibility per formula unit (f.u.) in absolute units, defined as χ^{″}(*ω*)=∫χ^{″}(**q**,ω)*d***q**/∫*d***q** (ref. 24), where , at different energies for Ba_{0.67}K_{0.33}Fe_{2}As_{2}, BaFe_{1.7}Ni_{0.3}As_{2}, and KFe_{2}As_{2}. Figure 1h shows the outcome together with previous data on optimally electron-doped superconductor BaFe_{1.9}Ni_{0.1}As_{2} (ref. 24). While electron doping up to BaFe_{1.7}Ni_{0.3}As_{2} does not change much the spectral weight of high-energy spin excitations from that of BaFe_{1.9}Ni_{0.1}As_{2}, hole-doping dramatically suppresses the high-energy spin excitations and shift the spectral weight to lower energies (Fig. 1h). For heavily hole-doped KFe_{2}As_{2}, spin excitations are mostly confined to energies below about *E*=25 meV (inset in Fig. 1h).

The solid green, red, black, and blue lines in Fig. 7 show calculated local susceptibility in absolute units based on a combined DFT and DMFT approach for KFe_{2}As_{2}, Ba_{0.67}K_{0.33}Fe_{2}As_{2}, BaFe_{2}As_{2}, and BaFe_{1.9}Ni_{0.1}As_{2}^{24}, respectively. This theoretical method predicts that electron doping to BaFe_{2}As_{2} does not affect the spin susceptibility at high energy (*E*>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 physics^{9,42}. 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 Ba_{0.67}K_{0.33}Fe_{2}As_{2}, 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 excitations^{18}, spin excitations at the resonance energy are essentially 2D without much *c* axis modulations. In previous work^{18}, 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 pockets^{40}. Figure 8h compares temperature dependence of the superconducting condensation energy^{43} 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 energy^{43}, thus indicating that AF spin excitations can be the major driving force for superconductivity in Ba_{0.67}K_{0.33}Fe_{2}As_{2}.

## Discussion

One way to quantitatively estimate the impact of hole/electron doping and superconductivity to spin waves of BaFe_{2}As_{2} 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 BaFe_{2}As_{2} and pushes the spectral weight to resonance at lower energies. The total fluctuating moment of Ba_{0.67}K_{0.33}Fe_{2}As_{2} below 300 meV is ‹*m*^{2}›=1.7±0.3 per Fe, somewhat smaller than per Fe for BaFe_{2}As_{2} and BaFe_{1.9}Ni_{0.1}As_{2} (ref. 24). For comparison, BaFe_{1.7}Ni_{0.3}As_{2} and KFe_{2}As_{2} 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*_{c}s. Table 1 summarizes the comparison of effective magnetic exchange couplings, total fluctuating moments, and spin excitation band widths for BaFe_{2−x}Ni_{x}As_{2} with *x*_{e}=0,0.1,0.3 and Ba_{1−x}K_{x}Fe_{2}As_{2} 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 Ba_{0.67}K_{0.33}Fe_{2}As_{2} is much larger than that of electron-doped BaFe_{1.9}Ni_{0.1}As_{2}. This is consistent with a large superconducting condensation energy in Ba_{0.67}K_{0.33}Fe_{2}As_{2} since its effective magnetic exchange coupling *J* is only ~10% smaller than that of BaFe_{1.9}Ni_{0.1}As_{2} (Fig. 1h)^{43,46}. For electron-overdoped nonsuperconducting BaFe_{1.7}Ni_{0.3}As_{2}, 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 large^{40,41}. 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 KFe_{2}As_{2} can dramatically reduce the effective magnetic exchange coupling in KFe_{2}As_{2} (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 superconductors^{15}, 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 Ba_{0.67}K_{0.33}Fe_{2}As_{2}, KFe_{2}As_{2}, and BaFe_{1.7}Ni_{0.3}As_{2} are grown using the flux method^{18,25}. The actual crystal compositions were determined using the inductively coupled plasma analysis. We coaligned 19 g of single crystals of Ba_{0.67}K_{0.33}Fe_{2}As_{2} (with in-plane and out-of-plane mosaic of 4°), 3 g of KFe_{2}As_{2} (with in-plane and out-of-plane mosaic of ~7.5°), and 40 g of BaFe_{1.7}Ni_{0.3}As_{2} (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, UK^{13,24}. 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 BaFe_{2}As_{2} (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 Ba_{0.67}K_{0.33}Fe_{2}As_{2}, *a*_{o}≈*b*_{o}=5.43 Å, and *c*=13.8 Å for KFe_{2}As_{2}, and *a*_{o}=*b*_{o}=5.6 Å, and *c*=12.96 Å for BaFe_{1.7}Ni_{0.3}As_{2}. The data are normalized to absolute units using a vanadium standard with 20% errors^{24} 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 BaFe_{1.3}Ni_{0.3}As_{2} with optimally electron-doped superconductor BaFe_{1.9}Ni_{0.1}As_{2} and AF BaFe_{2}As_{2}.

## 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 Wien2k^{48}. The DMFT method requires solution of the generalized quantum impurity problem, which is here solved by the numerically exact continuous-time quantum Monte Carlo method^{49,50}. 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 Ba_{0.67}K_{0.33}Fe_{2}As_{2}

In a neutron scattering experiment, we measure scattering function *S*(**Q**,*E*=*ħ*ω) 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 formula^{15}:

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 energy^{15}. 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 YBa_{2}Cu_{3}O_{6.95} 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 energy^{30}.

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 Fe^{2+} magnetic form factor, and (γ*r*_{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 BaFe_{2}As_{2} (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 Ba_{0.67}K_{0.33}Fe_{2}As_{2} can be calculated to be

from the specific heat data of Popovich *et al.*^{43} Therefore, we have the ratio Δ*E*_{ex}/*U*_{c}≈7.4, meaning that the change in the magnetic exchange energy is sufficient to account for the superconducting condensation energy in Ba_{0.67}K_{0.33}Fe_{2}As_{2}.

## 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).

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## Acknowledgements

The single crystal growth and neutron scattering work at Rice/UTK is supported by the US DOE BES under Grant No. DE-FG02-05ER46202. Work at IOP is supported by the MOST of China 973 programs (2012CB821400, 2011CBA00110) and NSFC. The LDA+DMFT computations were made possible by an Oak Ridge leadership computing facility director discretion allocation to Rutgers. The work at Rutgers is supported by DOE BES DE-FG02-99ER45761 (G.K.) and NSF-DMR 0746395 (K.H.). T.A.M. acknowledges the Center for Nanophase Materials Sciences, which is sponsored at ORNL by the Scientific User Facilities Division, BES, US DOE.

## Author information

## Author notes

- Meng Wang
- , Chenglin Zhang
- & Xingye Lu

These authors contributed equally to this work

## Affiliations

### Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

- Meng Wang
- , Xingye Lu
- , Huiqian Luo
- , Xiaotian Zhang
- & Pengcheng Dai

### Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA

- Chenglin Zhang
- , Yu Song
- & Pengcheng Dai

### Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996-1200, USA

- Chenglin Zhang
- , Xingye Lu
- , Guotai Tan
- , Yu Song
- , Miaoyin Wang
- & Pengcheng Dai

### ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, UK

- E.A. Goremychkin
- & T.G. Perring

### Center for Nanophase Materials Sciences and Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6494, USA

- T.A. Maier

### Department of Physics, Rutgers University, Piscataway, New Jersey 08854, USA

- Zhiping Yin
- , Kristjan Haule
- & Gabriel Kotliar

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## Contributions

This paper contains data from three different neutron scattering experiments in the group of P.D. lead by M.W. (Ba_{0.67}K_{0.33}Fe_{2}As_{2}), C.Z. (KFe_{2}As_{2}), and X.L. (BaFe_{1.7}Ni_{0.3}As_{2}). These authors made equal contributions to the results reported in the paper. For Ba_{0.67}K_{0.33}Fe_{2}As_{2}, 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 KFe_{2}As_{2}, 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 BaFe_{1.7}Ni_{0.3}As_{2}, 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.

## Competing interests

The authors declare no competing financial interests.

## Corresponding author

Correspondence to Pengcheng Dai.

## Supplementary information

## PDF files

- 1.
### Supplementary Information

Supplementary Figures S1-S9, Supplementary Tables S1-S2, Supplementary Note 1 and Supplementary Discussion

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