Physica B 407 (2012) Contents lists available at SciVerse ScienceDirect. Physica B. journal homepage:

Physica B 407 (2012) 4676 4685 Contents lists available at SciVerse ScienceDirect Physica B journal homepage: www.elsevier.com/locate/physb PKP simulation of size effect on interaction field distribution in highly ordered ferromagnetic nanowire arrays Costin-Ionut Dobrotă, Alexandru Stancu n Faculty of Physics, Alexandru Ioan Cuza University of Iasi, 700506, Romania article info Article history: Received 18 June 2012 Received in revised form 27 August 2012 Accepted 28 August 2012 Available online 4 September 2012 Keywords: Magnetic nanowires FORC diagram method Local interaction mean field abstract Perpendicular structured nanowire arrays interaction field distributions (IFDs), as revealed from firstorder reversal curves (FORC) diagrams, are related to the presence of the demagnetizing field in the system. Despite the similarity between the geometric properties of bit patterned media and mentioned nanowire arrays, FORC diagrams of these two types of systems are not similar essentially due to the different number of magnetic entities influencing the switch of an individual element. We show that one Preisach Krasnosel skii Pokrovskii (PKP) symmetrical hysteron can be representative of an ideal infinite nanowire array when the field is applied along the wires. Starting from this observation, we present a very simple model based on PKP symmetrical hysterons that can be applied to real finite ferromagnetic nanowire arrays, and is able to describe a wide class of experimentally observed FORC distributions, revealing features due to size effects. We also present IFDs modeled for different geometric characteristics such as array size, interwire distance, and nanowire dimensions, and an identification procedure for the proposed model. & 2012 Elsevier B.V. All rights reserved. 1. Introduction Arrays of ferromagnetic nanowires have been extensively studied due to their potential application in ultrahigh density magnetic recording media, as magnetic sensors and in microwave applications [1 3]. Nanowire arrays can be produced by electrodeposition of ferromagnetic metals, such as Fe, Co, Ni and their alloys, in nano-sized templates, and their macroscopic magnetic properties can be tuned by geometrical characteristics of the template and by the deposition time [4 7]. In spite of the relative simplicity of this type of magnetic system, the hysteretic behavior cannot be easily explained by means of the known models especially due to the strong correlations between the switching of the neighboring wires [8 10]. The correct understanding of the inter-wires interactions is a pre-condition in the appropriate theoretical description of these systems. For any ferromagnetic sample, the information about the interactions can be experimentally obtained from more sophisticated magnetization processes than the major hysteresis loop (MHL), when as considerably different ferromagnetic systems can have virtually the same MHLs. Higher-order magnetization curves covering states located inside the MHL offer additional information that can be used for magnetic interaction characterization. Several experimental magnetization n Corresponding author. Tel.: þ40 232 201175; fax: þ40 232 201205. E-mail address: alstancu@uaic.ro (A. Stancu). protocols have been designed for that purpose, such as dm plots [11], Henkel plots [12] and the first-order reversal curves (FORC) diagram method [13 15]. The first two methods require remanent magnetization measurements that are on second-order reversal curves [16] and are very sensitive to the way demagnetizing states are obtained [17]. These are the main reasons why these methods provide only a qualitative estimation of the intensity and the type of magnetic interactions. FORCs were first described by Mayergoyz [14] as an identification procedure for the Classical Preisach Model (CPM). Pike [15] suggested the use of this method as an experimental tool to evaluate magnetic interactions in virtually any hysteretic system. The relation between the experimental FORC analysis and other Preisach-type models, especially with the Moving Preisach Model (MPM) [18], was later clarified [19,20]. Advantages of the FORC method originate from the simple data measurement protocol that can be easily implemented in modern VSMs and the well-defined initial state (saturation) in this protocol. The FORC diagram offers a global image of the distributions of coercivities and interactions in the system and it is a very sensitive tool in evidencing the physical sources of the observed macroscopic hysteretic behavior of the magnetic ensembles. The FORC method was first used to characterize magnetic nanowire arrays by Spinu et al. [21] and many other subsequent studies have followed the idea to describe magnetic interactions as a function of various parameters like composition, geometrical dimensions of the individual wires and the interwire distance [22 24]. More complex nanowire arrays such as segmented (barcode) 0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2012.08.041

C.-I. Dobrotă, A. Stancu / Physica B 407 (2012) 4676 4685 4677 nanowires consisting of alternating magnetic and non-magnetic segments or complex wires with single modulated diameter structure have been also investigated [25,26]. If the applied field is along the wires, in most cases FORC diagrams have a specific shape consisting of a narrow distribution of coercive fields and a wide and flattened interaction field distribution (IFD) with steep extremities, denoting the presence of the mean interaction field in the system. Additional distributions that are less prominent can be observed for some samples, both in horizontal and vertical profiles of the FORC diagram. In this paper we propose a model based on a physical approach which can describe, in a very simple manner, typical FORC distributions for nanowire arrays. The CPM considers that any magnetic material consists of many elementary particles and each of them can be represented by a rectangular hysteresis loop, named hysteron [13,27]. Symmetric hysterons are associated with non-interacting particles and the shift of the non-symmetric hysterons along the field axis will quantify the interaction field. Krasnosels kii, Pokrovskii and co-workers first formulated the Preisach model in terms of hysteresis operators and elucidated the phenomenological character of these operators applied in the study of processes with hysteresis in any scientific field [28]. They have shown that the Preisach model is based on a superposition of specific elementary mathematical operators (the relay operators) and conducted a systematic analysis of the mathematical properties of these operators. Krasnosels kii and Pokrovskii also proposed other elementary hysteresis operators, the so-called play and stop operators. For example, play-type models and stop-type models are built by superposition of a finite number of play or stop hysterons, and continuous distributions of the same hysterons give Prandtl Ishilinskii s model [28 33]. In this work, after a brief presentation of the FORC diagram method, we show the details of a model based on Preisach Krasnosel skii Pokrovskii (PKP) symmetrical hysterons that describe the hysteretic processes of interacting nanowire arrays that can accurately reproduce the typical FORC distribution obtained in experiments. The physical basis of the model is Fig. 1. Preisach hysteron with a critical field H c and an interaction field H u. presented and a comparison of the computed interaction fields in patterned media and in nanowire arrays is performed to highlight the subtle differences which are reflected in FORC diagrams. Considering highly ordered nanowire arrays having finite dimensions, we present IFDs for specific sets of PKP symmetrical hysterons, provided by FORC diagrams, and an identification procedure for our model. 2. FORC diagram method A typical FORC measurement starts from the descending branch of MHL at a reversal field H r and includes the experimental points between the reversal field and a field sufficient to saturate the sample again. The sample s magnetic moment on a FORC for a given reversal field is a function of two variables, m FORC (H,H r ), where H is the applied field during the measurement. The FORC distribution is proportional to the second order mixed derivative of the magnetic moment measured for a set of FORCs and is given by r FORC ðh,h r Þ¼ 1 @ 2 m FORC ðh,h r Þ : ð1þ 2 @H@H r The FORC diagram is the contour plot of the corresponding distribution and is represented in the Preisach plane, using the coercive field axis [H c ¼(H H r )/2] and the interaction field axis [H u ¼(HþH r )/2¼ H i ]. Actually, H u is the bias field meaning the shift of the rectangular hysteresis loop (see Fig. 1), but usually it is named interaction field (the interaction field is H i ¼ H u ). For one hysteron as in Fig. 1, the FORC distribution is one Dirac peak described by the pair of parameters H c and H u which represents the coercive and the interaction fields characteristic to the particle associated to the hysteron, respectively. For a symmetrical Preisach hysteron, the peak is positioned on the H u ¼0 axis (see Fig. 2). In order to understand more complex systems it is useful to analyze simpler cases first, e.g. FORC diagrams for singular Preisach distributions in a mean field approach, using the Moving Preisach Model [18]. Fig. 3(a) shows the first case we have investigated: a system with negligible interactions but with uniform distribution of coercive fields within a given field range. In this system mean field interactions are added with demagnetizing effect (the moving constant in the MPM is negative, ao0, and the interaction field is given by am where m is the normalized magnetic moment). A set of FORCs is calculated with the MPM and it is seen that in this case, the FORC diagram shows a two-branch structure, as we see in Fig. 3(b). This so-called wishbone shape is specific to the FORC diagrams of patterned media [34]. Fig. 2. (a) FORC diagram and (b) normalized interaction field of one classical Preisach hysteron without interactions (H u ¼0).

4678 C.-I. Dobrotă, A. Stancu / Physica B 407 (2012) 4676 4685 even if the hysteron is rectangular, the FORC diagram has a slight width due to the numerical interpolation algorithm of the distribution (1). In what follows, we relax the conditions previously imposed to model realistic FORC distributions experimentally observed for a wider class of nanowire arrays. 3. Physical basis of the model Fig. 3. Modeled FORC diagrams for singular distributions of 10,000 Preisach hysterons in the CPM [uniformly distributed with coercive fields between 200 Oe and 600 Oe without interactions in (a), and with interaction fields between 500 Oe and 500 Oe and coercive fields 300 Oe in (c)], and in the MPM for the same distributions and the moving parameter a¼ 400 Oe [(b), (d)]. Another case we have considered is a system with very narrow coercivity distribution and a uniform distribution of interactions with the Preisach distribution presented in Fig. 3(c). We add a similar demagnetizing-type mean field interaction as in the previous case and calculate the FORCs with the MPM model. The FORC diagram of this system is presented in Fig. 3(d). It can be easily seen that the measured FORC distribution is wider along the interaction field axis but the coercivity is rigorously the same as the initial one. This is the only effect of the mean field interactions. This FORC diagram shape is typical for most nanowire arrays [21 26]. Unfortunately, in this type of FORC distribution one cannot distinguish between an intrinsic statistical distribution of interactions (as the one we have assumed in our case) and a mean field effect. Actually, the diagram can be reproduced in the MPM with multiple sets of parameters. Interaction fields introduced in the CPM are named statistical fields and essentially they are related to system non-uniformities like different reciprocal positions and orientations of the particles in the system. However, nanowire arrays are highly ordered systems, so statistical interaction fields should have a narrow distribution. While experimental FORC diagrams are extended along the H u axis, we may assume that the main reason for this is the strong demagnetizing mean field within the sample. Aiming to describe nanowire arrays via FORC diagrams, we first define an ideal non-interacting nanowire array as an infinite and perfectly ordered grid with identical and parallel wires with their axes perpendicular to a plane. Having the same coercive field, all wires switch simultaneously at the same value of the applied external field, so the normalized FORC distribution for such an array is also one Dirac peak located on the horizontal coercivity axis (see Fig. 2(b)). Consequently, a population of identical and symmetrical Preisach hysterons can describe the behavior of the mentioned ideal non-interacting nanowire array, whereas one hysteron can describe one isolated wire. Note that For a better insight into the nature of the interaction field, we consider finite rectangular nanowire arrays to account in this analysis for the size effects as well. We have assumed an ideal case in which the cylindrical nanowires have the same length (L) and radius (R) and are perfectly ordered in a 2-D square array with constant a (interwire distance). The field created at the center of each wire is evaluated as the sum of interaction fields created by all the other wires, when the sample is magnetically saturated in an axial applied field. If one considers each wire as a uniformly magnetized cylinder, the uncompensated magnetic charges are only at the two bases of the cylinder. An accurate calculation should be based on the field created by two discs with positive and negative uniform surface distributions. The axial component of the field at the distance x on the cylinder axis mediator is opposed to the applied field and is given by Z R Z 2p H s z ¼ LM s r dy dr 0 0 ðx 2 þl 2 =4þr 2 3=2 2xrcos yþ ð2þ where rdy dr represents the elementary area on the disk, and M s ¼485 emu/cm 3 is the saturation magnetization for Nickel used in the next simulations. When the point at which one calculates the field is far from the two discs, one can use point charges instead of disc charges, at the disc centers, and the nanowire can be considered as a dipole with the length L. In this case the field is given by pr 2 LM s H z ¼ ð3þ ðx 2 þl 2 =4Þ 3=2 Using Eq. (2), we have computed interaction fields generated by cylindrical wires with the same radius R¼50 nm and different values of the aspect ratio L/2R, covering typical values of both the particle types from patterned media and nanowire arrays. As seen in Fig. 4(a), the action radius of the interaction field is strongly dependent on the aspect ratio. In the case of patterned media (aspect ratio less than 10), the interaction field decreases abruptly with the distance and vanishes at small distances. By contrast, for nanowire arrays (aspect ratio greater than 10), the range of the interaction field is huge. So, for one nanowire from an array, the interaction field experienced is due to a collective effect and an extremely large number of neighboring wires must be considered. To save computational time we have tested the validity of the dipolar approximation (3) in order to use it in our simulations. As shown in Fig. 4(b), the relative variation between the two interaction fields, computed with Eqs. (2) and (3), tends to zero for aspect ratio values which correspond to nanowires, even at short distances x. Larger differences are observed for small aspect ratio and at small distances, so in the case of the dense patterned media the dipolar approximation (3) does not work properly and the distribution of the magnetic charge from the cylinder bases cannot be neglected. As a conclusion of this analysis one may say that the dipolar approximation is appropriate in the case of arrays of ferromagnetic nanowires. Different action radii of the interaction fields for patterned media versus nanowire arrays generate effects on IFD, apparent especially for a small number of particles. Fig. 5 shows IFDs histograms (number of particles with a computed value of the

C.-I. Dobrotă, A. Stancu / Physica B 407 (2012) 4676 4685 4679 is 4pM s (6095 Oe for Nickel), which corresponds to a demagnetizing factor N¼4p for perpendicular magnetization, and if we consider infinite nanowire arrays with different wire radii, the demagnetizing field is 4pM s P. We have found that H centr is approximately 4pM s P for very large arrays (2001 2001 in Fig. 6(a)). For small arrays (401 401 in Fig. 6(b)), H centr markedly decreases from the 4pM s P value as a function of wire length due to boundary effects. Note that for very short wires (less than about 1 mm) the dipolar approximation (3) actually does not work properly. This is in fact the case of patterned media in which the interaction fields can be computed taking into account a small number of neighboring particles (typically of the first order), but which should be computed with the relation (2). Computed interaction fields at the center of the wires, in saturation state of the nanowire arrays, will be used as parameters in a model based on PKP symmetrical hysterons, and will be calculated as the sum of the fields generated by all the wires from the system. For nanowire arrays with hexagonal order, one can rescale the calculus to obtain the same areal wire density. If a h is the lattice constant of a hexagonal nanowire array, the lattice constant of the equivalent qffiffiffiffiffiffiffiffiffiffiffiffi p rectangular array is a ¼ 3 =2 ah. As the results show that for nanowire arrays the most important element is the mean field interaction given by large areas from the system, this equivalence is a good approximation. When the local interactions are crucially important in the magnetic behavior of individual elements, like in the patterned media systems, the global behavior of the samples will dependonthetypeofsymmetry. 4. PKP hysteron Fig. 4. (a) Normalized interaction field (2); and (b) comparison between computed interaction fields using Eqs. (2) and (3) at the distance x4r. interaction field) in the demagnetized state [35], both for patterned media (left column) and for nanowire arrays (right column). As the size of the matrix decreases, the histograms for the two types of systems become more different due to the contribution of a different number of neighboring particles on the interaction field experienced by each particle (see the interaction field axes). While pattern media present the same dispersion in the histograms, for nanowire arrays a decrease of the dispersion is observed when the dimension of the array becomes smaller. In an axial applied field, both patterned media and nanowire arrays present mean field interactions of demagnetizing-type which can be considered uniform over the whole array for the first type of systems, but not for the second. For this reason, the MPM can successfully describe patterned media [34], whereas for nanowire arrays a model which takes into account local interaction mean fields is more appropriate. After the effective range of the interaction field (3) was established, we have evaluated the limits of the dipolar approximation applicability for different geometrical characteristics of the nanowire arrays. The computed interaction field in the saturated state at the center of the array H centr versus the membrane porosity defined as P¼pR 2 /a 2 is presented in Fig. 6. As the uncompensated magnetic charge density increases by reducing the lattice constant, the computed interaction field at the center of the system also increases. In the limit when the ferromagnetic material fills the entire array, it can be considered as a thin film and it is equivalent to two planar distributions of magnetic monopoles. In this case, the interaction field at the center of the system was computed considering a nanowire array with parallelepipedal wires with base area a 2. If an infinite thin film is saturated, the demagnetizing field A hysteron is a basic building element (also called kernel) for a hysteresis operator. The loop of a play hysteron, represented in Fig. 7(a), is counterclockwise oriented and it is characterized by only one constant: the threshold value h ck. The play hysteron has linear limiting branches with slope equal to unity and any horizontal path can be covered in both directions. We propose a modified play hysteron in order to limit the output evolution if the input exceeds a certain limit value h sk,as seen in Fig. 7(b). In literature, this kind of hysteron which is not necessarily symmetrical is known as PKP hysteron, with the nonhorizontal branches usually having a fixed slope for all hysterons in a PKP operator [36]. We will consider a PKP symmetrical hysteron parameterized by a pair of variables: the critical field h ck and the saturation field h sk. FORCs uniformly cover the interior of the MHL of PKP symmetrical hysterons and we can extract information about the interaction field even for one hysteron. The FORC distribution shown in Fig. 8 is characterized by two parameters: H ck is the position of the maximum of the distribution on the H c axis and DH uk is the width of its extension along the H u axis. The parameters of one PKP symmetrical hysteron are easily found from the FORC diagram to be h ck ¼ H ck, h sk ¼ H ck þdh uk =2: ð4þ So, the difference h uk ¼ h sk h ck ð5þ is the measure of an interaction field revealed in the FORC diagram. Starting from a population of Preisach symmetrical hysterons and adding a negative (demagnetizing-type) interaction mean field (am, wherea¼ h uk is the moving parameter), an extended FORC diagram as in Fig. 8 is obtained. Thus, one PKP hysteron describes the evolution of a population of symmetrical Preisach hysterons without a dispersion of coercivities in a negative mean field, as we have illustrated in Fig. 2. It is interesting to observe that one single PKP symmetrical hysteron can be associated

4680 C.-I. Dobrotă, A. Stancu / Physica B 407 (2012) 4676 4685 Fig. 5. IFDs for small arrays of patterned media (PM) and nanowires (NW) in the demagnetized state, for interparticle distance a¼250 nm, cylinder radius R¼50 nm and aspect ratio (L/2R) of 1 for PM and 200 for NW. with an ideal interacting nanowire array in which every wire is subjected to the same local interaction mean field created by all wires from the system. This is in fact the case of an infinite and perfectly ordered array. 5. Model based on PKP symmetrical hysterons A Preisach-type operator consists of a weighted superposition of a large number of hysterons often associated with the number of particles from a magnetic system. Subsequently, we will consider the PKP operator as a parallel connection of a set of N symmetrical PKP hysterons, which is able to provide a hysteretic output of M from an input H in the same way as the Preisach model does. The PKP operator is given as MðHÞ¼ XN k ¼ 1 f k ðm k ðhþþ, ð6þ where m k is the PKP hysteron operator and f k is the shape function that we consider in its simplest form as f k (m k )¼m k. While each PKP symmetrical hysteron describes one local interaction mean field, we associate these fields to the h uk parameter from Eq. (5), considering the same coercive field h ck ¼300 Oe for all hysterons. A single value of the coercive fields is an approximation, but it is consistent with narrow coercive field distributions observed in experimental FORC diagrams of nanowire arrays [21 26]. A real nanowire array is finite and consequently peripheral wires are subjected to lower demagnetizing fields in comparison with the rest of the wires. We associate PKP symmetrical hysterons to a rectangular array of nanowires dividing the matrix in regions (R1,R2,y), as seen in Fig. 9(a). For arrays of various sizes (see Fig. 9(b)), starting from the central nanowire to the peripheral ones, it is observed that the computed interaction field decreases both in diagonal direction (d.d.) and in median direction (m.d.), until the marginal rows are reached. The central

C.-I. Dobrotă, A. Stancu / Physica B 407 (2012) 4676 4685 4681 Fig. 6. Computed interaction field at the center of the system versus length of the wires and the superior limit, for different nanowire radii (the curves correspond to 25 nm changes in R, and a¼250 nm). Interaction fields are computed for large arrays (2001 2001) in (a), and for small arrays (401 401) in (b). Fig. 7. (a) Play hysteron, and (b) symmetrical PKP hysteron (saturated play hysteron). region (R1) is characterized by interaction fields not less than a step field value, DH compared to H centr. A good choice for DH is the field step used in the FORC measurements. The limits of the next regions (R2,R3,y) were set in the same way, but the range of DH is relative to the previous region limit. Then a local mean field for each region is computed as an average and used as the h uk parameter for one PKP symmetrical hysteron. We attach to each region one PKP symmetrical hysteron because the nanowires from one region felt the same interaction mean field in the saturate state, and one hysteron is characterized by an interaction field in the same state (5). The saturation magnetization of one PKP hysteron is set as the magnetization in the saturated state of the assembly of nanowires from the region it is associated to. If a matrix of 401 401 wires is considered, with a¼250 nm, R¼50 nm, L¼4 mm, for a field step value DH¼40 Oe, we find 30.37% from the total number of nanowires in the central region (R1). In this case, the matrix is divided into 10 rectangular regions for which we assign the same number of PKP symmetrical hysterons, and the normalized IFD obtained from FORC diagram is presented in Fig. 10(a). Modeled FORC distributions can be used to extract the local interaction mean fields specific to different regions of the array, as well as the qualitative variation of these fields from the center to the boundary, as presented in Fig. 10. The interaction mean field at the center of the array was computed with Eq. (3), by adding the interaction fields created by all the wires. Because the wires from the last row of the system periphery are subjected to different interaction fields (see Fig. 9(b)), peripheral interaction mean field (H periph ) was computed as the average of these fields. As seen in Fig. 10(a), the bottom half-width of the IFD obtained from FORC diagram represents the maximum of the mean interaction field associated to the center of the system (H max ), while the top half-width is the minimum of the mean interaction field, characteristic of the system periphery (H min ). The field variation between these two limits is revealed by the convexity of the lateral branches. In Fig. 10(b) we observe a very good agreement between computed and identified local interaction mean fields from the FORC diagram, within the limit errors due to FORC distribution fitting algorithm. Fig. 11 presents IFDs obtained from modeled FORC diagrams using small sets of PKP symmetrical hysterons for various geometrical characteristics of the nanowire arrays. As seen in Fig. 11(a), considering a few wires (an 11 11 array with the lattice constant a¼250 nm), the IFD shows the presence of weak interactions in the system, its shape being close to that for a single wire presented in Fig. 2(b). IFDs obtained for weak interactions are also observed in Fig. 11(c) for large arrays with a large interwire distance (a¼800 nm) and in Fig. 11(d) for thin wires

4682 C.-I. Dobrotă, A. Stancu / Physica B 407 (2012) 4676 4685 Fig. 8. (a) FORC diagram and (b) normalized interaction field distribution of one PKP symmetrical hysteron characterized by two parameters: h ck ¼300 Oe and h sk ¼800 Oe. Fig. 9. (a) A 101 101 nanowire array divided into regions for a convenient association of the hysterons, nuances signifying the decrease of the interaction fields from dark to bright. (b) Interaction fields acting on the nanowires located on the diagonal direction (d.d.) and on the median direction (m.d.), considering different array dimensions with parameters a¼250 nm, R¼50 nm, L¼4 mm, and M s ¼485 emu/cm 3. (R¼20 nm). Fig. 11(a) shows that as the array size increases, the maximum and minimum interaction mean fields converge respectively to the same limit values (about 800 Oe and 350 Oe), but the field variation is more abrupt for large arrays in which the identification of the lower limit becomes difficult. This is consistent with the case of an ideal infinite array with a rectangular shape of the IFD, like that from Fig. 8(b) described by a single mean interaction field associated to one PKP symmetrical hysteron. The dependence of the IFD on the nanowire lengths is presented in Fig. 11(b). Arrays with short wires (less than about 2 mm) have an almost rectangular shape of the IFD, so a unique interaction mean field can describe the system. IFDs obtained for longer wire arrays (L¼4 mm in Fig. 10 and L¼8 mm, L¼20 mm in Fig. 11(b)) show a strong variation of the local interaction mean fields from the center to the array periphery, which can reach about 400 Oe. The value of H centr when the wire lengths increase

C.-I. Dobrotă, A. Stancu / Physica B 407 (2012) 4676 4685 4683 Fig. 10. (a) Local interaction mean fields obtained from FORC distribution. (b) Central (H centr ) and peripheral (H periph ) interaction mean fields computed using Eq. (3) and obtained from FORC distribution (H max and H min ) for different array dimensions with a¼250 nm, R¼50 nm, and L¼4 mm. is determined by two factors: (i) wire s central point is farther from the uncompensated magnetic charges from the wires ends and as a result the value of the field decreases and (ii) as the wires are longer, the projection of the interaction magnetic field vector along the wire is bigger as the angle between the field vector and the wire direction is decreasing. As a combined effect, for very long wires, e.g. for L¼80 mm, the mean field decrease is substantial [23]. In Fig. 11(c) it is emphasized that an evident decrease of the maximum and minimum values of the local interaction mean fields occurs as the lattice constant increases. The radius dependence presented in Fig. 11(d) is, in fact, due to the increase of the wire magnetization with their radius [25]. 6. Identification procedure A nonparametric identification involves finding a small number of the PKP symmetrical hysterons and their parameters: the local interaction mean fields in the saturated state and the magnetic moments. For a given sample the magnetic moment in the saturation state and the geometrical characteristics of the array are easily evaluated. The procedure can be applied for any shape of the array, and we will present, as an example, how it can be applied for a modeled polygon-shaped array with an area of 10 2 mm 2, porosity P¼10% and nanowires length 10 mm. First, the interaction fields are computed and mapped over the nanowires using a color code for every interaction field change in DH Fig. 11. IFDs obtained from FORC diagrams for (a) different array dimensions with a¼250 nm, R¼50 nm, L¼4 mm; (b) different nanowire lengths L, with a¼250 nm, R¼50 nm; (c) different interwire distances a, with L¼4 mm, R¼45 nm; and (d) different nanowire radii R, and a¼250 nm, L¼4 mm. In (b) (d) the array dimension is 401 401 which covers a few thousand mm 2. steps. Then a rectangular grid with small squares is superimposed over the nanowire array, as in Fig. 12(a), for a convenient account of the regions with the interaction fields in the same range. The number of PKP hysterons depends on the difference between the central and peripheral interaction fields and on the chosen value as field step to divide the nanowire array into regions. Totalizing the squared regions with the same interaction fields, it

4684 C.-I. Dobrotă, A. Stancu / Physica B 407 (2012) 4676 4685 explanation for these results. Starting from the necessity to properly compute interaction fields, we have shown that the effective interaction radius is usually large in nanowire arrays in contrast with nanostructured systems like patterned media. A phenomenological model based on PKP symmetrical hysterons is adequate due to the fact that one hysteron can model a system characterized by one value of the mean field interactions, and only a small set of PKP hysterons can be conveniently attached to a real nanowire array. The number of PKP hysterons can be slightly increased to obtain a better fit of the experimental IFDs extracted from FORC diagrams. We have proven that a real nanowire array generates an uneven mean field which decreases towards the periphery of the system. This is the reason why IFDs found from FORC diagrams are more or less flattened and a unique moving parameter cannot be clearly identified, whereas a set of mean fields can correctly describe a given nanowire array. Consequently, the array can be characterized by local interaction mean fields with identifiable weights by the simple procedure described. Acknowledgments This work was partially supported by the European Social Fund in Romania, under the responsibility of the Managing Authority for the Sectorial Operational Program for Human Resources Development 2007 2013 (Grant POSDRU/88/1.5/S/47646). The authors also acknowledge the support given by Romanian CNCS-UEFISCDI Project IDEI-EXOTIC no. 185/25.10.2011. References Fig. 12. Identification procedure: (a) nanowire array mapped with computed interaction fields and (b) weights of the magnetic moment for 8 PKP hysterons. is easy to obtain the histogram of the local interaction mean fields, which is used to calculate the magnetic moment of each PKP hysteron by multiplying the weight of the regions with the saturation magnetic moment of the sample. Note that the proposed identification procedure can be carried out even for small arrays. This is an advantage of the model based on PKP hysterons since many practical applications require miniaturized components. Unfortunately, some experimental IFDs obtained from FORC diagrams for large nanowire arrays are less flattened, making local mean field identification difficult. Accurate FORC diagrams show supplemental IFDs which may be due to smaller interaction fields experienced by wire extremities or/and by the longer wires from the system [22,25,26]. 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