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Angular first-order reversal curves: an advanced method to extract magnetization reversal mechanisms and quantify magnetostatic interactions
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 116004 (http://iopscience.iop.org/0953-8984/26/11/116004) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 116004 (7pp)
doi:10.1088/0953-8984/26/11/116004
Angular first-order reversal curves: an advanced method to extract magnetization reversal mechanisms and quantify magnetostatic interactions M P Proenca1 , J Ventura1 , C T Sousa1 , M Vazquez2 and J P Araujo1 1
IFIMUP and IN—Institute of Nanoscience and Nanotechnology and Departamento de Física e Astronomia, Universidade de Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal 2 Instituto de Ciencia de Materiales de Madrid, CSIC, E-28049 Madrid, Spain E-mail: [email protected] Received 16 October 2013, revised 20 December 2013 Accepted for publication 13 January 2014 Published 3 March 2014
Abstract
The magnetic properties of ordered hexagonal arrays of Co nanowires (NWs) and nanotubes (NTs) with diameters of 50 nm and interwire/tube distances of 105 nm were studied using first-order reversal curves (FORCs). We report an advanced analysis of angle dependent first-order reversal curves (AFORCs), measured by changing the angle of the applied magnetic field from θ = 0◦ (parallel to the wire/tube axis) to 90◦ (perpendicular). This method allowed us to determine the magnetization reversal mode and to retrieve quantitative information on the magnetostatic interactions between NWs and between NTs. In particular, we found a sharp increase in the coercivity distribution of the NT arrays for θ > 70◦ , which is attributed to a transition between vortex and transverse reversal modes. Local magnetic interactions are found to prevail in the Co NT arrays, steadily increasing from θ = 0◦ to 90◦ . However, in the Co NW arrays the mean magnetic interactions decrease as θ increases, going from ones similar to local interactions to ones smaller than them. Keywords: nanowires, nanotubes, porous alumina, magnetization reversal, FORC (Some figures may appear in colour only in the online journal)
1. Introduction
combining experimental data and analytical calculations of the angular dependence of the coercivity measured from major magnetic hysteresis loops [9–11]. Co NW arrays with interwire distances of Dint ∼ 105 nm and diameters in the range of 40 6 d 6 65 nm were found to reverse through the nucleation and propagation of a transverse domain wall (DW). However, above a critical NT diameter of d ∼ 50 nm, a transition from a vortex to a transverse DW nucleation–propagation process is seen at a particular θ [10]. The use of arrays of NWs/NTs adds new degrees of freedom as the importance of inter-element coupling increases. The growth of highly ordered nanomagnet arrays by a template assisted method has aroused great interest and shown many potentialities, as it makes it possible to control the
Magnetic nanowires (NWs) and nanotubes (NTs) are attracting considerable attention due to their unique properties and potential applications, ranging from uses in magnetic storage and recording devices to uses in sensors and biomedical chips [1–5]. The high aspect ratio of these elongated nanomagnets gives rise to shape dependent magnetic properties, thus increasing the potential applicability of such structures. In particular, one can tune the magnetic reversal mechanisms of NW/NT arrays by means of external parameters such as the angle (θ ) of the applied magnetic field (H ) [6–10]. The magnetization (M) reversal modes in NW and NT arrays with different diameters have been recently studied by 0953-8984/14/116004+07$33.00
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dimensions (lengths, diameters, interwire/tube distances, and wall thicknesses) of the arrays in a low cost and high yield manner [12–14]. Most recent studies on Co NWs and NTs have been focused on the search for the conditions under which crystalline anisotropy can be tuned [15–18]. Additionally, when studying arrays of nanomagnets it is important to understand the magnetostatic interactions between individual elements, as they will significantly affect the overall magnetic properties of the array [19–21]. A simple method for use in the analysis of the magnetostatic properties of a system (coercivity, remanence, interactions, etc) is to measure major magnetic hysteresis loops [22–24]. However, that only provides information regarding the global (average) behavior of the magnetic system. To obtain the local magnetostatic properties, Mayergoyz proposed the analysis of multiple minor hysteresis loops, called first-order reversal curves (FORCs) [25]. This technique has proved to be particularly efficient and powerful in the study of highly interacting systems, such as magnetic nanoparticles [26–29], antidots [30–32], and ferromagnetic NW arrays [14, 19, 33– 36]. Recently, Co NT arrays were also studied by using FORCs measured with H parallel to the tube axis [10]. However, the magnetic interactions between the nanoelements of an array also depend on the direction of the applied magnetic field. Therefore, in this work we present a detailed study on the angular dependence of the magnetic properties of ordered hexagonal Co NW and NT arrays, with d ∼ 50 nm and Dint ∼ 105 nm, using the FORC method. We introduce the analysis of the angular dependence of first-order reversal curve (AFORC) data, measured by tuning θ from 0◦ (parallel to the wire/tube axis) to 90◦ (perpendicular) in 10◦ steps. The analysis of the coercivity distributions presented in the AFORC diagrams of the Co NW and NT arrays revealed a single magnetization reversal mode for the NW arrays (except at θ = 90◦ , where a nearly reversible magnetization process was found to occur through coherent rotation), and a transition between two reversal modes in the Co NT arrays at θ ≈ 70◦ . We were also able to extract the angular dependence of the magnetic interactions in high aspect ratio NW and NT arrays. Local magnetic interactions were found to prevail for the NT arrays, increasing when θ goes from 0◦ to 90◦ . On the other hand, the magnetic interactions between NWs were found to decrease with θ , illustrating a smooth transition from a behavior where both local and mean interaction fields are present with equivalent strengths, to a state where local interactions become dominant. Thus, using the AFORC method, we were able to simultaneously study magnetization reversal processes and magnetic interactions between the nanomagnets.
then performed for 24 h and 20 h, respectively, producing ordered arrays of pores with diameters of d ∼ 35 nm, interpore distances of Dint ∼ 105 nm and lengths of L ∼ 50 µm. The pores were then opened from both sides of the template by chemically etching the Al surface and the alumina barrier layer present at the bottom of the pores. During this process, the pores’ diameters were also enlarged to obtain d ∼ 50 nm. For the subsequent potentiostatic electrodeposition of Co inside the pores, an Au metallic contact was sputtered at the bottom of the template to serve as the working electrode in a three-electrode cell. The thickness of the Au film was tuned to obtain NWs (∼100 nm) or NTs (∼50 nm). A Pt mesh and Ag/AgCl (in 4 M KCl) were used as the counter and reference electrodes, respectively. Further details on the membrane preparation for subsequent electrodeposition of NW/NT-like structures can be found in [38]. Co electrodeposition was performed in an aqueous solution of 0.89 M CoSO4 ·7H2 O and 0.49 M H3 BO3 , at 30 ◦ C, and applying a constant potential of −1.5 V versus Ag/AgCl, using a Solartron 1480 MultiStat. Morphological characterizations were performed using a scanning electron microscope (SEM; FEI Quanta 400FEG). Prior to bottom SEM imaging, ion milling was performed to remove the Au contact (200 nm) and smooth the sample surface. The milling process was carried out using an ion beam sputter deposition system from Commonwealth Scientific Corporation. The AFORC measurements were performed in a vibrating sample magnetometer (VSM; LOT-Oriel EV7) at room temperature. To perform a FORC measurement we first saturated the sample under a high positive magnetic saturation field of Hsat = 15 kOe. Then we decreased H until it reached the so-called reversal field Hr , and subsequently the magnetization M(H, Hr ) was measured by increasing H from Hr up to a maximum applied field Hmax . The minor hysteresis loop measured from Hr to Hmax is what is known as a first-order reversal curve (FORC). To complete the characterization of the system, Hr was ramped from a minimum applied field Hmin up to Hmax , and a set of FORCs were measured starting at each Hr value. The AFORCs were then obtained by changing θ from 0◦ to 90◦ , and measuring a set of FORCs at each θ . In this work, the maximum amplitude of the magnetic field (Hmax ) was tuned between 5 and 12 kOe depending on θ , and the reversal field (Hr ) steps corresponded to the respective values of Hmax divided by 50. 3. Results and discussion 3.1. Morphological characterization
Ordered hexagonal arrays of Co NWs and NTs with interwire/tube distances of Dint ∼ 105 nm, outer diameters of d ∼ 50 nm, NT wall thicknesses of t ∼ 10 nm, and aspect ratios (length/diameter) higher than 50, were obtained by electrodeposition inside the nanopores. Figure 1 shows bottom (after 200 nm ion milling) SEM images of Co NW and NT arrays in nanoporous alumina templates, evidencing the high order of the hexagonal array within domains of a few µm2 . In particular, figure 1(b) illustrates the tubular shape and the hollow cores that typically form a NT-like structure [39].
2. Experimental details
Nanoporous alumina templates were prepared by a two-step process of anodization [37] of high purity (99.999%) Al disks in 0.3 M oxalic acid at 40 V and 4 ◦ C. The Al surface was first cleaned in acetone and ethanol, and electropolished in a solution of 75% ethanol and 25% perchloric acid by applying 20 V for 2 min. First and second anodizations were 2
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Figure 1. SEM bottom (after 200 nm milling) images of Co ((a), (b)) NWs and (c) NTs in nanoporous alumina templates.
For the Co NTs extra FORCs were also measured at θ = 85◦ and 88◦ . Analyzing the AFORC diagrams represented in figure 2, one can observe that the NW arrays exhibit a large coercivity distribution (1Hc ), which can be seen as an enlargement of the FORC distribution along the Hc axis, or even the existence of a second branch, as depicted for θ = 0◦ . In this work, since we compare the magnetization reversal processes of polycrystalline Co NW and NT arrays with a mixture of fcc and hcp phases, the differences found between the coercivity distributions of the FORC diagrams of the two kinds of nanostructures were interpreted as mainly arising from different magnetization reversal modes. A broad coercivity distribution is usually attributed to the presence of reversal mechanisms, via nucleation and propagation of a DW [15, 40, 43, 44]. In particular, the second distribution of coercivity near saturation, well illustrated for the NW arrays at θ = 0◦ , can be associated with the annihilation of the DWs [32, 40, 43]. On the other hand, the NT arrays exhibit a much narrower coercivity distribution at lower angles (θ < 70◦ ), which then widens with increased θ , evidencing a transition between two different reversal mechanisms. The AFORC diagrams in figure 2 also illustrate a magnetic field of interaction, 1Hu , between the nanoelements of NW arrays much higher than those of the NT arrays, especially for H applied parallel to the wire/tube axis. However, these are seen to strongly decrease for the NW arrays as H becomes perpendicular to the wire axis (θ > 70◦ ).
3.2. Magnetic properties: the FORC method
The magnetic properties of the Co NW and NT arrays were studied by using angular first-order reversal curves (AFORCs). Graphical representations of the FORC dis2 r) tribution (ρFORC (H, Hr ) = −( 12 ) ∂ ∂M(H,H H ∂ Hr , with H > Hr ), called FORC diagrams, are illustrated in figure 2, consisting of contour plots of ρFORC , with a scale going from blue (minimum ρFORC ) to red (maximum ρFORC ) [40]. To simplify the analysis of the FORC diagrams, we performed a change of coordinates, defining a coercive field (Hc = (H − Hr )/2) axis and an interaction field (Hu = −(H + Hr )/2) axis. One can extract, from a quantitative point of view, from the FORC diagram, the interaction field at saturation (1Hu ) as the halfwidth of the FORC distribution elongation parallel to the Hu axis. Using the moving Preisach model [26, 41], which takes into consideration the interactions between the individual magnetic elements, one can approximate the interaction field at saturation as Hint,sat = 2σint − k, (1) where σint and k are the local and mean interaction fields, respectively [26, 42]. A positive (negative) value of k would correspond to a mean interaction field parallel (antiparallel) to M. In particular, an array of high aspect ratio NWs is known to have an antiparallel field of interaction (opposite to the magnetization) between every two elements, due to the strong demagnetizing field produced by the array. This effect can be seen in the FORC diagrams as a large elongation of the FORC distribution along the Hu axis (figure 2). Additionally, the shape of the FORC diagram along the Hc axis also provides information on the magnetization reversal processes. When two peaks appear along the Hc axis, the reversal occurs by domain wall (DW) motion, and the two irreversible regions are usually associated with the nucleation (at lower Hc ) and annihilation (at higher Hc ) of the DWs [32, 43].
To study the effect of the magnetization reversal mechanisms on the coercivity distribution of the AFORC diagrams, we performed a quantitative analysis of the AFORC measurements, estimating 1Hc (θ ) for the Co NW and NT arrays (figure 3). To estimate 1Hc we fitted the peak profile of the FORC diagram observed along the Hc axis (see the insets of figure 3) using a Gaussian function, and extracted the values of the full width at half-maximum. For the Co NT arrays, all the FORC diagrams exhibit a single peak along the Hc axis (figure 2). However, a broader coercivity distribution can be observed on the FORC diagrams of the NW arrays, evidencing two different peaks. For both kinds of nanostructured arrays, 1Hc is seen to increase overall with the angle of the applied magnetic field 3.2.2. Magnetization reversal.
3.2.1. AFORC measurements. To better understand the angular dependence of the magnetic properties of Co NW and NT arrays, we measured AFORCs at different angles of H with respect to the wire/tube axis, from θ = 0◦ to 90◦ (figure 2(c)), in 10◦ steps. Figure 2 shows selected FORC diagrams of the Co NW and NT arrays measured at θ = 0◦ , 40◦ , 70◦ , 80◦ and 90◦ . 3
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Figure 2. Comparative table of selected AFORC diagrams measured for Co NW and NT arrays with θ ranging from 0◦ to 90◦ (note the higher field scales for θ = 80◦ to 90◦ for the NT arrays). For the NT arrays extra FORC diagrams were also measured at θ = 85◦ and 88◦ . (a) and (b) show the FORCs measured at θ = 0◦ for the Co NT and NW arrays, respectively. (c) Schematic representation of the angle of the
applied magnetic field H with respect to the tube axis.
(figure 3). Note that the apparent existence of two regimes for the NW arrays arises from the overestimation of 1Hc for θ < 30◦ , due to the presence of high negative mean interaction fields at lower angles (see below), that broaden the coercivity distribution [36]. When comparing the two nanostructures, the higher 1Hc values are found for the NW arrays, indicating more complex and irreversible processes of magnetization
reversal and higher negative mean interactions. However, at θ = 90◦ the NW arrays exhibit a sharp decrease of 1Hc , which should be seemingly attributed to a nearly reversible magnetization process occurring by coherent rotation and to an increase in local interactions. For the Co NT arrays, one can clearly distinguish the existence of two different regimes: up to 70◦ , one observes 4
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Figure 3. Angular dependence of the coercivity distribution width (1Hc ) measured from the AFORC diagrams of Co NWs (closed
symbols) and NTs (open symbols). The respective linear fits are guides to the eye. Insets show the cross-section of the FORC diagram along the Hc axis for the NW (blue solid lines) and NT (green dashed lines) arrays, at θ = 0◦ , 40◦ , 85◦ and 90◦ , illustrating the different shapes of the coercivity distribution.
Figure 4. Angular dependence of the interaction field distribution (1Hu ) measured from the AFORC diagrams of Co NWs (closed symbols) and NTs (open symbols). Insets show the cross-section of the FORC diagram along the Hu axis for the NW (blue) and NT (green) arrays, at θ = 0◦ , 60◦ and 80◦ , illustrating the different shapes of the interaction field distributions.
a slow increase of 1Hc , while above 70◦ , 1Hc is sharply enhanced. This observation correlates well with the previously measured magnetization reversal processes for the same Co NT arrays [9, 10], where a transition between two different reversal modes (vortex to transverse) was found, lying between ∼55◦ and ∼65◦ . Therefore, we can attribute the two 1Hc (θ ) regimes to the formation of two different kinds of DWs: vortex, for 0 6 θ < 70◦ ; and transverse, for 70 6 θ 6 90◦ . Additionally, taking into consideration that the NW arrays reverse their magnetization through the nucleation and propagation of a transverse DW [10], the behavior of the angular dependence of 1Hc observed for both NW and NT arrays allows us to conclude that such a transverse DW enhances the coercivity distribution of the AFORC diagrams, while a vortex DW hardly affects its behavior.
A quantitative analysis of the AFORC diagrams was also performed along the Hu axis, providing additional information on the magnetic interactions (1Hu ) between the nanoelements of NW and NT arrays. Figure 4 shows the results obtained for the angular dependence of 1Hu in the NW (closed symbols) and NT (open symbols) arrays. For the NT arrays the interaction field distributions (green dashed insets in figure 4) exhibit a sharp peak, that can be well fitted using a Gaussian function. However, for the NW arrays these only appear at high angles (θ > 70◦ ), while at very low angles (θ < 40◦ ) one observes broader interaction field distributions with flatter tops (blue insets in figure 4). To better understand the kinds of magnetic interactions present in our samples, we compared our experimental data with previously reported theoretical results using the moving 3.2.3. Magnetic interactions.
5
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Figure 5. (a) Schematic representation of the magnetic interactions in NW arrays for θ = 0◦ (top) and θ = 90◦ (bottom). Cross-sections of the FORC diagrams along the Hu axis for (b) NWs and (c) NTs, at θ = 0◦ (black solid lines) and θ = 80◦ (blue dashed lines).
the array, and, as its width increases with θ , one can conclude that the dominant local interactions present in NT arrays are higher at θ = 80◦ . Therefore, at θ = 0◦ the interactions between nanoelements differ according to whether we have NWs (with both local and mean interactions) or NTs (with dominant local interactions); while at θ = 80◦ , local magnetic interactions prevail for both NW and NT arrays.
Preisach model (equation (1)) [42, 45]. Sharp peaks along the Hu axis suggest that local interactions are dominant with respect to the magnetostatic interactions between nanoelements (σint |k|; figure 5(a)) [10, 32, 42]. This behavior is seen in the Co NT arrays, independently of θ (green dashed insets in figure 4). Additionally, these local magnetic interactions tend to increase with θ , exhibiting higher values when H becomes closer to the perpendicular of the tube axis. The NW arrays exhibited a completely different behavior. In particular, the angular dependence of 1Hu appears to display a transition between a regime where broad peaks and high 1Hu values are found (θ < 40◦ ), and one where a sharp peak and low 1Hu values are observed (θ > 70◦ ; figure 4). Again, comparing with theoretical FORC diagrams previously reported [42], the existence of a broad interaction field distribution with a flattened top indicates that all NWs experience almost the same magnetostatic interactions and hence a higher negative mean interaction field (k < 0 and |k| σint ; figure 5(a)). Therefore, for θ < 40◦ , 1Hu has a strong k-component, which means that magnetostatic interactions between neighboring NWs are dominant (figure 5(a)). For 40◦ 6 θ 6 70◦ there is a crossover phase, where the interactions between nanoelements start to decrease, and a sharp peak appears in the interaction field distribution (inset of figure 4), corresponding to the decrease of |k| and the increase of σint . Finally, for θ > 70◦ only a sharp distribution is observed, like for the NT case, which means that the NW arrays have dominant local interactions when H approaches the perpendicular direction (|k| → 0 and σint |k|; figure 5(a)). Figures 5(b) and (c) compare the interaction field distributions measured at θ = 0◦ and 80◦ , for the Co NW and NT arrays, respectively. In the NW arrays the interaction field distribution clearly has a different shape for θ = 0◦ and 80◦ (figure 5(b)). For θ = 0◦ the distribution is broader and exhibits a slightly flattened top, while for θ = 80◦ a sharp peak is observed. Therefore, local interactions are dominant for θ = 80◦ , while for θ = 0◦ there is an additional component coming from the negative mean field of interaction between the NWs. For the NT arrays the two distributions have similar shapes, illustrating a sharp peak (figure 5(c)). This suggests that similar magnetic interactions occur, independently of the angle of the applied magnetic field. The sharp peak is associated with the existence of local magnetic interactions in
4. Conclusions
The magnetic properties of Co NW and NT arrays with d ∼ 50 nm and Dint ∼ 105 nm were studied using angular firstorder reversal curves. The quantitative analysis of the AFORC measurements allowed us to obtain the angular dependence of the coercivity distributions (1Hc ) and of the magnetic interaction field distributions (1Hu ). 1Hc was found to increase with θ for both NW and NT arrays. In particular, for the NT arrays two different regimes were found, associated with different magnetization reversal mechanisms. Therefore, AFORC measurements allowed us to conclude that the magnetization reversal occurring via the nucleation and propagation of a transverse domain wall leads to an increase in the coercivity distribution of the FORC diagrams, while the vortex mode of reversal leaves the coercivity distribution almost unaffected. From the analysis of 1Hu we observed that the NT arrays exhibit dominant local magnetic interactions that increase with θ , while the magnetic interactions decrease with θ for the NW arrays. In particular, for the NW arrays we found that with increasing θ one goes from the presence of a strong negative mean interaction field to an increased importance of local magnetic interactions. AFORC analysis is thus an advanced technique for unveiling the magnetization reversal mode and for quantifying magnetostatic interactions in nanomagnet arrays. Acknowledgments
MPP and CTS are grateful to FCT for grants SFRH/BPD/ 84948/2012 and SFRH/BPD/82010/2011, respectively. JV acknowledges financial support through FSE/POPH. MV thanks the Spanish Ministry of Economia y Competitividad, MEC, under project MAT2010-20798-C05-01. The authors acknowledge funding from FCT through the Associated Laboratory— IN. 6
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Angular first-order reversal curves: an advanced method to extract magnetization reversal mechanisms and quantify magnetostatic interactions
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 116004 (http://iopscience.iop.org/0953-8984/26/11/116004) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 129.108.9.184 This content was downloaded on 20/10/2014 at 06:21
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 116004 (7pp)
doi:10.1088/0953-8984/26/11/116004
Angular first-order reversal curves: an advanced method to extract magnetization reversal mechanisms and quantify magnetostatic interactions M P Proenca1 , J Ventura1 , C T Sousa1 , M Vazquez2 and J P Araujo1 1
IFIMUP and IN—Institute of Nanoscience and Nanotechnology and Departamento de Física e Astronomia, Universidade de Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal 2 Instituto de Ciencia de Materiales de Madrid, CSIC, E-28049 Madrid, Spain E-mail: [email protected] Received 16 October 2013, revised 20 December 2013 Accepted for publication 13 January 2014 Published 3 March 2014
Abstract
The magnetic properties of ordered hexagonal arrays of Co nanowires (NWs) and nanotubes (NTs) with diameters of 50 nm and interwire/tube distances of 105 nm were studied using first-order reversal curves (FORCs). We report an advanced analysis of angle dependent first-order reversal curves (AFORCs), measured by changing the angle of the applied magnetic field from θ = 0◦ (parallel to the wire/tube axis) to 90◦ (perpendicular). This method allowed us to determine the magnetization reversal mode and to retrieve quantitative information on the magnetostatic interactions between NWs and between NTs. In particular, we found a sharp increase in the coercivity distribution of the NT arrays for θ > 70◦ , which is attributed to a transition between vortex and transverse reversal modes. Local magnetic interactions are found to prevail in the Co NT arrays, steadily increasing from θ = 0◦ to 90◦ . However, in the Co NW arrays the mean magnetic interactions decrease as θ increases, going from ones similar to local interactions to ones smaller than them. Keywords: nanowires, nanotubes, porous alumina, magnetization reversal, FORC (Some figures may appear in colour only in the online journal)
1. Introduction
combining experimental data and analytical calculations of the angular dependence of the coercivity measured from major magnetic hysteresis loops [9–11]. Co NW arrays with interwire distances of Dint ∼ 105 nm and diameters in the range of 40 6 d 6 65 nm were found to reverse through the nucleation and propagation of a transverse domain wall (DW). However, above a critical NT diameter of d ∼ 50 nm, a transition from a vortex to a transverse DW nucleation–propagation process is seen at a particular θ [10]. The use of arrays of NWs/NTs adds new degrees of freedom as the importance of inter-element coupling increases. The growth of highly ordered nanomagnet arrays by a template assisted method has aroused great interest and shown many potentialities, as it makes it possible to control the
Magnetic nanowires (NWs) and nanotubes (NTs) are attracting considerable attention due to their unique properties and potential applications, ranging from uses in magnetic storage and recording devices to uses in sensors and biomedical chips [1–5]. The high aspect ratio of these elongated nanomagnets gives rise to shape dependent magnetic properties, thus increasing the potential applicability of such structures. In particular, one can tune the magnetic reversal mechanisms of NW/NT arrays by means of external parameters such as the angle (θ ) of the applied magnetic field (H ) [6–10]. The magnetization (M) reversal modes in NW and NT arrays with different diameters have been recently studied by 0953-8984/14/116004+07$33.00
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dimensions (lengths, diameters, interwire/tube distances, and wall thicknesses) of the arrays in a low cost and high yield manner [12–14]. Most recent studies on Co NWs and NTs have been focused on the search for the conditions under which crystalline anisotropy can be tuned [15–18]. Additionally, when studying arrays of nanomagnets it is important to understand the magnetostatic interactions between individual elements, as they will significantly affect the overall magnetic properties of the array [19–21]. A simple method for use in the analysis of the magnetostatic properties of a system (coercivity, remanence, interactions, etc) is to measure major magnetic hysteresis loops [22–24]. However, that only provides information regarding the global (average) behavior of the magnetic system. To obtain the local magnetostatic properties, Mayergoyz proposed the analysis of multiple minor hysteresis loops, called first-order reversal curves (FORCs) [25]. This technique has proved to be particularly efficient and powerful in the study of highly interacting systems, such as magnetic nanoparticles [26–29], antidots [30–32], and ferromagnetic NW arrays [14, 19, 33– 36]. Recently, Co NT arrays were also studied by using FORCs measured with H parallel to the tube axis [10]. However, the magnetic interactions between the nanoelements of an array also depend on the direction of the applied magnetic field. Therefore, in this work we present a detailed study on the angular dependence of the magnetic properties of ordered hexagonal Co NW and NT arrays, with d ∼ 50 nm and Dint ∼ 105 nm, using the FORC method. We introduce the analysis of the angular dependence of first-order reversal curve (AFORC) data, measured by tuning θ from 0◦ (parallel to the wire/tube axis) to 90◦ (perpendicular) in 10◦ steps. The analysis of the coercivity distributions presented in the AFORC diagrams of the Co NW and NT arrays revealed a single magnetization reversal mode for the NW arrays (except at θ = 90◦ , where a nearly reversible magnetization process was found to occur through coherent rotation), and a transition between two reversal modes in the Co NT arrays at θ ≈ 70◦ . We were also able to extract the angular dependence of the magnetic interactions in high aspect ratio NW and NT arrays. Local magnetic interactions were found to prevail for the NT arrays, increasing when θ goes from 0◦ to 90◦ . On the other hand, the magnetic interactions between NWs were found to decrease with θ , illustrating a smooth transition from a behavior where both local and mean interaction fields are present with equivalent strengths, to a state where local interactions become dominant. Thus, using the AFORC method, we were able to simultaneously study magnetization reversal processes and magnetic interactions between the nanomagnets.
then performed for 24 h and 20 h, respectively, producing ordered arrays of pores with diameters of d ∼ 35 nm, interpore distances of Dint ∼ 105 nm and lengths of L ∼ 50 µm. The pores were then opened from both sides of the template by chemically etching the Al surface and the alumina barrier layer present at the bottom of the pores. During this process, the pores’ diameters were also enlarged to obtain d ∼ 50 nm. For the subsequent potentiostatic electrodeposition of Co inside the pores, an Au metallic contact was sputtered at the bottom of the template to serve as the working electrode in a three-electrode cell. The thickness of the Au film was tuned to obtain NWs (∼100 nm) or NTs (∼50 nm). A Pt mesh and Ag/AgCl (in 4 M KCl) were used as the counter and reference electrodes, respectively. Further details on the membrane preparation for subsequent electrodeposition of NW/NT-like structures can be found in [38]. Co electrodeposition was performed in an aqueous solution of 0.89 M CoSO4 ·7H2 O and 0.49 M H3 BO3 , at 30 ◦ C, and applying a constant potential of −1.5 V versus Ag/AgCl, using a Solartron 1480 MultiStat. Morphological characterizations were performed using a scanning electron microscope (SEM; FEI Quanta 400FEG). Prior to bottom SEM imaging, ion milling was performed to remove the Au contact (200 nm) and smooth the sample surface. The milling process was carried out using an ion beam sputter deposition system from Commonwealth Scientific Corporation. The AFORC measurements were performed in a vibrating sample magnetometer (VSM; LOT-Oriel EV7) at room temperature. To perform a FORC measurement we first saturated the sample under a high positive magnetic saturation field of Hsat = 15 kOe. Then we decreased H until it reached the so-called reversal field Hr , and subsequently the magnetization M(H, Hr ) was measured by increasing H from Hr up to a maximum applied field Hmax . The minor hysteresis loop measured from Hr to Hmax is what is known as a first-order reversal curve (FORC). To complete the characterization of the system, Hr was ramped from a minimum applied field Hmin up to Hmax , and a set of FORCs were measured starting at each Hr value. The AFORCs were then obtained by changing θ from 0◦ to 90◦ , and measuring a set of FORCs at each θ . In this work, the maximum amplitude of the magnetic field (Hmax ) was tuned between 5 and 12 kOe depending on θ , and the reversal field (Hr ) steps corresponded to the respective values of Hmax divided by 50. 3. Results and discussion 3.1. Morphological characterization
Ordered hexagonal arrays of Co NWs and NTs with interwire/tube distances of Dint ∼ 105 nm, outer diameters of d ∼ 50 nm, NT wall thicknesses of t ∼ 10 nm, and aspect ratios (length/diameter) higher than 50, were obtained by electrodeposition inside the nanopores. Figure 1 shows bottom (after 200 nm ion milling) SEM images of Co NW and NT arrays in nanoporous alumina templates, evidencing the high order of the hexagonal array within domains of a few µm2 . In particular, figure 1(b) illustrates the tubular shape and the hollow cores that typically form a NT-like structure [39].
2. Experimental details
Nanoporous alumina templates were prepared by a two-step process of anodization [37] of high purity (99.999%) Al disks in 0.3 M oxalic acid at 40 V and 4 ◦ C. The Al surface was first cleaned in acetone and ethanol, and electropolished in a solution of 75% ethanol and 25% perchloric acid by applying 20 V for 2 min. First and second anodizations were 2
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Figure 1. SEM bottom (after 200 nm milling) images of Co ((a), (b)) NWs and (c) NTs in nanoporous alumina templates.
For the Co NTs extra FORCs were also measured at θ = 85◦ and 88◦ . Analyzing the AFORC diagrams represented in figure 2, one can observe that the NW arrays exhibit a large coercivity distribution (1Hc ), which can be seen as an enlargement of the FORC distribution along the Hc axis, or even the existence of a second branch, as depicted for θ = 0◦ . In this work, since we compare the magnetization reversal processes of polycrystalline Co NW and NT arrays with a mixture of fcc and hcp phases, the differences found between the coercivity distributions of the FORC diagrams of the two kinds of nanostructures were interpreted as mainly arising from different magnetization reversal modes. A broad coercivity distribution is usually attributed to the presence of reversal mechanisms, via nucleation and propagation of a DW [15, 40, 43, 44]. In particular, the second distribution of coercivity near saturation, well illustrated for the NW arrays at θ = 0◦ , can be associated with the annihilation of the DWs [32, 40, 43]. On the other hand, the NT arrays exhibit a much narrower coercivity distribution at lower angles (θ < 70◦ ), which then widens with increased θ , evidencing a transition between two different reversal mechanisms. The AFORC diagrams in figure 2 also illustrate a magnetic field of interaction, 1Hu , between the nanoelements of NW arrays much higher than those of the NT arrays, especially for H applied parallel to the wire/tube axis. However, these are seen to strongly decrease for the NW arrays as H becomes perpendicular to the wire axis (θ > 70◦ ).
3.2. Magnetic properties: the FORC method
The magnetic properties of the Co NW and NT arrays were studied by using angular first-order reversal curves (AFORCs). Graphical representations of the FORC dis2 r) tribution (ρFORC (H, Hr ) = −( 12 ) ∂ ∂M(H,H H ∂ Hr , with H > Hr ), called FORC diagrams, are illustrated in figure 2, consisting of contour plots of ρFORC , with a scale going from blue (minimum ρFORC ) to red (maximum ρFORC ) [40]. To simplify the analysis of the FORC diagrams, we performed a change of coordinates, defining a coercive field (Hc = (H − Hr )/2) axis and an interaction field (Hu = −(H + Hr )/2) axis. One can extract, from a quantitative point of view, from the FORC diagram, the interaction field at saturation (1Hu ) as the halfwidth of the FORC distribution elongation parallel to the Hu axis. Using the moving Preisach model [26, 41], which takes into consideration the interactions between the individual magnetic elements, one can approximate the interaction field at saturation as Hint,sat = 2σint − k, (1) where σint and k are the local and mean interaction fields, respectively [26, 42]. A positive (negative) value of k would correspond to a mean interaction field parallel (antiparallel) to M. In particular, an array of high aspect ratio NWs is known to have an antiparallel field of interaction (opposite to the magnetization) between every two elements, due to the strong demagnetizing field produced by the array. This effect can be seen in the FORC diagrams as a large elongation of the FORC distribution along the Hu axis (figure 2). Additionally, the shape of the FORC diagram along the Hc axis also provides information on the magnetization reversal processes. When two peaks appear along the Hc axis, the reversal occurs by domain wall (DW) motion, and the two irreversible regions are usually associated with the nucleation (at lower Hc ) and annihilation (at higher Hc ) of the DWs [32, 43].
To study the effect of the magnetization reversal mechanisms on the coercivity distribution of the AFORC diagrams, we performed a quantitative analysis of the AFORC measurements, estimating 1Hc (θ ) for the Co NW and NT arrays (figure 3). To estimate 1Hc we fitted the peak profile of the FORC diagram observed along the Hc axis (see the insets of figure 3) using a Gaussian function, and extracted the values of the full width at half-maximum. For the Co NT arrays, all the FORC diagrams exhibit a single peak along the Hc axis (figure 2). However, a broader coercivity distribution can be observed on the FORC diagrams of the NW arrays, evidencing two different peaks. For both kinds of nanostructured arrays, 1Hc is seen to increase overall with the angle of the applied magnetic field 3.2.2. Magnetization reversal.
3.2.1. AFORC measurements. To better understand the angular dependence of the magnetic properties of Co NW and NT arrays, we measured AFORCs at different angles of H with respect to the wire/tube axis, from θ = 0◦ to 90◦ (figure 2(c)), in 10◦ steps. Figure 2 shows selected FORC diagrams of the Co NW and NT arrays measured at θ = 0◦ , 40◦ , 70◦ , 80◦ and 90◦ . 3
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Figure 2. Comparative table of selected AFORC diagrams measured for Co NW and NT arrays with θ ranging from 0◦ to 90◦ (note the higher field scales for θ = 80◦ to 90◦ for the NT arrays). For the NT arrays extra FORC diagrams were also measured at θ = 85◦ and 88◦ . (a) and (b) show the FORCs measured at θ = 0◦ for the Co NT and NW arrays, respectively. (c) Schematic representation of the angle of the
applied magnetic field H with respect to the tube axis.
(figure 3). Note that the apparent existence of two regimes for the NW arrays arises from the overestimation of 1Hc for θ < 30◦ , due to the presence of high negative mean interaction fields at lower angles (see below), that broaden the coercivity distribution [36]. When comparing the two nanostructures, the higher 1Hc values are found for the NW arrays, indicating more complex and irreversible processes of magnetization
reversal and higher negative mean interactions. However, at θ = 90◦ the NW arrays exhibit a sharp decrease of 1Hc , which should be seemingly attributed to a nearly reversible magnetization process occurring by coherent rotation and to an increase in local interactions. For the Co NT arrays, one can clearly distinguish the existence of two different regimes: up to 70◦ , one observes 4
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Figure 3. Angular dependence of the coercivity distribution width (1Hc ) measured from the AFORC diagrams of Co NWs (closed
symbols) and NTs (open symbols). The respective linear fits are guides to the eye. Insets show the cross-section of the FORC diagram along the Hc axis for the NW (blue solid lines) and NT (green dashed lines) arrays, at θ = 0◦ , 40◦ , 85◦ and 90◦ , illustrating the different shapes of the coercivity distribution.
Figure 4. Angular dependence of the interaction field distribution (1Hu ) measured from the AFORC diagrams of Co NWs (closed symbols) and NTs (open symbols). Insets show the cross-section of the FORC diagram along the Hu axis for the NW (blue) and NT (green) arrays, at θ = 0◦ , 60◦ and 80◦ , illustrating the different shapes of the interaction field distributions.
a slow increase of 1Hc , while above 70◦ , 1Hc is sharply enhanced. This observation correlates well with the previously measured magnetization reversal processes for the same Co NT arrays [9, 10], where a transition between two different reversal modes (vortex to transverse) was found, lying between ∼55◦ and ∼65◦ . Therefore, we can attribute the two 1Hc (θ ) regimes to the formation of two different kinds of DWs: vortex, for 0 6 θ < 70◦ ; and transverse, for 70 6 θ 6 90◦ . Additionally, taking into consideration that the NW arrays reverse their magnetization through the nucleation and propagation of a transverse DW [10], the behavior of the angular dependence of 1Hc observed for both NW and NT arrays allows us to conclude that such a transverse DW enhances the coercivity distribution of the AFORC diagrams, while a vortex DW hardly affects its behavior.
A quantitative analysis of the AFORC diagrams was also performed along the Hu axis, providing additional information on the magnetic interactions (1Hu ) between the nanoelements of NW and NT arrays. Figure 4 shows the results obtained for the angular dependence of 1Hu in the NW (closed symbols) and NT (open symbols) arrays. For the NT arrays the interaction field distributions (green dashed insets in figure 4) exhibit a sharp peak, that can be well fitted using a Gaussian function. However, for the NW arrays these only appear at high angles (θ > 70◦ ), while at very low angles (θ < 40◦ ) one observes broader interaction field distributions with flatter tops (blue insets in figure 4). To better understand the kinds of magnetic interactions present in our samples, we compared our experimental data with previously reported theoretical results using the moving 3.2.3. Magnetic interactions.
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Figure 5. (a) Schematic representation of the magnetic interactions in NW arrays for θ = 0◦ (top) and θ = 90◦ (bottom). Cross-sections of the FORC diagrams along the Hu axis for (b) NWs and (c) NTs, at θ = 0◦ (black solid lines) and θ = 80◦ (blue dashed lines).
the array, and, as its width increases with θ , one can conclude that the dominant local interactions present in NT arrays are higher at θ = 80◦ . Therefore, at θ = 0◦ the interactions between nanoelements differ according to whether we have NWs (with both local and mean interactions) or NTs (with dominant local interactions); while at θ = 80◦ , local magnetic interactions prevail for both NW and NT arrays.
Preisach model (equation (1)) [42, 45]. Sharp peaks along the Hu axis suggest that local interactions are dominant with respect to the magnetostatic interactions between nanoelements (σint |k|; figure 5(a)) [10, 32, 42]. This behavior is seen in the Co NT arrays, independently of θ (green dashed insets in figure 4). Additionally, these local magnetic interactions tend to increase with θ , exhibiting higher values when H becomes closer to the perpendicular of the tube axis. The NW arrays exhibited a completely different behavior. In particular, the angular dependence of 1Hu appears to display a transition between a regime where broad peaks and high 1Hu values are found (θ < 40◦ ), and one where a sharp peak and low 1Hu values are observed (θ > 70◦ ; figure 4). Again, comparing with theoretical FORC diagrams previously reported [42], the existence of a broad interaction field distribution with a flattened top indicates that all NWs experience almost the same magnetostatic interactions and hence a higher negative mean interaction field (k < 0 and |k| σint ; figure 5(a)). Therefore, for θ < 40◦ , 1Hu has a strong k-component, which means that magnetostatic interactions between neighboring NWs are dominant (figure 5(a)). For 40◦ 6 θ 6 70◦ there is a crossover phase, where the interactions between nanoelements start to decrease, and a sharp peak appears in the interaction field distribution (inset of figure 4), corresponding to the decrease of |k| and the increase of σint . Finally, for θ > 70◦ only a sharp distribution is observed, like for the NT case, which means that the NW arrays have dominant local interactions when H approaches the perpendicular direction (|k| → 0 and σint |k|; figure 5(a)). Figures 5(b) and (c) compare the interaction field distributions measured at θ = 0◦ and 80◦ , for the Co NW and NT arrays, respectively. In the NW arrays the interaction field distribution clearly has a different shape for θ = 0◦ and 80◦ (figure 5(b)). For θ = 0◦ the distribution is broader and exhibits a slightly flattened top, while for θ = 80◦ a sharp peak is observed. Therefore, local interactions are dominant for θ = 80◦ , while for θ = 0◦ there is an additional component coming from the negative mean field of interaction between the NWs. For the NT arrays the two distributions have similar shapes, illustrating a sharp peak (figure 5(c)). This suggests that similar magnetic interactions occur, independently of the angle of the applied magnetic field. The sharp peak is associated with the existence of local magnetic interactions in
4. Conclusions
The magnetic properties of Co NW and NT arrays with d ∼ 50 nm and Dint ∼ 105 nm were studied using angular firstorder reversal curves. The quantitative analysis of the AFORC measurements allowed us to obtain the angular dependence of the coercivity distributions (1Hc ) and of the magnetic interaction field distributions (1Hu ). 1Hc was found to increase with θ for both NW and NT arrays. In particular, for the NT arrays two different regimes were found, associated with different magnetization reversal mechanisms. Therefore, AFORC measurements allowed us to conclude that the magnetization reversal occurring via the nucleation and propagation of a transverse domain wall leads to an increase in the coercivity distribution of the FORC diagrams, while the vortex mode of reversal leaves the coercivity distribution almost unaffected. From the analysis of 1Hu we observed that the NT arrays exhibit dominant local magnetic interactions that increase with θ , while the magnetic interactions decrease with θ for the NW arrays. In particular, for the NW arrays we found that with increasing θ one goes from the presence of a strong negative mean interaction field to an increased importance of local magnetic interactions. AFORC analysis is thus an advanced technique for unveiling the magnetization reversal mode and for quantifying magnetostatic interactions in nanomagnet arrays. Acknowledgments
MPP and CTS are grateful to FCT for grants SFRH/BPD/ 84948/2012 and SFRH/BPD/82010/2011, respectively. JV acknowledges financial support through FSE/POPH. MV thanks the Spanish Ministry of Economia y Competitividad, MEC, under project MAT2010-20798-C05-01. The authors acknowledge funding from FCT through the Associated Laboratory— IN. 6
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