The behaviour of a group cannot be simply reduced to the sum of the behaviour of individuals (Le Bon, 1895). Examples abound in nature: school of fish or swarms of insect act like a single individual, but with emergent properties, i.e. properties which none of the individuals in the group possess, but which appear when these individuals are grouped into a system. A key element characterising a crowd is that the individuals should know that they are part of an ensemble (Le Bon, 1895). In other words, there must be interactions between them, on a short or long range. Otherwise they only constitute a system of several isolated individuals without emergent properties. Whatever the system, it is usually much easier to access its global behaviour, compared to the behaviour of individuals and their method of communication. However, the understanding of the global behaviour, in order to be able to control it subsequently, necessarily requires the knowledge of individual behaviour and of their interactions.
According to these criteria, the magnetic behaviour of structures composed of magnetic entities embedded in a non-magnetic matrix must be considered as a crowd: entities "talking" to each other via exchange or dipolar interactions depending upon the distance that separates them. Therefore, the behaviour of the global (or collective) structure may differ from that of the local (or individual) entities. This difference will be accentuated in the presence of non-uniformity (geometric, structural, etc.) of the entities. To access the magnetostatic properties of a system (coercivity, remanence, interactions, etc.), one usually uses a magnetometer to measure the major hysteresis curve. It is possible to use the same experimental setup to obtain the local magnetostatic properties of the system, by measuring multiple minor hysteresis curves, called first-order reversal curves (FORC) (Mayergoyz, 1985).
This technique can be particularly efficient and powerful in the case of highly interacting systems, like ferromagnetic nanowire arrays (Fig. 1). The large dipolar interaction field between the wires, associated with small interwire distance, strongly affects the overall array behaviour. Array properties (high anisotropy, resonance frequency in the gigahertz, etc.) make them promising candidates for high frequency devices (Saib et al., 2005), high-density magnetic memories (Almawlawi et al., 1991; Ross, 2001) and sensors (Lindeberg & Hjort, 2003). Moreover, with a multilayer structure (i.e. alternation of magnetic and non-magnetic nanodiscs), controlling the ratio between the magnetic and non-magnetic nanodiscs provides easy control of the effective anisotropy of the array (Tang et al., 2007; Carignan et al., 2007). This possibility makes the multilayer nanowire array particularly interesting for high frequency devices (Ye et al., 2007), especially circulators (Saib et al., 2001), although the initial goal of using multilayer nanowires was to obtain giant magnetoresistance devices (Blondel et al., 1994, Piraux et al., 1994; Piraux et al., 2007).
Despite the advantages of the FORC method for nanowire array characterisation, few studies exist on the subject (Spinu et al., 2004; Béron et al., 2006; Lavín et al., 2008; Peixoto & Cornejo, 2008; Béron et al., 2008b). This is mainly due to the lack of interpretative models suitable for highly interacting systems. This confines the analysis of the FORC result, which graphically represents the magnetisation reversal, to qualitative conclusions. This problem motivated the elaboration of the physical analysis model (Béron et al., 2008a), which is based on physically meaningful hypotheses. It is therefore possible to extract the intensity and spatial distribution of the interaction field at saturation, the coercivity distribution of individual nanowires (Béron et al., 2008b) and a quantitative evaluation of the overall magnetic anisotropy (Béron et al., 2007). In addition, insights about the magnetisation reversal mechanism can also be obtained. Here, we demonstrate the utility of FORC characterisation of nanowire arrays by showing how to extract these local properties from a global measurement. The versatility of the FORC method is demonstrated by presenting experimental examples of nanowire arrays with different saturation magnetisations (Ni = 490 emu/cm3, Co94Fe5B1 = 1500 emu/cm3), structures (uniform or multilayer) and applied field directions (parallel or perpendicular to the nanowire axis).
For systems such as arrays of nanoparticles with high dispersion of such properties as particle size, spacing and geometry, it may be more difficult to analyse the effects of these parameters, and to develop the kind of physical models described here. For systems such as sheets of transformer steel, it may even be difficult to identify physically meaningful objects whose properties and interactions may be modelled. However, we believe that the FORC method is sufficiently powerful, and the physical modelling approach presented here is sufficiently promising, that effort in this direction is likely to be rewarded.
2. First-order reversal curve (FORC) method
Application of the FORC method requires that the system exhibits hysteretic behaviour. A phenomenon is called hysteretic when, for a given value of input, there exist several possible output values. The principal consequence is the division into two paths between the saturation points, depending on the direction of variation of the input. The set of the two branches is called hysteresis cycle or major hysteresis curve, while the term hysteresis area refers to the interval between the saturation points, i.e. where the two path are not superposed (Fig. 2). The remanence represents the state for a null input, while the coercivity is the input value needed for the system to give a null output. Finally, the slope of the hysteresis curve is called susceptibility (
From a mathematical point of view, a hysteretic phenomenon can be described as a transducer for which the input-output relationship has multiple branches, a branch transition occurring for each applied extreme input value. A hysteresis mathematical model thus needs to keep in memory the previous extreme applied input values (Mayergoyz, 1985).
The nature of the physical phenomena exhibiting a hysteresis is exceedingly diverse: magnetic, mechanical, optical, etc. However, because of its mathematical nature, the following model and the conclusions derived from it remain valid regardless of the physical phenomenon. One just needs to adjust the input and output variables with the correct parameters, for example, the applied magnetic field and the magnetisation in case of magnetic materials, or the applied tension and the deformation generated for a plastic material.
2.1. Measurement and calculations
The FORC method was based, at its origin, on the classical Preisach model (Preisach, 1938), a mathematical model where hysteresis can be modelled as a set of elementary processes, or operators, called hysterons and characterized by two parameters,
The goal of any FORC measurements is to retrieve the
In order to obtain the complete representation of the magnetisation reversal, the reversal field values cover the hysteretic area of the major hysteresis curve, from
Before calculating equation (1), the FORCs are extrapolated in a way which minimises the discontinuity at
A value of η = 0 corresponds to a completely irreversible process and of η = 1 to a fully reversible process.
2.2. Analysis of results
Information about both reversible and irreversible processes can be combined in a graphical representation called “FORC result” ( Fig. 6 ) (Béron, 2008). It consists of a contour plot of the FORC distribution ρFORC, with a scale going from blue (minimum ρFORC) to red (maximum ρFORC), which directly indicates the statistical distribution of the mathematical hysterons. In order to read the
In the classical Preisach model approach, the
The FORC result analysis is generally the most difficult step in the use of the FORC method.
The three major problems are 1) the FORC distribution represents the statistical distribution of entities that do not necessarily have a physical meaning (the mathematical hysterons), 2) the FORC distribution can be deformed by the presence of state-dependent interaction field during the reversal, such as a mean interaction field, and 3) different physical systems can exhibit identical FORC results. Several approaches to FORC interpretation, designed to overcome these problems, exist. One is called the physical analysis model (Béron et al., 2008a), which allows us to establish some quantitative relationships between the characteristics of a physical system and of the FORC result. It is achieved by using physically meaningful hysterons, representing the supposed magnetic behaviour of the system entities, coupled with an interaction term, to simulate the global behaviour of the system. For example, the behaviour of each nanowire can theoretically be represented by the coherent rotation model (Stoner & Wohlfarth, 1948). The nanowire is approximated as an infinite cylinder where all the spins remain parallel to each other during the magnetisation reversal. This defines two different hysterons, depending upon the applied field direction, that can be used in the physical analysis model ( Fig. 7 ).
The FORC behaviour strongly depends upon the type of hysterons. For example, for easy axis hysterons, a coercivity distribution will spread the FORC distribution along the
An antiparallel interaction field (opposite to the magnetisation) will elongate the FORC distribution along the
From a quantitative point of view, the average individual coercivity of easy hysterons can be extracted from the FORC result by taking the
Finally, in the presence of a mean parallel interaction field (
3. Fabrication of nanowire arrays
Nanowire arrays may be fabricated by electrodeposition of metallic ions into the cylindrical pores of a dielectric template (Masuda & Fukuda, 1995). This deposition technique is economic, fast, versatile and does not require high vacuum facilities. A three electrodes configuration is typically used, the sample being used as the working electrode while the counter electrode is in platinum (Pt) ( Fig. 9 ). The applied potential is given with respect to that of the reference electrode, usually a saturated calomel, because of the stability of its potential.
The nanoporous dielectric template is typically made of anodised aluminium (alumina) (O’Sullivan & Wood, 1970), ion bombarded polycarbonate (Ferain & Legras, 2003) or diblock copolymer (Thurn-Albrecht et al., 2000). It is important that the pores be of constant diameter, parallel to each other, and perpendicular to the template surface. The most widely used template is currently alumina, mainly due to the ease of tailoring the template geometry (pore diameter, length and interpore distance) in laboratory, to the hexagonal arrangement of the pores, and to the rigidity of the template. They are fabricated in a two-step anodisation process ( Fig. 10 ) (Masuda & Fukuda, 1995; Pirota et al., 2004; Zhao et al., 2007). A cleaned aluminium substrate is first slowly anodised in an acidic bath under constant voltage. Since the pore order is getting more regular during this process, the initial layer of alumina (aluminium oxide) is chemically etched, before a second anodisation is performed, under the same conditions as the first. Finally, a chemical etch can enlarge the pores, if needed.
The samples discussed here were all fabricated in commercially available alumina templates (thickness = 60 μm, pore diameter = 175 nm, interpore distance = 300 nm, pore density = 109 pores/cm2, figure 11a) . However, due to the filtration purpose of these templates, the pores split into multiple smaller pores in the first 500 to 700 nm ( Fig. 11b) . This region was therefore removed by mechanical polishing before magnetic characterisation, in order to make the magnetic behaviour more uniform.
The pulse-current electrodeposition technique was used for the fabrication of the uniform nanowires. This technique consists of sending short current pulses (8 ms), during which the deposition occurs, followed by a pause (152 ms) (Ciureanu et al., 2005) ( Fig. 12a) . It avoids the formation of composition gradients along the nanowires (Nielsh et al., 2000), prevents formation of hydrogen in the pores and therefore allows uniform filling of the pores. Neither composition (Ni and CoFeB) exhibited large magnetocrystalline anisotropy, since the CoFeB was amorphous and the magnetocrystalline anisotropy of Ni is low. Multilayer nanowires can also be fabricated in an electrolytic bath containing all the elements to be deposited (Blondel et al., 1994; Piraux et al., 1994). In this case, the potential is alternated between two values, where the deposition of each material is favoured for one value of potential (Alper et al., 1993) (Fig. 12b). The non-magnetic metal is kept dilute in the solution to avoid excessive inclusion in the magnetic layer. Ni/Cu nanowires were fabricated by this method (-1 V for Ni, -0.56 V for Cu), varying the Cu/Ni ratio from 0.3 to 1.17 (Carignan et al., 2007).
4. Experimental FORC results
The experimental FORC results for four different nanowire arrays, two uniform (Ni and CoFeB) and two multilayer, are presented in Figs 13-16 as typical examples. Despite the fact that the nanowire arrays have the same geometry, one may easily see that they exhibit different magnetic behaviour. They were therefore chosen in order to exhibit the versatility of the FORC method. They are all characterised by the presence of both reversible and irreversible processes, which complicates their quantitative analysis.
In an axial applied field, the FORC distribution of elongated cylinders always exhibits the same kind of shape, i.e. narrow in the
5. Extracting individual nanowire properties
5.1. Individual nanowire coercivity/interaction field
In strongly interacting systems, like ferromagnetic nanowire arrays, the extraction of the interaction field and of individual coercivities have to be done together. This is because the interaction field modifies the FORC distribution, which can then no longer be analysed using the classical Preisach model, in which the coercive and interaction fields are independent. In the specific case of nanowire arrays, both FORC results obtained with axial and with transverse applied fields should be considered simultaneously.
5.1.1. Axial applied field
When the axial FORC distribution is qualitatively equivalent to that presented in Fig. 8b, as is the case for all the experimental distributions presented above (Figs 13-16), it suggests a narrow coercivity distribution and a net antiparallel interaction field. This allows us to use
If we assume a uniform, antiparallel interaction field, described by Eq. (4), we expect a FORC distribution that is straight and parallel to the
First, a spatial distribution of interaction field caused by the array border can induce a central peak in the
Second, if the antiparallel interaction field decrease allows the preponderance of a parallel interaction field (for example, the dipolar field between nanodiscs within a multilayer nanowire), then, in addition to the central peak, another distribution can appear, reflecting this parallel interaction field effect (ex.: distribution of higher coercivity, circled on figure 16a) .
Finally, for arrays of radius significantly larger than the nanowire length and for which the condition
5.1.2. Transverse applied field
With a transverse applied field on an infinite array, the interaction field at saturation is theoretically parallel to M, and equal to the half of the axial interaction field (Carignan et al., 2007). This value has been successfully used to fit the transverse major hysteresis curves of uniform nanowires (Béron, 2008). For FORC distributions that are narrow on the Hc axis, as in figures 15b and 16b, the transverse individual coercivity is indirectly accessible through use of the analysis illustrated in Fig. 8c. Subtracting the half value of the axial ΔHu from the transverse Hc FORC value allows us to retrieve the coercivity of individual nanodiscs in multilayer nanowires (Béron et al., 2008c).
5.2. Effective nanowire array anisotropy
In cases where the sum of the minimum values of the reversibility indicator η (Béron et al., 2007, see fig. 19a) in axial and transverse applied field is around 1, an indication of the overall anisotropy of the nanowire array can be obtained by the FORC method.
Otherwise, other processes contribute to the reversibility indicator and therefore the minimum value of η cannot be used to adequately characterise the effective nanowire array anisotropy. When the sum is around 1, the reversibility being lower along an easy direction, the lowest η minimum value indicates an easy direction. A cross-over between the η minimum value curves, occurring when the array is magnetically isotropic, implies a change in the easy axis direction ( Fig. 19b) (Béron et al., 2008c). This result for multilayer nanowires is in agreement with those found from the major hysteresis curves and ferromagnetic resonance measurements, as well as demagnetisation factor modelling (Carignan et al., 2007).
5.3. Magnetisation reversal process
Quantitative knowledge of the interaction field, average individual coercivity and overall array anisotropy, along with the FORC distribution shape, can help us determining the magnetisation reversal process. However, one has to remember that the FORC method does not directly give the type of magnetisation reversal occurring in the sample and so has to be used as a tool which suggests a probable mechanism, since systems reversing by different means can exhibit the same FORC result.
5.3.1. Axial applied field
For uniform nanowire arrays and axial applied field, coherent rotation reversal of individual nanowires leads to purely irreversible behaviour, with a unique value of Hc FORC equal to the shape anisotropy field (as in Fig 8b). Another possible reversal mechanism is by nucleation-propagation of a domain wall, which has been observed for the reversal of individual nanowires (Wernsdorfer et al., 1996; Hertel, 2001) (Fig. 20a). Then, the FORC result can exhibit some reversibility, a coercivity lower than the shape anisotropy field and a second distribution of coercivity near saturation, associated with annihilation of the domain walls (Béron, 2008) (circled on Fig. 20b). The FORC results presented in Figs 13a and 14a exhibit these three characteristics and, therefore, nucleation-propagation magnetisation reversal mechanism is more likely in these cases than coherent rotation.
5.3.2. Transverse applied field
For an applied field transverse to the nanowire axis, the coherent rotation model predicts fully reversible behaviour for a uniform individual nanowire, being along a hard direction, and no other FORC distribution. The existence of irreversible processes in Figs 13b and 14b, even in low proportion (1 - 4%), indicates that at least one other mechanism is present. This process could be the irreversible reversal of the domain walls created between regions decoupled enough to allow the magnetisation to rotate coherently freely toward one or the other nanowire extremity (Fig. 21) (Henry et al., 2002; Hertel, 2001). Other phenomena, such as closure domains at nanowire extremities, could also contribute to the non-null FORC distribution.
Other types of FORC results are possible and have to be analysed individually, based on the available information concerning the transverse magnetic behaviour of the array. For example, for multilayer nanowires, the FORC result presented in Fig. 15b was attributed to a mixture of coherent and incoherent rotation, the favourable conditions for one or the other changing during the reversal, while Fig. 16b was interpreted as an abrupt, irreversible reversal, caused by field applied along an easy direction (Béron et al., 2008c).
The unique structure of ferromagnetic nanowire arrays gives them interesting properties for various applications, ranging from high-density memory to high frequency devices. They may be fabricated by electrodeposition, a process which is fast, cheap and versatile. From a magnetic point of view, their behaviour is mainly governed by a competition between the shape anisotropy (an individual property of each nanowire) and the dipolar interaction field (an effect of the network). Therefore, it is important to be able to characterise both the individual and global properties of nanowire arrays, in order to understand their magnetic behaviour and to implement them adequately in devices.
This experimental characterisation can be achieved in one single step on the whole array, by using the very promising first-order reversal curve (FORC) method. The acquisition of several minor hysteresis curves allows us to record and discriminate the magnetisation reversal of each nanowire, ultimately yielding a representation of the reversal of all the nanowires. From this representation (the FORC result) and the help of the physical analysis model (based on predicted behaviour), one can extract valuable information, usually difficult to obtain experimentally. The half-width of the FORC distribution elongation (ΔHu) along the interaction field axis (Hu) gives the value of the interaction field at saturation, while a spatial distribution of this field modifies its shape. The spatial non-homogeneity can be in the array (border effect), which induces a central peak in the FORC distribution, or along the nanowires (extremity effect), which breaks the FORC distribution linearity along the Hu axis. An approximation of the average individual coercivity of the nanowires can be obtained by taking the position of the maximum of the FORC distribution on the coercivity axis (Hc) (Hc FORC). This approximation remains valid if the coercivity distribution is narrow, and both the reversibility indicator and the antiparallel interaction field are not too large. Also, in the presence of a parallel interaction field, its saturation value has to be subtracted from Hc FORC in order to reflect the individual coercivity. Finally, a quantitative evaluation of the effective anisotropy of the array is accessible through the minimum value of the reversibility indicator function of the reversal field, if the sum from axial and transverse applied fields is close to 1.
The versatility of the FORC method has been demonstrated here by discussing experimental results for nanowire arrays with different compositions (Ni and CoFeB), and thus different saturation magnetisation, and different structures (uniform and multilayer), all under both axial and transverse external applied fields. However, the application range of the FORC method is not confined to the two principal directions (axial and transverse) of the array. The physical analysis model can also be employed for the study of the magnetic behaviour as a function of the applied field angle (Béron et al., 2009a). The tridimensional and highly reversible magnetisation reversal process in this case led to the development of a novel variant of the FORC method, vector FORC, where both magnetisation components, parallel and perpendicular to the applied field direction, are used. This allows us to adequately follow and characterise reversible magnetisation rotation in three dimensions (Béron et al., 2009b).