Main

Since the discovery of static antiferromagnetic order (with a spin structure as in Fig. 1a) in the parent compounds of iron pnictide superconductors5,6, much effort has been focused on understanding the role of spin dynamics in the superconductivity of these materials11,12,13,14,15,16,17,18,19,20. A determination of the effective magnetic exchange coupling and ground-state Hamiltonian in the parent compounds of these materials is important because such an understanding will provide the basis against which superconductivity-induced changes can be identified. Using inelastic neutron scattering, we have measured the dispersion of spin-wave excitations in CaFe2As2 (refs 26, 27), one of the parent compounds of the FeAs-based superconductors, and determined the effective magnetic exchange interactions. If the static long-range antiferromagnetic order shown in Fig. 1a for the parent compounds of iron-based superconductors originates from a collective spin-density-wave order instability of itinerant electrons like in chromium, the velocity of spin-wave excitations c should be c=(vevh/3)1/2, where ve and vh are the electron and hole Fermi velocity, respectively9. Furthermore, spin-wave excitations should exhibit longitudinal and transverse polarization, and damp into single-particle excitations (Stoner continuum) through the transfer of an electron (spin) from the majority to the minority band at high energies as shown schematically in Fig. 1c (ref. 10). On the other hand, if magnetic order in iron pnictides has a local moment origin as in the parent compounds of the copper oxides1, one should observe well-defined (essentially instrumental resolution limited) spin waves throughout the Brillouin zone and magnetic coupling between local moments should be dominated by direct and super-exchange interactions (Fig. 1d)11,12,13,14,15. In recent neutron scattering experiments, the presence of itinerant magnetic excitations and a Stoner continuum have been suggested in BaFe2As2(ref. 24) and CaFe2As2 (ref. 25). Whereas low-energy spin waves in CaFe2As2 can be described by a classical Heisenberg Hamiltonian, a Stoner line broadening was reported to develop above 100 meV (or wave vector Q=(1.2,0,1) reciprocal lattice units (r.l.u.) or ∼0.2 r.l.u. in reduced wave vector from the zone centre (1,0,1)) with no localized spin waves near the zone boundary25. Furthermore, the authors find that a Heisenberg Hamiltonian with effective in-plane nearest-neighbours (Fig. 1a, J1a and J1b), next-nearest-neighbour (Fig. 1a, J2) and out-of-plane (Fig. 1a, Jc) exchange couplings of S(J1a+J1b)=44, S J2=31±3 and S Jc=4.5±1 meV (where spin S=1) can best describe spin waves of CaFe2As2 below 100 meV (ref. 25). Although these results are interesting, they are similar to earlier work21,22,23,24 and have not determined the effective ground-state Hamiltonian because the signs of the effective change coupling constants (Fig. 1a, J1a and J1b) can be determined only by zone-boundary spin-wave data, which are lacking in ref. 25. A correct determination of all exchange coupling constants (J1a, J1b and so on) is important because it enables the formation of an appropriate ground-state Hamiltonian from which superconductivity can be derived.

Figure 1: Magnetic structure, calculated spin-wave dispersion and wave-vector dependence of spin-wave excitations at different energies for CaFe2As2.
figure 1

Our inelastic neutron scattering experiments were carried out on the MERLIN time-of-flight chopper spectrometer at the Rutherford Appleton Laboratory, Didcot, UK. We co-aligned 6.4 g of single crystals of CaFe2As2 grown by self-flux (with in-plane mosaic of 2∘and out-of-plane mosaic of 3∘). The incident beam energies were Ei=50, 80, 150, 200, 250, 450, 600 meV, and mostly with Ei parallel to the c axis. Spin-wave intensities were normalized to absolute units using a vanadium standard (with 30% error). We define the wave vector Q at (qx, qy, qz) as (H,K,L)=(qxa/2π,qyb/2π,qzc/2π) r.l.u., where a=5.506, b=5.450 and c=11.664 Å are the orthorhombic cell lattice parameters at 10 K (ref. 27). a, Schematic diagram of the Fe spin ordering in CaFe2As2. b, Calculated three-dimensional spin-wave dispersions using S J1a=49.9, S J1b=−5.7, S J2=18.9 and S Jc=5.3 meV. c, Schematic diagram for how spin-wave dispersion enters into the Stoner continuum. d, Dispersion of spin waves in a classical Heisenberg Hamiltonian. e–l, Wave-vector dependence of the spin waves for energy transfers of E=48±6 meV [Ei=150 meV and Q=(1,0,3)] (e); E=65±4 meV [Ei=250 meV and Q=(1,0,3)] (f); E=100±10 meV [Ei=450 meV and Q=(1,0,3.5)] (g); E=115±10 meV [Ei=450 meV and Q=(1,0,4)] (h); E=137±15 meV [Ei=600 meV and Q=(1,2,4)] (i); E=135±10 meV [Ei=450 meV and Q=(1,0,4.5)] (j); E=144±15 meV [Ei=450 meV and Q=(1,0,5)] (k); E=175±15 meV [Ei=600 meV and Q=(1,0,5.2)] (l).

Full size image

We used inelastic neutron scattering to study low-temperature (T=10 K) spin waves of single crystals of CaFe2As2, which has a Néel temperature of TN≈170 K (refs 26, 27). Figure 1e–l shows two-dimensional constant-energy (E) images of spin-wave excitations of CaFe2As2 around the antiferromagnetic zone centre in the (H, K) scattering plane21,22,23,24,25. Previous low-energy measurements23 revealed that spin waves in CaFe2As2 are three-dimensional and centred at antiferromagnetic wave vector Q=(1,0,L=1,3,5,…)r.l.u. For energy transfers of E=48±6 (Fig. 1e) and 65±4 meV (Fig. 1f), spin waves are still peaked at Q=(1,0,L=1,3,5) r.l.u. in the centre of the Brillouin zone (shown as dashed rectangles). As the energy increases to E=100±10 (Fig. 1g), 115±10 (Fig. 1h), 137±15 (Fig. 1i), 135±10 (Fig. 1j) and 144±15 meV (Fig. 1k), counter-propagating spin-wave modes become apparent. The scattering changes from ring-like at 100 meV (Fig. 1g) to ellipses elongated along the K-direction for energies above 110 meV (Fig. 1h–k). For an energy transfer of 175±15 meV (Fig. 1l), spin waves show a broad square-like scattering already reaching the zone boundary in the K-direction.

To quantitatively determine the spin-wave dispersion, we cut through the two-dimensional images similar to Fig. 1 for various incident beam energies (Ei) aligned along the c axis. Figure 2a–g shows the outcome for different spin-wave energies in the form of constant-E scans along the K-direction around the antiferromagnetic zone centre. As the excitation energy increases from 25 meV (Fig. 2g) to 144 meV (Fig. 2a), well-defined counter-propagating spin waves approach the zone boundary. To illustrate the general feature of the high-energy spin waves, we have used the scattering near (2, 0, 0) r.l.u. as a background and assumed that the positive scattering at wave vectors below (2, 0, 0) r.l.u. is entirely magnetic. Figure 3a shows the outcome of the background-subtracted scattering for the Ei=450 meVdata projected in the wave vector (Q=[1,K]) and energy space. In spite of the spin-wave intensity modulation along the L-direction due to the exchange interaction Jc between the FeAs planes23 (Fig. 1a), one can see three clear plumes of scattering arising from the in-plane antiferromagnetic zone centres Q=(1,−2), (1, 0) and (1, 2) r.l.u. The spin-wave scattering disperses for energies above 100 meV and extends up to about 200 meV. As spin waves become less dispersive as the zone boundary is approached, we locate the spin-wave excitations through energy scans at a fixed wave vector. Figure 3c–h summarizes a series of such scans at different wave vectors that reveal clear dispersions near the zone boundary and a maximum spin-wave bandwidth of about 200 meV.

Figure 2: Constant-energy cuts of the spin-wave dispersion as a function of increasing energy and our model fit using the Heisenberg Hamiltonian.
figure 2

A series of constant-energy cuts through the antiferromagnetic spin-wave zone centre as a function of decreasing energy E=144±20 (a); E=135±10 (b); E=115±15 (c); E=100±10 (d); E=64±10 (e); E=48±6 (f); E=25±5 meV (g). The solid lines are model fits to the data after convoluting the cross-section to the instrumental resolution. Typical instrumental resolutions are shown as dotted lines in a and d. Error bars indicate one sigma.

Full size image
Figure 3: Observed and calculated spin waves at 10 K, and constant-Q cuts near the antiferromagnetic zone boundary.
figure 3

a, The projections are in the scattering plane formed by the energy transfer axis and (1,K) direction (with integration of H from 0.8 to 1.2 r.l.u.) after subtracting the background integrated from 1.8<H<2.2 and from −0.25<K<0.25. Data were obtained with Ei=450 meV. b, Calculated spin-wave excitations using the model specified in the text. c–h, Constant-Q cuts at various wave vectors near the zone boundary obtained with Ei=600 meV. The solid (S J1a>0,S J1b<0) lines are our model fits to the data and the dashed lines are calculations assuming S J1a≈S J1b. The error bars indicate one sigma.

Full size image

In addition to the results presented in Figs 1–3, we have also collected similar data at other wave vectors throughout the Brillouin zone. The filled circles in Fig. 4a,b summarize our measured spin-wave dispersions along the [H,0,1], [1,0,L] and [1,K,1] directions. To understand these data as well as the wave vector/energy (Q−E) dependence of the spin-wave intensities, we consider a Heisenberg Hamiltonian consisting of effective in-plane nearest-neighbours (Fig. 1a, J1a and J1b), next-nearest-neighbour (Fig. 1a, J2) and out-of-plane (Fig. 1a, Jc) exchange interactions. The dispersion relations are given by21,22,23,24,25: , where

Js is the single-ion anisotropy constant and q is the reduced wave vector away from the antiferromagnetic zone centre. The neutron scattering cross-section can be written as22:

where (γ r0/2)2=72.65 mb sr−1, g is the g-factor (≈2), f(Q) is the magnetic form factor of iron Fe2+, e−2W is the Debye–Waller factor (≈1 at 10 K), Qα is the α component of a unit vector in the direction of Q, Sαβ(Q,E) is the response function that describes the α βspin–spin correlations and ki and kf are incident and final wave vectors, respectively. Assuming that only the transverse correlations contribute to the spin-wave cross-section and finite excitation lifetimes can be described by a damped simple harmonic oscillator with inverse lifetime Γ (refs 28, 29, 30), we have

where kB is the Boltzmann constant, E0 is the spin-wave energy and Seff is the effective spin. We analysed our data by keeping S and Seff distinct following the practice of ref. 22.

Figure 4: Spin-wave dispersion relation along high-symmetry directions in the three-dimensional Brillouin zone and energy dependence of the local susceptibility.
figure 4

a,b, The filled circles are extracted from constant-E(−Q) cuts of various Ei data. The horizontal bars indicate the E(Q) integration range and vertical bars are errors calculated from least-square fittings. Solid (dashed) lines are fits to the spin-wave models discussed in the text. The lengths of the blue vertical bars indicate the wave-vector dependence of Γ; the Γ/E∼0.15 is much smaller than that of metallic ferromagnet La2−2xSr1+2xMn2O7 where Γ/E∼0.33–0.46 (refs 28, 29), thus suggesting a smaller influence of itinerant electrons in CaFe2As2. The blue dotted line is a guide to the eye. c, Energy dependence of the local susceptibility2 obtained by integrating raw intensities above the background from 0.5<H<1.5; −0.5<K<0.5, and L from L−0.5 to L+0.5, where L=1, 3, 5 in the (1,0,L) zone. The twinning effect has not been taken out. In our experimental set-up, the energy, magnetic form factor and polarization factors are all weakly Q dependent within the Brillouin zone. For simplicity, we used appropriate values for these factors at the zone centre Q=(1,0,L). Solid and dashed lines are the expected energy dependence of the local susceptibility for the two models discussed in the text with consideration of the twinning effect.

Full size image

We fitted the measured absolute intensity of spin-wave excitations and their dispersions in Figs 1–4 by convoluting the above-discussed neutron scattering spin-wave cross-section with the instrument resolution using the Tobyfit program28,29,30. As CaFe2As2 exhibits tetragonal to orthorhombic lattice distortion below the TN (ref. 27), care was taken to include the (H,K)/(K,H) twin domains in the computed scattering cross-section. We find that the Heisenberg Hamiltonian with only the nearest-neighbours effective exchange couplings (J1a and J1b are finite, and J2=0) cannot explain the data. Theoretically, it has been argued that the observed collinear spin structure in Fig. 1a is consistent with either S J1a≈J1b≈(1/2)S J2 or S J1a≈2S J2≫S J1b, and distinguishing these two models requires spin-wave data near the zone boundary22.

The red dashed lines in Fig. 3f–h show the expected zone-boundary spin waves assuming S J1a=27,S J1b=25,S J2=36 and S Jc=5.3 meV. It is obvious that such a model failed to describe the zone-boundary data. Our best fits to both the low-energy and zone-boundary spin waves by independently varying the effective exchange parameters are shown as solid black lines in Figs 2 and 3 with S J1a=49.9±9.9, S J1b=−5.7±4.5, S J2=18.9±3.4 and S Jc=5.3±1.3 meV. The broadening of the spin waves with increasing energy is accounted for through Γ∝0.15E and shown as a blue dotted line in Fig. 4a. From our best fit to all spin-wave data, we find Seff=0.22±0.06, which is smaller than previous measurements on powder samples of BaFe2As2 (ref. 22). The value of Seff and the measured 0.8 μB/Fe static moment27 suggest a S∼1/2 system.

From the fitting results in Figs 2–4, we see that the spin-wave dispersion and intensity in CaFe2As2 throughout the Brillouin zone can be well described by a Heisenberg Hamiltonian with effective nearest-neighbours and next-nearest-neighbour exchange interactions. Figure 4a,b summarizes the spin-wave dispersions along all three high-symmetry directions and Fig. 4c shows the energy dependence of the local susceptibility7, together with calculations using S J1a≈S J1b (red dashed lines) or our (solid lines) models. The former model clearly fails to describe the data. To test whether the spin-wave branch crosses the Stoner continuum as schematically illustrated in Fig. 1c, we plot spin-wave damping Γ versus E as a blue dotted line in Fig. 4a. Although Γ is approximately proportional to 0.15E, there is no steep increase in Γ at any wave vector indicative of a Stoner continuum (Fig. 1c). Instead, the observed spin-wave broadening at high energies may arise from magnon–electron scattering due to the low-temperature metallic nature of the system, similar to ferromagnetic metallic manganites28,29,30. Although these results may be consistent with ab initio calculations presented in ref. 25, our data show well-defined spin waves near the zone boundary, in contrast to a simple picture of an electron–hole Stoner continuum extending to very high energies as in the case of pure metal Cr (refs 9, 10).

The central message of our work is that one can fit spin waves of CaFe2As2 throughout the Brillouin zone with a simple Heisenberg Hamiltonian without the need for a Stoner continuum—the hallmark of an itinerant electron system. In a spin-density-wave state driven by Fermi surface nesting of itinerant electrons, a Stoner continuum is expected to have an energy scale around 2Δ, where Δ is the quasiparticle gap in the spin-density-wave state. Spin waves should be well-defined below 2Δ, and quickly damp into a particle–hole continuum above the characteristic energy. From Figs 1–4, we notice that there is no particular energy scale above which damped spin waves appear. This observation is in direct conflict with ref. 25, where a Stoner continuum is believed to set in above 100 meV. As our experiments were carried out on samples more than three times the mass and on an instrument with more neutron flux, the diminishing spin-wave scattering above 100 meV in ref. 25 may simply arise from a poor signal-to-noise ratio of the measurement due to insufficient sample mass. The lack of direct evidence for a Stoner continuum below 200 meV suggests weak low-energy electron–hole particle excitations. One local density approximation calculation has predicted essentially the correct in-plane magnetic exchange couplings20; these results, however, are obtained within the tetragonal and collinear antiferromagnetic ordered structures contrary to the experiments. Furthermore, band-structure calculations suggest that the Fermi velocity a/b anisotropy in CaFe2As2 is less than 8% in the low-temperature orthorhombic phase (D. J. Singh, personal communication). If spin-wave velocities in CaFe2As2 are proportional to (vevh/3)1/2 such as those in chromium9, they should be similar along the a/b directions. Although our results seem to favour a localized moment picture, a spin-1/2 model cannot be produced if all orbitals in iron are localized because there are even numbers of electrons per iron. Moreover, it is difficult to understand why direct and super-exchange interactions within the Fe–As–Fe plane are so different along the a/b directions of the orthorhombic structure because the tetragonal to orthorhombic lattice distortion below TN is small and only weakly affects the Fe–As–Fe bond distances/angles5,6. The observed large difference may hint at the involvement of other electronic degrees of freedom, such as orbital, in the magnetic transition. To achieve a comprehensive understanding of spin excitations, one must consider both the localized and itinerant electrons in these materials.