research papers
Magnetic field dependent smallangle neutron scattering on a Co nanorod array: evidence for intraparticle spin misalignment
^{a}Physics and Materials Science Research Unit, University of Luxembourg, 162A Avenue de la Faïencerie, L1511 Luxembourg, Luxembourg, ^{b}Institut Laue–Langevin, 6 Rue Jules Horowitz, BP 156, F38042 Grenoble Cedex 9, France, ^{c}HelmholtzZentrum Berlin für Materialien und Energie GmbH, HahnMeitnerPlatz 1, D14109 Berlin, Germany, ^{d}Jülich Centre for Neutron Science JCNS, Forschungszentrum Jülich GmbH, Outstation at MLZ, Lichtenbergstrasse 1, D85747 Garching, Germany, and ^{e}Experimentalphysik, Universität des Saarlandes, Postfach 151150, D66041 Saarbrücken, Germany
^{*}Correspondence email: annegret.guenther@web.de
The structural and magnetic properties of a cobalt nanorod array have been studied by means of magnetic field dependent smallangle neutron scattering (SANS). Measurement of the unpolarized SANS cross section dΣ/dΩ of the saturated sample in the two scattering geometries where the applied magnetic field H is either perpendicular or parallel to the wavevector k_{i} of the incoming neutron beam allows one to separate nuclear from magnetic SANS, without employing the usual sectoraveraging procedure. The analysis of the SANS data in the saturated state provides structural parameters (rod radius and centretocentre distance) that are in good agreement with results from Between saturation and the coercive field, a strong field dependence of dΣ/dΩ is observed (in both geometries), which cannot be explained using the conventional expression of the magnetic SANS cross section of magnetic nanoparticles in a homogeneous nonmagnetic matrix. The origin of the strong field dependence of dΣ/dΩ is believed to be related to intradomain spin misalignment, due to magnetocrystalline and magnetoelastic anisotropies and magnetostatic stray fields.
Keywords: smallangle neutron scattering; magnetism; magnetic materials; nanorods.
1. Introduction
As a consequence of their interesting magnetic properties, magnetic transitionmetal nanorod arrays are attracting much scientific attention (Fert & Piraux, 1999; Sellmyer et al., 2001; Kou et al., 2011; Greaves et al., 2012). Essentially, it is their pronounced magnetic shape anisotropy which largely determines the magnetization process in these systems and which renders them potential candidates for perpendicular magnetic storage media (Ross et al., 1999; Greaves et al., 2012). Owing to the technological relevance of such functional magnetic materials, a better understanding of the microstructure–property relationship is crucial (Goolaup et al., 2005; Zighem et al., 2011; Chumakov et al., 2011).
Smallangle neutron scattering (SANS) is a powerful volumesensitive technique for probing structural and magnetic properties of such nanorod arrays. In particular, SANS provides access to nanoscale spatial variations of the local orientation and magnitude of the magnetization vector field (Wagner & Kohlbrecher, 2005; Wiedenmann, 2005; Michels & Weissmüller, 2008).
Previous SANS studies on ordered arrays of Co and Ni nanowires embedded in Al_{2}O_{3} matrices have employed polarized incident neutrons for studying the structural and magnetic correlations (Napolskii et al., 2007, 2009; Grigoryeva et al., 2007; Chumakov et al., 2011; Maurer et al., 2013). It is worth mentioning that for Ni nanowires (of average length 50 µm) the validity of the Born approximation has been questioned (Napolskii et al., 2009), while for Co nanowires an anomalously low magnetic scattering contribution (relative to the nuclear SANS) has been reported (Chumakov et al., 2011). The nonnegligible but relevant influence of magnetostatic stray fields on the magnetization distribution inside the wires has been pointed out by Napolskii et al. (2009) and Maurer et al. (2013).
In this paper, we provide a SANS study of a (shortrangeordered) Co nanorod array using unpolarized neutrons. The focus of our study is on the field dependence of the cross section in the two scattering geometries that have the applied magnetic field either perpendicular or parallel to the wavevector of the incoming neutrons. In particular, the discussion addresses the validity of the standard expression for the magnetic SANS cross section, which assumes uniformly magnetized particles.
2. Experimental
2.1. Sample preparation and characterization
The Co nanorod array was prepared by pulsed et al., 2008, 2011; Klein et al., 2009); here, we present only a brief outline of the sample preparation. The porous alumina template was synthesized by a twostep anodization process (Masuda & Fukuda, 1995; Masuda & Satoh, 1996). The anodization was carried out in 2 M sulfuric acid at constant cell voltages of 15 and 20 V (first and second anodization step, respectively). A total charge density of 2 C cm^{−2} during the second anodization and a final treatment of the alumina templates in 0.1 M phosphoric acid resulted in an oxide layer thickness of ∼1200 nm, a pore diameter of d ≃ 27 nm and a centretocentre distance of the pores of d_{cc} ≃ 48 nm.
of Co into a nanoporous aluminium oxide layer. A detailed description of the synthesis of porous alumina templates and their filling with metals can be found elsewhere (GüntherThe pores were filled with Co by pulsed et al., 2000) from an aqueous solution composed of 0.3 M CoSO_{4}·7H_{2}O and 45 g l^{−1} H_{3}BO_{3} at room temperature and a pH value of 6.4 (Ramazani et al., 2012). Such a Cofilled alumina template observed with (SEM) is shown in Fig. 1. As can be seen in Fig. 1(b), the pores were not homogeneously filled up to the level of the surface.
(NielschAs a consequence, it was necessary to remove alumina (and partly Co) in order that most of the nanorods end at the alumina surface. This was realized by an etching process, which was performed with an Arion beam milling system (Leica EM RES101) under etching conditions of 6 kV voltage, 2.2 mA current and 30° milling angle. Owing to sample oscillation during the etching process, an area with a diameter of ∼8 mm could be homogeneously etched. In Fig. 2 the top view of the etched Co sample is shown. The white circles represent the crosssectional areas of the nanorods, which sit flush with the alumina surface. The nanorods with average diameter nm and length nm are hexagonally arranged with a centretocentre distance of nm (see Fig. 2).
Magnetic characterization of the array was carried out using a vibrating sample magnetometer (VSM, LakeShore VSM 7400). The magnetization loops were recorded at room temperature for different angles γ between the magnetic field and the long rod axes in the field range from −0.8 to +0.8 T (see Fig. 3).
The magnetization measurements reveal that the Co nanorod array exhibits an effective anisotropy (due to magnetocrystalline and shape anisotropy) with the easy axis along the long rod axis (Ramazani et al., 2012; Srivastav & Shekhar, 2014).
2.2. SANS experiment
SANS experiments were performed at KWS1 (Jülich Centre for Neutron Science, Outstation at MLZ, Garching, Germany), at V4 (HelmholtzZentrum Berlin, Germany) and at the D33 instrument at the Institut Laue–Langevin (ILL, Grenoble, France); here, we only show ILL data. At ILL, we used unpolarized incident neutrons with a mean wavelength of Å [% (FWHM)] and two sampletodetector distances of 12.8 and 2.5 m, resulting in an accessible q range of nm^{−1}. Magnetic field dependent measurements were carried out by first applying a large positive field ( T), which is assumed to saturate the sample (compare Fig. 3), and then reducing the field to the experimental value (following the magnetization curve). This procedure was executed for two different scattering geometries, namely Hk_{i} geometry (Fig. 4a) and Hk_{i} geometry (Fig. 4b). All data were collected at room temperature. SANS data reduction (correction for background scattering, transmission, detector efficiency) was carried out using the GRAS_{ans}P software package (Dewhurst, 2001).
3. SANS cross sections
For the scattering geometry where the applied magnetic field He_{z} is perpendicular to the wavevector ke_{x} of the incoming neutron beam (Hk_{i}), the unpolarized elastic differential SANS cross section of a ferromagnet can be written as (Michels & Weissmüller, 2008)
whereas for Hke_{z} one obtains (Michels et al., 2011)
In equations (1) and (2), V denotes the scattering volume, is the nuclear scattering amplitude, and represents the Fourier coefficient of the magnetization ; the asterisks `^{*}' mark the complexconjugated quantity. The atomic magnetic form factor in the expression for the atomic magnetic scattering length was set to unity, which is permissible along the forward direction (: atomic : Bohr magneton). The above relation defines the quantity b_{H} = 2.9 ×10^{8} A^{1} m^{1}, which is independent of the material (Michels & Weissmüller, 2008); was absorbed into the expression for the saturation magnetization M_{s}, which enters the expression for the Fourier coefficients. Note that is assumed to be parallel to in both geometries, so that in both equations (1) and (2) denotes the corresponding longitudinal magnetization Fourier coefficient, while and are the respective transverse components, giving rise to spinmisalignment scattering. For Hk_{i}, the angle θ is measured between and , whereas for Hk_{i}, θ is the angle between and (compare Fig. 4).
At magnetic saturation, when the magnetization of the rods is perpendicular (Hk_{i}) or parallel (Hk_{i}) to the rod axes, equations (1) and (2) reduce to
for Hk_{i} and to
for Hk_{i}.
4. Results and discussion
The experimental differential SANS cross sections of the Co nanorod array for the two scattering geometries are shown in Fig. 5 for selected applied magnetic fields between saturation (left images) and the respective coercive fields (right images).
At saturation in Hk_{i} geometry, an intensity ring occurs with maxima perpendicular to H (seen as two darkred halfmoons; Fig. 5a, left). With decreasing magnetic field, scattering due to transverse spin components emerges at smaller q (see below) and a maximum (overall) intensity can be observed at the coercive field T (Fig. 5a, right). The same qualitative behaviour is detected in Hk_{i} geometry (Fig. 5b), except that the scattering at saturation (Fig. 5b, left) is isotropically distributed on the ring.
The intensity rings that occur in both scattering geometries arise from the fact that the hexagonal order of the rods is not perfect over the whole scattering (coherence) volume, but is rather restricted to domains with a size of a few hundred nanometres (see Fig. 2). This gives rise to Debye–Scherrer diffraction rings. The halfmoon intensity maxima in Hk_{i} geometry reflect the angular anisotropy of the SANS cross section at saturation, which follows the well known dependence [compare equation (3) and the discussion below]. By contrast, for the Hk_{i} geometry, the SANS cross section at saturation exhibits an isotropically distributed intensity, i.e. depends only on the magnitude q of the scattering vector ; the slight intensity asymmetry that can be detected in Fig. 5(b) is due to a small misalignment of the sample relative to the incident beam. By comparison to equation (4), isotropy of implies that the sum of and is isotropic. In the later data analysis, we will assume that both Fourier coefficients are isotropic (see below).
The resulting radially averaged data of the differential SANS cross sections of the Co nanorod array are displayed in Fig. 6. The intensity rings observed in both geometries on the twodimensional detector images at 2 T can be identified in the radially averaged data (black open squares in Fig. 6) as the lowq peak at nm^{−1} ( nm). Moreover, two additional peaks were detected at higher q values ( nm^{−1} and nm^{−1}), which can also be related to the hexagonal shortrange order of the rods.
Before discussing the field dependence of , we provide an analysis of the SANS data in the saturated state. For fully saturated particles, like the Co nanorod array under study at a magnetic field of T, equations (1) and (2) reduce to equations (3) and (4). We now assume that both Fourier coefficients and are independent of the orientation of [as supported by the twodimensional data shown in Figs. 5(a) and (b)]. Radial averaging of the scattering cross section at saturation in Hk_{i} geometry [equation (3)] then results in , whereas for Hk_{i} geometry we obtain . By assuming that at saturation is independent of the orientation of the externally applied magnetic field, one can combine these two equations and separate the nuclear from the longitudinal magnetic SANS:
The sodetermined experimental nuclear and longitudinal magnetic SANS cross sections are shown in Fig. 7(a); for simplicity, we will omit the constant prefactors and in the following.
For the quantitative description of and as well as the SANS data at saturation (Fig. 7b), we consider a magnetic field independent model,
where I_{inc} denotes the background, A is a scaling constant, which is proportional to the and the respective scatteringlength density contrast, V_{p} is the particle volume, and F(q,R) is the form factor of a cylinder for q being perpendicular to the long rod axes; F(q,R) = 2J_{1}(qR)/(qR), where J_{1}(qR) is the spherical Bessel function of first order with R = d/2 being the rod radius. The is modelled as a sum of Gaussians, S(q) = , with the Bragg peak positions given by the twodimensional hexagonal lattice at , where (hk) = (10), (11), (20), (21), (30) and (22).
The data fits by this model with I_{inc}, A, a_{i}, , d_{cc} and R as adjustable parameters are shown as the solid lines in Fig. 7. Obviously, the considered model, equation (7), does provide an excellent description of the measurements. The resulting values of the structural fit parameters are listed in Table 1 and are in good agreement with each other as well as being consistent with the results from where we have found nm and nm.

The magnetic scattering contribution is larger than the nuclear SANS (see Fig. 7a), and the averaged experimental ratio is in good agreement with the theoretically calculated value of the nucleartomagnetic scatteringlength density contrasts . For the computation of the latter, we used with m^{−2} and m^{−2}, and 4.06×10^{14} m^{−2} with M_{s} = 1400 kA m^{−1} for Co (Skomski, 2003) and M_{s} = 0 for the nonmagnetic Al_{2}O_{3} matrix. This finding suggests that the nuclear and magnetic form factors of the nanorods are not too different from each other, in agreement with the observations in Fig. 7(a) and the fit results listed in Table 1.
Let us now discuss the field dependence of . By reducing the field from the saturation value of T to smaller fields, the total nuclear and magnetic SANS cross sections in both scattering geometries increase at smaller nm^{−1}, and the total intensity in the first Bragg peak is slightly reduced and washed out (compare Fig. 6). The intensity increase continues until the coercive fields ( T in Hk_{i} geometry and 0.25 T in Hk_{i} geometry) are reached. Further reduction of the fields to more negative values leads again to a decrease of the scattering intensity (see data at T in Fig. 6).
The conventional `standard' expression for describing magnetic SANS data of magnetic nanoparticles that are embedded in a homogeneous nonmagnetic matrix considers the particles to be homogeneously (or stepwise homogeneously) magnetized (Heinemann et al., 2000; Wagner & Kohlbrecher, 2005; Wiedenmann, 2005; Disch et al., 2012). The possible continuous spatial dependence of the magnetization of the particles is ignored. For a dilute assembly of N monodisperse magnetic nanoparticles in the scattering volume V, the magnetic part of the total unpolarized SANS cross section is usually expressed as (Heinemann et al., 2000; Wagner & Kohlbrecher, 2005; Wiedenmann, 2005; Disch et al., 2012)
The only dependency on the applied magnetic field in equation (8) is contained in the function , which takes into account the dipolar character of the neutron–magnetic interaction (Halpern & Johnson, 1939; Shull et al., 1951). One may also include a in equation (8) [compare equation (7)], but (for rigid nanoparticles in a rigid matrix) this would only affect the q dependence of the scattering (similar to a particlesize distribution), not its field dependence. We also note that different definitions regarding the angle α can be found in the literature (Shull et al., 1951; Heinemann et al., 2000; Wagner & Kohlbrecher, 2005; Wiedenmann, 2005; Disch et al., 2012).
If α is taken to be the angle between and the local direction of the magnetization of a uniformly magnetized nanoparticle, then, for Hk_{i} geometry, the of the function varies between a value of 1/2 at saturation and a value of 2/3 in the demagnetized state; for Hk_{i}, the of varies between a value of 1 at saturation and a value of 2/3 in the demagnetized state (Halpern & Johnson, 1939; Shull et al., 1951). In other words, the above definition of α in combination with the standard expression for the SANS cross section of (dilute) nanoparticles, equation (8), can only explain an intensity increase by a factor of 4/3 (between saturation and the case of random domain orientation) in Hk_{i} geometry, whereas it predicts an intensity decrease with decreasing field for Hk_{i}. This is, however, inconsistent with the experimental observations in this work.
The measured radially averaged SANS cross sections in Hk_{i} geometry change at least by a factor of 4 at nm^{−1} with decreasing applied magnetic field (see Fig. 6a); in the `pocket' at nm^{−1} the scattering changes by a factor of about 5. For Hk_{i} geometry, the situation is even more striking, since here we observe an intensity increase (at least by a factor of 8 at small q) with decreasing field (see Fig. 6b).
As mentioned before, the obvious reason why equation (8) is not suited for describing the magnetic field dependent SANS cross section of the Co nanorod array is related to the fact that it describes magnetic scattering from homogeneously magnetized domains (particles). For magnetic microstructures where the magnetization vector field depends on the position inside the sample, i.e. M_{z}(x,y,z)], the corresponding SANS cross sections are given by equations (1) and (2), where the angle θ specifies the orientation of the scattering vector on the twodimensional detector. Besides its spatial dependence, depends of course on the applied magnetic field, the magnetic interaction parameters and the details of the microstructure.
At saturation, equations (1) and (2) reproduce the anisotropy (Hk_{i}) and the isotropic scattering pattern (Hk_{i}) (Fig. 5). At lower fields, spinmisalignment SANS with related transverse Fourier coefficients and contributes to the total , and, at least for bulk may give rise to a variety of angular anisotropies (Michels et al., 2006, 2014; Döbrich et al., 2012). In Fig. 5, the spinmisalignment SANS is observed as the intensity that emerges with decreasing field at the smallest q values. The analysis of the SANS data at saturation suggests an average nanorod diameter of about 30 nm. The existence of intraparticle spin misalignment would then give rise to magnetic SANS at , in agreement with our observations in Fig. 6. We note that in nanocrystalline bulk the field dependence of spinmisalignment SANS can be several orders of magnitude between a field close to saturation and the coercive field (Honecker et al., 2011; Bick, Honecker et al., 2013; Bick, Suzuki et al., 2013; Honecker et al., 2013).
The origin of the spin misalignment within the individual Co nanorods, which gives rise to the strong field dependence of , may be related to the polycrystalline nature of the rods: besides the dipolar shape anisotropy, which prefers an alignment of along the long rod axis, there are magnetocrystalline and magnetoelastic anisotropies (due to stressactivate microstructural defects) which give rise to internal spin disorder. Additionally, the magnetostatic stray field that emerges from neighbouring rods may produce inhomogeneous spin structures inside a given rod. A rigorous calculation of the magnetization distribution of such a nanorod array (and of the ensuing magnetic SANS) by means of numerical micromagnetics (Hertel, 2001; Nielsch et al., 2002; Zighem et al., 2011; Kulkarni et al., 2013; Bran et al., 2013) is a very complicated problem and is beyond the scope of this paper.
5. Summary and conclusion
We have reported the results of magnetic field dependent unpolarized SANS experiments on a Co nanorod array. Measurement of the SANS cross section in a saturating applied field of 2 T for two different scattering geometries (Hk_{i} and Hk_{i}) allows us to separate nuclear from magnetic SANS without employing the usual sector averaging in unpolarized SANS. The ratio of the experimentally determined nucleartomagnetic scattering is in good agreement with the theoretically expected value. The total SANS data in the saturated state (as well as the corresponding nuclear and magnetic contributions) could be well described by a model that combines a with the form factor of a cylinder. The obtained structural parameters (cylinder radius and centretocentre distance) of the Co nanorod array are consistent with the results from Between 2 T and the respective coercive fields, we observe a relatively strong field dependence of , for instance, by a factor of 4 for Hk_{i}. This cannot be explained by the standard expression for , which assumes uniformly magnetized domains. It seems obvious that the strong field dependence of is related to intraparticle spin misalignment.
Acknowledgements
Financial support by the National Research Fund of Luxembourg (ATTRACT project No. FNR/A09/01 and AFR project No. 1164011) is gratefully acknowledged. We thank Dominic Rathmann and Jörg Schmauch for assistance in using the ion beam milling system at the Materials Science and Engineering Department at the Universität des Saarlandes. We thank André Heinemann for critically reading the manuscript.
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