Extended X-ray absorption fine structure (EXAFS) and X-ray absorption near-edge structure (XANES) study of melt-spun Fe 85 Ga 15 ribbons

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Journal of Magnetism and Magnetic Materials 32 (28) e578 e582 www.elsevier.com/locate/jmmm Extended X-ray absorption fine structure (EXAFS) and X-ray absorption near-edge structure (XANES) study of melt-spun Fe 85 Ga 15 ribbons R. Sato Turtelli a,, S. Pascarelli b, M. Ruffoni b, R. Gro ssinger a, C. Eisenmenger-Sittner a a Institut f. Festkörperphysik, Technische Universität Wien, Wiedner Haupts. 8-1, A-1 Vienna, Austria b European Synchroton Radiation Facility, BP 22, 383 Grenoble, Cedex, France Available online 1 April 28 Abstract The local structure in melt-spun Fe 85 Ga 15 ribbons with a width 3 mm and thickness 6 mm produced in argon atmosphere was studied by analyzing EXAFS and XANES data. The following results were obtained: Ga Ga bonds were not detected excluding the tendency to form clusters of Ga atoms; Ga substitutes Fe creating a local strain of about +1% on the first shell Fe Ga bond, whereas on the second Fe Ga shell strain quickly relaxes down to +.3%; XANES spectra are compatible with a random substitution of Fe atoms by Ga atoms in the A2 structure. From the AFM investigation, we observed that at the surface (free side) of the ribbon the particles are elongated along the ribbon (2 mm 5 mm) and each particle is formed by small grains of average size of 2 nm. r 28 Elsevier B.V. All rights reserved. PACS: 75.5.Bb; 75.8.+q; 61.1.Ht Keywords: Fe Ga alloy; Magnetostriction; EXAFS; XANES 1. Introduction Fe Ga alloys can provide low-cost magnetostricitve materials with high mechanical strength, good ductility, large magnetostrictive at low saturation fields with negligible magnetostrictive hysteresis and attractive magnetic properties to be used in acoustic sensors and actuators. The large magnetostriction of Fe Ga alloys depends strongly on sample production methods and thermal history, which result in different crystallographic texture and grain morphology [1 5]. Single crystal [1 ] oriented Fe 1 x Ga x alloys, where 13pxp23, can exhibit magnetostriction in the l 1 direction of up to 27 ppm [3]. Ductility of Fe Ga alloys has been largely improved by the melt-spun method, producing ribbons. In addition, thin ribbon sample minimizes ac-eddy current losses and be well suited for working at high frequency. However, reports of Corresponding author. Tel.: +3 1 5881 1315; fax: +3 1 5881 13199. E-mail address: reiko.sato@ifp.tuwien.ac.at (R. Sato Turtelli). different magnetostriction values in the literature, for similar compositions, are not uncommon. These include a value of 13 ppm in strained ribbons of Fe 83 Ga 17 [6], a giant magnetostriction up to 13 ppm in stacked ribbons of Fe 85 Ga 15 [7], and a value of 3 ppm in annealed Fe 81 Ga 19 ribbons [8]. Some reported values are larger than that of Terfenol-D. The enhancement of magnetostriction in melt-spun samples has been attributed to a change in local structure due to the melt spinning method itself, and also to the short-range preferential ordering of Ga atoms in the [ 1] direction, which supposedly form clusters. However, different researchers have reported contradicting results of crystallographic phases. For melt-spun Fe Ga alloys with Ga contents lower than 21 at%, Ref. [6] reports the existence of only a disordered A2 phase, while Ress. [7,9] report the appearance of a D 3 structure. In order to understand these different results in similar ribbons, more microstructure and local structure investigations are necessary. Therefore in this work, the local structure of a melt-spun Fe 85 Ga 15 ribbon with a width 3 mm and 3-8853/$ - see front matter r 28 Elsevier B.V. All rights reserved. doi:1.116/j.jmmm.28..36

thickness 6 mm was produced in an argon atmosphere and was studied by analyzing extended X-ray absorption fine structure (EXAFS) and X-ray absorption near-edge structure (XANES) data. Additionally a Topometrix Explorer atomic force microscope (AFM) was used to investigate the grain structure of the ribbon. R. Sato Turtelli et al. / Journal of Magnetism and Magnetic Materials 32 (28) e578 e582 e579 2. Experimental procedure An ingot of Fe 8 Ga 15 alloy with high purity (99.9%) was prepared by arc melting. Ejecting the melt onto a rotating copper wheel produced a rapidly quenched ribbon with a thickness of 5 mm and width of 3 mm. From the energy dispersive X-ray analysis, the as-quenched ribbon exhibits a composition similar to that of the ingot. The grain structure of the ribbon was investigated by means of a Topometrix Explorer AFM with a Si3N tip mounted on a cantilever of spring constant.3 N/m. Local structural investigations were performed on a thin sample, 7 mm thick, obtained by polishing both the wheel and free sides of the ribbon. EXAFS spectra were recorded at the Fe and Ga K-edges at beam line BM29 of the European synchrotron radiation facility (ESRF). A mm thick highpurity Fe foil (from Goodfellow) was used as a standard. 3. Results and discussions Fig. 1 shows AFM pictures obtained on the wheel side of the Fe 85 Ga 15 ribbon. As can be seen in the upper picture, the grains are elongated along the direction of the ribbon with an average grain size of 2 3 by 5 1 mm. It is interesting to note that each grain is composed of fine elongated particles around 2 nm in the direction of the thickness of the ribbon. This kind of small particle also exists in melt-spun Fe 8 Ga 2 samples [1]. The grain size determined by X-ray diffraction is of same order of magnitude and presents a texture [1]. As magnetostriction depends strongly of structural properties of grains (such as shape and texture [1 5]), magnetostriction measurements in each direction may result in different values. X-ray absorption fine structure (XAFS) spectroscopy is particularly suited to probe the local environment of the absorber atom. We have independently analyzed the extended fine structure region of the spectra (or EXAFS), and the near-edge (or XANES) region. The former, characterized by relatively large photoelectron energy (5 ev) with a relatively low mean free path, can be interpreted in a simplified manner using a single-scattering formalism, whereby the emitted photoelectron wave is scattered once by the electronic potentials of the neighboring atoms. The latter, where the photoelectron energy is low with a large mean free path, arises from complex multiple scattering processes and its interpretation is difficult because details in the description of the potentials cannot be neglected so easily as can be done in EXAFS region. Fig. 1. AFM pictures obtained on the wheal side of the Fe 85 Ga 15 ribbon. Up: sensor picture; bottom: topography picture with high amplification. Consequently, EXAFS is very sensitive to the first neighbor shells and can yield information on the chemical nature of the nearest neighbors, their number, distance from the absorber atom and thermal and static disorder relative to the absorber atom. XANES, on the other hand, is very sensitive to the local bonding geometry and electronic structure. We based our EXAFS analysis on a structural model obtained from crystallographic data [], which found that Fe 85 Ga 15 ribbons crystallize in the A2 phase [bcc phase] with lattice parameter a ¼ 2.9 A. Based on these input structural parameters, we built atomic clusters (one centered on Fe and one centered on Ga) for the EXAFS simulations. Due to the short-range sensitivity of the EXAFS probe, we considered only the region of R-space that extends up to about 3 A from the absorber, i.e. the first two coordination shells of the bcc lattice. The software package FEFF8.2 [11] was used to calculate ab initio scattering phases and amplitudes for a Fe and a Ga absorber using the atomic clusters defined above, and

e58 R. Sato Turtelli et al. / Journal of Magnetism and Magnetic Materials 32 (28) e578 e582 simulated the EXAFS signal using a high-order path expansion. For the Fe6¼K-edge we considered the following: (i) a first coordination shell composed of a total of N Fe + N Ga ¼ 8 atoms, with an unknown x ¼ N Ga /(N Fe +N Ga ). The bond distances and mean square relative displacements (msrd) of Fe Fe and Fe Ga are unknown; (ii) a second shell composed, in a first approximation, of 6 Fe atoms at an unknown distance (linked through crystallographic constrains to the first shell Fe Fe distance) and msrd. As a second step, the presence of some Ga was also considered in the second shell, but we did not have sufficient sensitivity to confirm the number of Ga atoms in this shell. For the Ga K-edge both the first and second shell are composed of only Fe atoms (8+6) at an unknown distance. The presence of Ga in the first shell was tested and was excluded in this analysis. The only paths contributing to the EXAFS signal in the R-range up to 3 A are the two single scattering (SS) paths corresponding to the first (total path length 2R 1 ) and second shell (total path length 2R 2 ). Fig. 2 illustrates the results of ab initio calculations for the first shell SS paths, plotted as a function of photoelectron wavevector k. These signals show that, besides a small and almost constant phase shift, the scattering amplitudes of Ga and Fe differ mainly in the low k region. Fig. 3 shows the normalized EXAFS spectra obtained on the Fe 85 Ga 15 ribbon at the Fe and Ga K-edges, respectively. The Fe K-edge EXAFS on a pure Fe foil standard is also shown. By comparing the Fe K-edge EXAFS oscillations of the Ga-containing sample to that of pure Fe foil, besides a reduction in amplitude, there is also a small change in the shape of the first oscillations, at k3 A. The difference in calculated signals.5..3.2.1 -.1 -.2 Ga-Fe Ga-Ga Fe-Fe Fe-Ga 6 8 1 12 1 Fig. 2. Ab initio calculations for the first shell SS paths: Blue curve: Fe (absorber) Fe (scatterer). Red curve: Ga(absorber) Ga (Scatterer). EXAFS χ(k)..3.2.1 -.1 -.2 -.3 -. Fe 85 Ga 15 Fe K-edge the shape of the Fe K-edge EXAFS between the sample and pure Fe at low k values, seen in Fig. 3, is likely to be attributed to the detection of Ga in the first coordination shell. In order to extract the local structural parameters (bond distance, R, msrd, Ds 2, and ratio between Ga/Fe scatters, x, we performed a least-square fitting of the first peak of the Fourier transform (FT) of the EXAFS signals, where the FT was performed in the range 2.6pkp15. A 1. Using the calculated phases and amplitudes, we constructed a model EXAFS signal as the sum of two SS paths (first and second shell) with variable fitting parameters for energy offsets, bond distances and msrds for two shells, and chemical composition of the first shell. The Fe and Ga K-edges data were fitted simultaneously. A more detailed description of the fitting is given elsewhere [1]. Fig. shows the results of the best fits at the Fe and Ga edges for the Fe 85 Ga 15 ribbon and the standard Fe foil. The structural parameters obtained from the best fits are consistent with a picture whereby Ga substitutes Fe atoms randomly, creating local strain. The obtained value of the Fe Ga bond is 2.52 A, whereas the Fe Fe bond is 2.5 A, implying a +1% expansion. On the second shell, strain quickly relaxes down to +.3%, since we find Fe Ga bonds of 2.89 2.9 A and Fe Fe bonds of 2.88 2.89 A. From Ga edge data we can conclude that no clustering occurs since we do not detect any Ga Ga first shell bonds. We also performed fits at the Fe K-edge with a second shell Fe Ga contribution of R Fe Ga and Ds 2 Fe Ga, equal to Ga K-edge values. However, these additional parameters did not improve the quality of the fits. To simulate the X-ray absorption near-edge structure region of the spectra at the Fe and Ga K-edges, we again Fe Fe 85 Ga 15 Ga K-edge 6 8 1 12 1 Fig. 3. EXAFS oscillations plotted as function of photoelectron wavevector k.

R. Sato Turtelli et al. / Journal of Magnetism and Magnetic Materials 32 (28) e578 e582 e581 kχ(k) 1.5 1.5 -.5 Ga K-edge Fe K-edge Fe 85 Ga 15 Fe 85 Ga 15 Fe 6 8 1 12 1 Fig.. Best fits for the Ga K-edge data on Fe 8 Ga 15 (top) and at the Fe K-edges on Fe 8 Ga 15 and pure Fe foil (bottom). All scattering paths within the same atomic cluster (approximately 6 atoms) were summed to infinite order. The imaginary part of the potential was not optimized and our simulations underestimate this damping at the absorption edge. Moreover, no disorder factor was introduced in the calculations, so that the high-energy fine structure is also under-damped in the simulations, compared to the experimental data. The Fe K-edge simulations were done for (i) the pure Fe bcc phase (a ¼ 2.87 A ); (ii) the A2 phase (a ¼ 2.9 A ) where 9 Fe atoms out of 59 (about 15%) have been substituted by Ga and (iii) the DO3 Fe 3 Ga phase (a ¼ 2.9 A ); and Ga K-edge simulations (see Fig. 5) for (i) A2 phase where Ga is surrounded only by Fe atoms ( Ga impurity x- ) (ii) the A2 phase where Ga is surrounded by Fe and 15% Ga randomly distributed ( Ga impurity x.15 ) (iii) the A2 phase where Ga is surrounded only by Ga in the first shell ( Ga cluster ), and by Fe in further shell and (iv) the DO3 Fe 3 Ga phase. A qualitative comparison of our XANES data with ab initio simulations indicates that all the observed features are reproduced by the A2 model, therefore the DO3 phase and clustering of Ga can be excluded. 2. Conclusions absorption 1.5 1.5 -.5-1 -1.5-2 1.36 D3 A2 (Ga cluster) A2 (x ~.15) A2 (x-> ) experimental data 1.38 1. 1.2 1. Energy (kev) (i) EXAFS: (a) No Ga Ga bonds are detected, excluding the tendency to form clusters of Ga atoms; (b) Ga substitutes Fe, creating a local strain of about +1% on the first shell Fe Ga bond. On the second Fe Ga shell, strain quickly relaxes down to +.3%. (ii) XANES: (a) the spectra are compatible with a random substitution of Fe atoms by Ga atoms in the A2 structure; (b) the presence of Ga clusters or of a D 3 local environment around Ga has been tested and excluded. (iii) AFM: (a) the particles observed at the surface (wheel side) of the ribbon are elongated (width 3 mm, length 5 1 mm) along the ribbon; (b) these particles are formed by fine grains elongated perpendicular to the surface of the ribbon with average width size of 2 nm. Fig. 5. Ab initio full multiple scattering simulations of Ga K-edge data. The following models are used for the local environment of Ga, from top to bottom: the D3 phase, the A2 phase with only Ga surrounding Ga in the first shell (Ga cluster), the A2 phase where the first shell around Ga is composed of Fe atoms, and some Ga (x.15) distributed randomly, the A2 phase where Ga is surrounded only by Fe atoms (x-). The bottom curve is the experimental data. used FEFF8.2, but this time to perform full multiple scattering calculations. Ab initio scattering amplitudes and phases were calculated using complex Hedin-Lundvquist potentials, with the electronic density calculated selfconsistently within a radius of 5.5 A from the absorber. References [1] A.E. Clark, M. Wun-Fogle, J.B. Restorff, T.A. Lograsso, J.R. Cullen, IEEE Trans. Magn. 37 (21) 2678. [2] N. Srisukhumbowornchai, S. Guruswany, J. Appl. Phys. 9 (21) 568. [3] A.E. Clark, K.B. Hathaway, M. Wun-Fogle, J.B. Restorff, T.A. Lagrasso, V.M. Keppens, G. Petculescu, R.A. Taylor, J. Appl. Phys. 93 (23) 8621. [] C. Bormio-Nunes, R. Sato Turtelli, R. Grössinger, H. Mu ller, G. Wiesinger, H. Sassik, M. Reissner, M.A. Tirelli, J. Appl. Phys. 97 (25) 3391. [5] R. Sato Turtelli, C. Bormio-Numes, J.P. Sinnecker, R. Gro ssinger, Phyisica B 38 (26) 265.

e582 R. Sato Turtelli et al. / Journal of Magnetism and Magnetic Materials 32 (28) e578 e582 [6] S.F. Cheng, B.N. Das, M. Wun-Fogle, P. Lubitz, A.E. Clark, IEEE Trans. Magn. 38 (22) 2838. [7] G.D. Liu, L.B. Liu, Z.H. Liu, M. Zhang, J.L. Chen, J.Q. Li, G.H. Wu, Y.X. Li, J.P. Qu, T.S. Chin, Appl. Phys. Letters 8 (2) 212. [8] M.C. Zhang, X.X. Gao, H.L. Jiang, Y. Qiao, S.Z. Zhou, J. Alloys Comp. 31 (27) 2. [9] M.C. Zhang, H.L. Jiang, X.X. Gao, J. Zhu, S.Z. Zhou, J. Appl. Phys. 99 (26) 2393. [1] S. Pascarelli, M. Ruffoni,R. Sato Turtelli, F. Kubel, R. Gro ssinger, Phys. Rev. B, in print. [11] S. Pascarelli, G. Aquilanti, W. Crichton, T. Le Bihan, M. Mezouar, S. De Panfilis, J.P. Itié, A. Polian, Europhys. Lett. 61 (23) 55.