Spin waves in metallic iron and nickel measured by soft x-ray resonant inelastic scattering

The spin-wave dispersions in iron and nickel along the [111] direction are determined using soft x-ray resonant inelastic x-ray scattering (RIXS). For iron, a 10-nm thin film was studied and, over the limited $q$ range accessible, the peaks disperse as expected for a spin wave and in agreement with inelastic neutron scattering (INS) results. At the higher $q$ values damping is observed with the peaks weakening and broadening. This damping is less pronounced than in the INS studies. The RIXS results are also compared with ab initio spin fluctuation calculations. The calculations slightly underestimate the energy dispersion and the damping is larger than in the measurement. Nevertheless, the agreement between the RIXS results, INS studies, and the theory is quite satisfactory. For the single crystal of nickel, the measured $q$ dispersion flattens out rapidly and the peaks broaden. The strong damping effect is reproduced by the spin fluctuation calculations but the energy of the peaks is largely overestimated. Nevertheless, the flattening of the dispersion is not reproduced by the calculations and, although similar effects were observed in early INS experiments, they are not seen in more recent work. Possible reasons for this are discussed. These measurements show that using soft x-ray RIXS to study spin fluctuations in metallic systems, which are in general very challenging for the technique, has much promise. More interestingly, since the iron measurements were performed on a 10-nm thin film, the study opens the possibility to study tailor-made thin-film samples, which cannot be easily studied by other techniques. Combining these studies with state-of-the-art ab initio calculations opens up interesting prospects for testing our understanding of spin waves in metallic systems.

Figure 1

(a) Generic RIXS measurement geometry, showing the rotation angles θ, ϕ, and χ. (b) Measurement geometry for the [111] dispersion, after rotation of the sample around χ. (c) The unit cell for fcc nickel showing the 111 plane.

Figure 2

Wide energy scan RIXS spectra from iron. The inset shows the iron XAS spectra and the arrow marks the energy used to take the RIXS spectra.

Figure 3

Wide energy scan RIXS spectra from nickel. The inset shows the nickel XAS spectra and the arrows mark the energies used to take the RIXS spectra.

Figure 4

(a) $q$-dependent RIXS spectra along the [111] direction in iron. The vertical bars mark the peak position found by peak fitting. (b) Peak fitting for $q=0.692{\AA }^{-1}$ and (c) for $q=0.311{\AA }^{-1}$.

Figure 5

Spin-wave dispersion for iron along the [111] direction. RIXS data (black squares), INS [28] open triangles. The dashed line is a parabolic curve with a stiffness constant $D=230\mathrm{meV}{\AA }^2$; see text. Inset: Intensity and deconvoluted width of the loss peaks.

Figure 6

Calculated $\mathrm{Im}{\chi }^{+-}(q,\omega )$ for iron along the [111] direction. The experimental results are shown as black squares. The right-hand panel shows cuts corresponding to the vertical lines in the left-hand panel.

Figure 7

(a) $q$-dependent RIXS spectra along the [111] direction in nickel. The vertical bars mark the peak position found by peak fitting. (b) Peak fitting for $q=0.787{\AA }^{-1}$ and (c) for $q=0.372{\AA }^{-1}$.

Figure 8

Spin-wave dispersion for nickel along the [111] direction. RIXS data (black squares), INS data (triangles [30] and dots [33]). The dashed line represents a spin-wave curve with $D=433\mathrm{meV}{\AA }^2$. Inset: Peak intensity and width of the loss peaks.

Figure 9

Calculated $\mathrm{Im}{\chi }^{+-}(q,\omega )$ for nickel along the [111] direction. The experimental results are shown as black squares. The right-hand panel shows two cuts given by the vertical lines in the left-hand panel.

Figure 10

Calculated Re and Im parts of ${\chi }^{\mathrm{xx}}(q,\omega )$, ${\chi }^{\mathrm{xy}}(q,\omega )$, ${\chi }^{\mathrm{zz}}(q,\omega )$, ${\chi }^{00}(q,\omega )$ functions for Fe(111). The top-left panel shows the spin fluctuations given by $\mathrm{Im}{\chi }^{+-}(q,\omega )$. The color scale is on a linear scale but each panel has been scaled to 1. The multiplication factor is given in each panel. The green line gives the spin-wave fit derived from this and previous work.

Figure 11

Calculated Re and Im parts of ${\chi }^{\mathrm{xx}}(q,\omega )$, ${\chi }^{\mathrm{xy}}(q,\omega )$, ${\chi }^{\mathrm{zz}}(q,\omega )$, ${\chi }^{00}(q,\omega )$ functions for Ni(111). The top-left panel shows the spin fluctuations given by $\mathrm{Im}{\chi }^{+-}(q,\omega )$. The color scale is on a linear scale but each panel has been scaled to 1. The multiplication factor is given in each panel. The green line gives the spin-wave fit derived from this and previous work.

Figure 12

The calculation of the RIXS cross section for the 8 [111] magnetic domains and their average for Ni(111). The scale bar shows the intensity of five decades on a log scale. The center panel shows one of the examples where the cross section is reduced in the center of the zone. The $q$ region measured in the experiment is marked by the two vertical lines. The $q$ scale is in ${\AA }^{-1}$.