Magnetic field dependent small-angle neutron scattering on a Co nanorod array: evidence for intraparticle spin misalignment

The results of magnetic field dependent small-angle neutron scattering measurements on a cobalt nanorod array are reported. The data provide evidence for the existence of intraparticle spin disorder.


Introduction
As a consequence of their interesting magnetic properties, magnetic transition-metal 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 microstructureproperty relationship is crucial (Goolaup et al., 2005;Zighem et al., 2011;Chumakov et al., 2011).
Small-angle neutron scattering (SANS) is a powerful volume-sensitive 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 MðrÞ (Wagner & Kohlbrecher, 2005;Wiedenmann, 2005;Michels & Weissmü ller, 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., , 2009Grigoryeva et al., 2007;Chumakov et al., 2011;Maurer et al., 2013). It is worth mentioning that for Ni nanowires (of average length 50 mm) 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 non-negligible 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 (short-rangeordered) 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.

Sample preparation and characterization
The Co nanorod array was prepared by pulsed electrodeposition 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 (Gü nther et al., 2008(Gü nther et al., , 2011Klein et al., 2009); here, we present only a brief outline of the sample preparation. The porous alumina template was synthesized by a two-step 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 À2 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 centre-to-centre distance of the pores of d cc ' 48 nm.
The pores were filled with Co by pulsed electrodeposition (Nielsch et al., 2000) from an aqueous solution composed of 0.3 M CoSO 4 Á7H 2 O and 45 g l À1 H 3 BO 3 at room temperature and a pH value of 6.4 (Ramazani et al., 2012). Such a Co-filled alumina template observed with scanning electron microscopy (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.
As 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 Ar-ion 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 cross-sectional areas of the nanorods, which sit flush with the alumina surface. The nanorods with average diameter d ' 27 AE 3 nm and length l ' 480 AE 45 nm are hexagonally arranged with a centre-to-centre distance of d cc ' 48 AE 5 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 between the magnetic field H and the long rod axes in the field range from À0.8 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    Magnetization measurements of the Co nanorod array, with being the angle between the applied magnetic field H and the long rod axes. along the long rod axis (Ramazani et al., 2012;Srivastav & Shekhar, 2014).

SANS experiment
SANS experiments were performed at KWS-1 (Jü lich Centre for Neutron Science, Outstation at MLZ, Garching, Germany), at V4 (Helmholtz-Zentrum 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 ¼ 8 Å [Á= ¼ 10% (FWHM)] and two sample-to-detector distances of 12.8 and 2.5 m, resulting in an accessible q range of 0:03 < $ q < $ 1:3 nm À1 . Magnetic field dependent measurements were carried out by first applying a large positive field ( 0 H ¼ 2 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 H?k i geometry ( Fig. 4a) and Hjjk 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).

SANS cross sections
For the scattering geometry where the applied magnetic field Hjje z is perpendicular to the wavevector k i jje x of the incoming neutron beam (H?k i ), the unpolarized elastic differential SANS cross section dAE ? =d of a ferromagnet can be written as (Michels & Weissmü ller, 2008) whereas for Hjjk i jje z one obtains In equations (1) and (2) was set to unity, which is permissible along the forward direction ( a : atomic magnetic moment; B : Bohr magneton). The above relation b m ¼ b H a defines the quantity b H ¼ 2:9 Â 10 8 A À1 m À1 , which is independent of the material (Michels & Weissmü ller, 2008); a was absorbed into the expression for the saturation magnetization M s , which enters the expression for the Fourier coefficients. Note that H is assumed to be parallel to e z in both geometries, so thatM M z ðqÞ in both equations (1) and (2) denotes the corresponding longitudinal magnetization Fourier coefficient, whileM M x ðqÞ andM M y ðqÞ are the respective transverse components, giving rise to spin-misalignment scattering. For H?k i , the angle is measured between H and q ffi q ð0; sin ; cos Þ, whereas for Hjjk i , is the angle between e x and q ffi q ðcos ; sin ; 0Þ (compare Fig. 4). At magnetic saturation, when the magnetization of the rods is perpendicular (H?k i ) or parallel (Hjjk i ) to the rod axes, equations (1) and (2) reduce to for H?k i and to for Hjjk i .

Results and discussion
The experimental differential SANS cross sections dAE=d of the Co nanorod array for the two scattering geometries are shown in Fig. 5   The two different scattering geometries for magnetic field dependent SANS. (a) H?k i geometry: the long rod axes are aligned parallel to the incident neutron beam k i jje x and perpendicular to the applied magnetic field H. (b) Hjjk i geometry: the long rod axes are aligned parallel to the incident neutron beam k i jje z and parallel to the applied magnetic field H. With reference to equations (1) and (2) we emphasize that in both geometries the applied-field direction H defines the e z direction of a Cartesian laboratory coordinate system and thatM M z ðqÞ denotes the respective longitudinal magnetization Fourier coefficient, whileM M x ðqÞ andM M y ðqÞ are the respective transverse components, varying in the e x e y plane. The angle specifies the orientation of the scattering vector on the two-dimensional detector; it is measured between Hjje z and q ffi ð0; q y ; q z Þ (a) and between e x and q ffi ðq x ; q y ; 0Þ (b).
saturation (left images) and the respective coercive fields (right images). At saturation in H?k i geometry, an intensity ring occurs with maxima perpendicular to H (seen as two dark-red 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 0 H c ¼ À0:05 T (Fig. 5a, right). The same qualitative behaviour is detected in Hjjk 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 half-moon intensity maxima in H?k i geometry reflect the angular anisotropy of the SANS cross section at saturation, which follows the well known sin 2 dependence [compare equation (3) and the discussion below]. By contrast, for the Hjjk i geometry, the SANS cross section at saturation exhibits an isotropically distributed intensity, i.e. dAE jj;sat =d depends only on the magnitude q of the scattering vector q; 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 dAE jj;sat =d implies that the sum of jÑ Nj 2 and jM M z j 2 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 two-dimensional detector images at 2 T can be identified in    Fig. 6) as the low-q peak at q 1 ffi 0:14 nm À1 (2=q 1 ffi 45 nm). Moreover, two additional peaks were detected at higher q values (q 2 ffi 0:25 nm À1 and q 3 ffi 0:38 nm À1 ), which can also be related to the hexagonal short-range order of the rods.
Before discussing the field dependence of dAE=d, 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 0 H ¼ 2 T, equations (1) and (2) reduce to equations (3) and (4). We now assume that both Fourier coefficients jÑ Nj 2 and jM M z j 2 are independent of the orientation of q [as supported by the two-dimensional data shown in Figs. 5(a) and (b)]. Radial averaging of the scattering cross section at saturation in H?k i geometry [equation (3)] then results in dAE ?;sat =d / b À2 H jÑ NðqÞj 2 þ 1=2 jM M z ðqÞj 2 , whereas for Hjjk i geometry we obtain dAE jj;sat =d / b À2 H jÑ NðqÞj 2 þ jM M z ðqÞj 2 . By assuming that jM M z j 2 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 so-determined experimental nuclear jÑ NðqÞj 2 and longitudinal magnetic jM M z ðqÞj 2 SANS cross sections are shown in Fig. 7(a); for simplicity, we will omit the constant prefactors 8 3 =V and ð8 3 =VÞb 2 H in the following. For the quantitative description of jÑ Nj 2 and jM M z j 2 as well as the SANS data at saturation (Fig. 7b), we consider a magnetic field independent model, where I inc denotes the incoherent scattering background, A is a scaling constant, which is proportional to the particle density and the respective scattering-length density contrast, V p is the particle volume, and Fðq; RÞ is the form factor of a cylinder for q being perpendicular to the long rod axes; Fðq; RÞ ¼ 2J 1 ðqRÞ=ðqRÞ, where J 1 ðqRÞ is the spherical Bessel function of first order with R ¼ d=2 being the rod radius. The structure factor is modelled as a sum of Gaussians, SðqÞ ¼ P i a i ð2 2 i Þ À1=2 exp½Àðq À q i Þ 2 =2 2 i , with the Bragg peak positions given by the two-dimensional hexagonal lattice at q i ¼ 4=ðd cc 3 1=2 Þ ðh 2 þ k 2 þ hkÞ 1=2 , where (hk) = (10), (11), (20), (21), (30) and (22).
The data fits by this model with I inc , A, a i , 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 electron microscopy, where we have found R ' 13:5 AE 1:5 nm and d cc ' 48 AE 5 nm.
The magnetic scattering contribution jM M z j 2 is larger than the nuclear SANS jÑ Nj 2 (see Fig. 7a), and the averaged experimental ratio jÑ Nj 2 =ðb H jM M z jÞ 2 ' 0:5 AE 0:2 is in good agreement with the theoretically calculated value of the nuclear-to-magnetic scattering-length density contrasts ðÁÞ 2 nuc =ðÁÞ 2 mag ' 0:7. For the computation of the latter, we used ðÁÞ nuc ¼ Al 2 O 3 nuc À Co nuc with Al 2 O 3 nuc ¼ 5:66 Â 10 14 m À2 and Co nuc ¼ 2:26 Â 10 14 m À2 , and ðÁÞ mag ¼ b H ÁM s ¼ 4:06 Â 10 14 m À2 with M s ¼ 1400 kA m À1 for Co (Skomski, 2003) and M s ¼ 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) Table 1 Resulting structural parameters obtained by fitting equation (7) to the nuclear jÑ Nj 2 and longitudinal magnetic jM M z j 2 SANS cross sections as well as to the SANS data at saturation dAE ?;sat =d and dAE jj;sat =d.
R denotes the rod radius and d cc the centre-to-centre distance of the rods in the alumina layer.
Let us now discuss the field dependence of dAE=d. By reducing the field from the saturation value of 0 H ¼ 2 T to smaller fields, the total nuclear and magnetic SANS cross sections dAE=d in both scattering geometries increase at smaller q < $ 0:25 nm À1 , 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 ( 0 H c ¼ À0:05 T in H?k i geometry and 0 H c ¼ À0:25 T in Hjjk 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 0 H ¼ À0:5 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 MðrÞ 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 sin 2 , which takes into account the dipolar character of the neutron-magnetic interaction (Halpern & Johnson, 1939;Shull et al., 1951). One may also include a structure factor 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 particle-size distribution), not its field dependence. We also note that different definitions regarding the angle can be found in the literature (Shull et al., 1951;Heinemann et al., 2000;Wagner & Kohlbrecher, 2005;Wiedenmann, 2005;Disch et al., 2012).
If is taken to be the angle between q and the local direction of the magnetization M of a uniformly magnetized nanoparticle, then, for H?k i geometry, the expectation value of the function sin 2 varies between a value of 1/2 at saturation and a value of 2/3 in the demagnetized state; for Hjjk i , the expectation value of sin 2 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 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 H?k i geometry, whereas it predicts an intensity decrease with decreasing field for Hjjk i . This is, however, inconsistent with the experimental observations in this work.
The measured radially averaged SANS cross sections in H?k i geometry change at least by a factor of 4 at q < $ 0:1 nm À1 with decreasing applied magnetic field (see Fig. 6a); in the 'pocket' at q ffi 0:2 nm À1 the scattering changes by a factor of about 5. For Hjjk 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 r inside the sample, i.e. M ¼ ½M x ðx; y; zÞ; M y ðx; y; zÞ; M z ðx; y; zÞ, the corresponding SANS cross sections are given by equations (1) and (2), where the angle specifies the orientation of the scattering vector on the two-dimensional detector. Besides its spatial dependence, M 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 sin 2 anisotropy (H?k i ) and the isotropic scattering pattern (Hjjk i ) (Fig. 5). At lower fields, spin-misalignment SANS with related transverse Fourier coefficientsM M x ðqÞ andM M y ðqÞ contributes to the total dAE=d, and, at least for bulk ferromagnets, may give rise to a variety of angular anisotropies (Michels et al., 2006(Michels et al., , 2014Dö brich et al., 2012). In Fig. 5, the spin-misalignment 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 q < $ 2=ð30 nmÞ ffi 0:21 nm À1 , in agreement with our observations in Fig. 6. We note that in nanocrystalline bulk ferromagnets the field dependence of spin-misalignment SANS can be several orders of magnitude between a field close to saturation and the coercive field Bick, Honecker 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 dAE=d, may be related to the polycrystalline nature of the rods: besides the dipolar shape anisotropy, which prefers an alignment of M 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.

Summary and conclusion
We have reported the results of magnetic field dependent unpolarized SANS experiments on a Co nanorod array.

research papers
Measurement of the SANS cross section dAE=d in a saturating applied field of 2 T for two different scattering geometries (H?k i and Hjjk i ) allows us to separate nuclear from magnetic SANS without employing the usual sector averaging in unpolarized SANS. The ratio of the experimentally determined nuclear-to-magnetic 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 structure factor with the form factor of a cylinder. The obtained structural parameters (cylinder radius and centre-to-centre distance) of the Co nanorod array are consistent with the results from electron microscopy. Between 2 T and the respective coercive fields, we observe a relatively strong field dependence of dAE=d, for instance, by a factor of 4 for H?k i . This cannot be explained by the standard expression for dAE=d, which assumes uniformly magnetized domains. It seems obvious that the strong field dependence of dAE=d is related to intraparticle spin misalignment.
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 Jö rg Schmauch for assistance in using the ion beam milling system at the Materials Science and Engineering Department at the Universitä t des Saarlandes. We thank André Heinemann for critically reading the manuscript.