Angular Dependence of Magnetization Behavior in Ni 81 Fe 19 Nanowires by Micromagnetic Simulations

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Te origin of the magnetic state within such ferromagnetic materials, however, arises from the alignments of internal neighboring electron spins, which are strongly dependent on the quantum mechanical exchange interaction.As it is well known from the Pauli Exclusion Principle, any two electrons cannot occupy the same orbital if they have the same quantum numbers [13,14].Tus, repulsive or attractive energy appears when the electron spins are parallel or antiparallel, respectively.Tus, there is electrostatic energy between the nearest neighboring electrons, known as exchange energy or force.Tis energy, however, is not the only energy responsible for the overall magnetic state within such ferromagnetic materials due to the anisotropic energies arising from diferent origins that compete with each other to give the overall magnetic state within these materials.Tere are two main sources of magnetic anisotropy.Te frst is due to the spin-orbit coupling, including magnetoelastic and magnetocrystalline anisotropies [15,16].Te second class is termed magnetostatic or shape anisotropy, which is the most signifcant anisotropy in measuring ferromagnetic nanostructures and is related to dipole-dipole interactions.Tese anisotropies strongly infuence the overall magnetic state of ferromagnetic nanomaterials.
With the modern progress in computing technology and their programming applications, it has become encouraging to utilize micromagnetic simulation to theoretically explore the magnetic properties and explain experimental observations or predict new experimental designs of an extensive variety of ferromagnetic nanostructures, including nanowires [43][44][45][46][47][48][49][50][51], nanodots, and nanotubes [52][53][54].Te most widely used micromagnetic simulation software is the OOMMF package.Tis is because the OOMMF package has the capability to show magnetic hysteresis loops and visualize the magnetic moment at any period throughout a hysteresis loop cycle.Tese studies, however, have shown that magnetization reversal in ultrathin nanostructures can happen uniformly via coherent rotation, in which all the spins rotate immediately upon applying an external magnetic feld [44].In thicker nanostructures (<100 nm), two unlike models have been described, known as vortex and transverse wall modes [44,45,50].During the submission of an external magnetic feld, a reversed domain can nucleate at the ends of the nanostructure or at imperfections [45,55] separated from the old domain by a newly formed domain wall.Consequential propagation of the domain walls throughout the nanostructure leads to the nanowire switching from one magnetic confguration to another and decreasing the magnetostatic energy.Te nucleation at these sites, however, is due to the large demagnetizing feld at these places [45].Te magnetic moments precede a spiral motion as the wall spreads parallel to the wire's long axis because of the torque employed on the magnetic spins.Te transition between transverse and vortex modes was discussed elsewhere [50] and was found to depend on various elements, including element composition, dimensions, crystalline structure, and surface morphology of the nanostructures, as well as the preliminary magnetic feld applied [50].
Nevertheless, most of these studies concentrated on a small number of two-dimensional arrays of nanowires or individual nanowires of smaller diameter/thickness in which the magnetization structures show a simple magnetic state compared to the thick isolated nanostructures, which support more sophisticated reversal behavior.Tus, the magnetization reversal of relatively thick individual nanowires is still confusing and needs more examination.Terefore, the aim of the work presented here is to perform OOMMF modeling to discuss the angular dependence of magnetization behavior in Ni 81 Fe 19 nanowires with diferent T up to 150 nm.Te fndings were then investigated and compared with the literature and theoretical models of domain wall reversal.

Methodology
Te OOMMF software was utilized here to explore the magnetic properties of permalloy wires.Tis package is public domain software available freely from the National Institute of Standards and Technology (NIST) website [56].It is a group of programs working together to analyze magnetic problems, and each program can be modifed or redesigned without varying the entire OOMMF software.Te OOMMF code has a good magnetic output fle that can display the magnetic state of a simulated sample in diferent formats, such as data tables, graphs, or magnetic confgurations [56][57][58][59][60].It can also easily calculate the magnetic state dynamics for any arbitrarily shaped material composed of one or more diferent ferromagnetic elements.Te T of permalloy nanowires was chosen to be 10-150 nm with a permanent stretch of 1 µm.Tis stretch was designated in accordance with the space accessible in the memory and the time required for computation [61].
In the OOMMF platform, it is essential to describe the micromagnetic problem under discussion, which contains nanowires' geometrical shape and dimensions as well as cell size, magnetocrystalline anisotropy, magnetostriction anisotropy, and magnetic considerations for which the calculation is executed.For simplifcation in the micromagnetic analysis, some of the magnetic anisotropy terms can be neglected depending upon the magnetic system in use.

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Journal of Nanotechnology Tus, micromagnetic simulations were performed by allocating a rectangular piece divided into threedimensional arrays of cubic cells, as shown in Figure 1.Te cell size should be less than or equal to the exchange length to permit the exchange interaction to be appropriately performed [15,[57][58][59] and to give more accurate and reliable results.Exchange length is of primary importance because it governs the length of the transition between magnetic domains [62].
If the cell size is larger than the exchange length, the simulation would not have enough cells to do a realistic simulation, whereas if the cell size is much smaller than the exchange length, then it slows down the computation speed without any additional gains.Tus, the cell dimensions were chosen to be less than the exchange length (5.3 nm [62]) of permalloy: 2 × 2 × 2 nm 3 and 5 × 5 × 5 nm 3 for nanowires <50 nm and ≥50 nm, respectively.Te magnetic confguration was supposed to be fxed and positioned at the center of each cell.
Te factors of exchange stifness and the saturation magnetic feld were nominated to be 13 × 10 -12 J/m and 8.6 × 10 5 A/m, respectively [15,62].Te magnetostriction energy and magnetocrystalline anisotropy were ignored from the micromagnetic simulation because the magnetostriction energy is very small and adjacent to zero, and permalloy has no magnetocrystalline anisotropy [13][14][15].Te gyromagnetic ratio defned as default in the OOMMF platform was equal to 17.59 MHz/Oe.Further details can be found elsewhere [15,43].
Te integration of the Landau-Lifshitz-Gilbert (LLG) equation, Runge-Kutta evolver was performed, in which the application of magnetic feld was accomplished in steps to achieve energy minimization for each feld step.Te damping factor was responsible for reducing the energy of the system under discussion.Accordingly, an appropriate stopping condition (1 × 10 −5 A/m), which is specifed by the torque M × H, was defned in the micromagnetic input fle to permit the simulation to fnish or progress to the consequent feld step [63].If the stopping step is very low, the simulation will remain for longer periods without any progress.If it is used very high, the simulation will not minimize the energy state, and the results will not be accurate.Tus, to decrease the time of micromagnetic simulation, the damping parameter was taken to be ∼0.5, and the stopping step used was dM_dt 1. Te feld steps and the extreme magnetic feld were selected according to initial simulations executed to estimate the shape of the loop.In complete micromagnetic simulations, the feld strength was reduced from saturation to the negative sign in steps of 5-10 Oe.Comprehensive information on the micromagnetic simulation and the OOMMF code can be found in the studies in [57][58][59][63][64][65].
In order to gain a full understanding of the magnetic properties and the mechanism of the magnetization reversal in such nanowires, a series of micromagnetic simulations were performed by applying two orthogonal magnetic felds to these nanowires.When the feld is applied parallel to the nanowire's long axis, the magnetic moments return through the nucleation and propagation of the walls of the domain.Whereas when the feld is applied orthogonally to the nanowire's long axis, the magnetic moments return by pseudocoherent rotation.Te results of this investigation may provide a guide to control the magnetic properties of such nanowires for potential technological applications.Tus, the frst was applied parallel to the nanowires' long axis (H x ), and the other was applied orthogonally to the nanostructures' long axis (H y ), as presented in Figure 2. Using mathematical equations, the vector sum of the magnetization feld along the designated angle β with respect to the nanowire long axis was calculated using the following equation [63]: With respect to the magnetic feld H β , the magnetization will change to a net state of M α .Te component of M α , called M β , along H β , can be determined, and the angle ∝ can be calculated using the following equation: Te magnetization along the angles α and β is acquired using the following relations: Tus, the magnetic feld H β was identifed in the micromagnetic input fle (MIF) to simulate the magnetic hysteresis loops at various wire angles with respect to the feld applied.Accordingly, the hysteresis loops and the angular dependence of remanent magnetization (M β ) and switching felds (H β ) were determined using equations (1)-(4).Te remanent magnetization M R , at any angle β, can be found by the projection of magnetization towards the measurement direction using the following equation [21]: where M R (‖) is the remanent magnetization; at β = 0, the external magnetic feld H β is with the nanowire long axis.
To theoretically calculate the critical thickness, the switching feld for the Stoner-Wohlfarth model, and the curling models for an infnite cylinder, the following equations were utilized [30,31,35,41,63]: where β o is the angle between the externally applied magnetic feld and the nanowire's long axis, M is the saturation magnetization of permalloy, and h sw , is the reduced switching feld.t and t o are the thickness/width of the nanowire and the exchange length, respectively.Te t o /t is known as the reduced radius R. Te exchange length is given by the following equation [62,63]: Figure 2: Schematic diagrams presenting the permalloy wire with the coordinate system.Two perpendicular magnetic felds were applied parallel (H x ) and orthogonal (H y ) to the nanowire's long axis with the projections of their components of magnetization.Te determined magnetic feld at any angle was then defned within the OOMMF platform to examine the angular dependence of the remanent magnetization and switching felds [63].
where E is the exchange constant, which is dependent on the critical size of the nanowires and independent on the shape and real size of the considerable wires [36,41].

Results and Discussion
Te normalized magnetic hysteresis loops of angular dependent of permalloy wires with various T (30 nm, 50 nm, 75 nm, 100 nm, and 150 nm) at diferent nanowire long axis angles (1 °, 15 °, 30 °, 45 °, 60 °, 75 °, and 90 °) with respect to the magnetic feld applied externally are shown in Figure 3. From all measurements examined here, the magnetic loops have a distinct switching behavior.For all T, the magnetic loops have a lower squareness ratio (shared loops in the feld) by increasing the nanowire T and angle.
Te remanent magnetization against nanowires' long axis angle for various T of nanowires is displayed in Figure 4. Clearly, for all T of wires investigated here, the remanent magnetization is at its maximum when the magnetic feld is parallel to the nanowire's long axis.Ten, it decreased constantly by increasing the wire angle and disappeared for a magnetic feld at an angle of 90 °.Comparing this result with the theoretical predictions using relation 5 presented in the methodology section, an excellent agreement was obtained, as established by the ftting dashed lines presented in Figure 4.
For each angle of investigation and for nanowires with T up to 100 nm, the remanent magnetization is nearly close to each other and is higher than 150 nm thick nanowires.At high angles of measurement (≥75 °), the remanent magnetization values are almost the same for all T of wires, and they are at minimum values.For all of the angles discussed here, a reduction in the remanent magnetization was noticed with increasing nanowire T, and this was analyzed in detail in other research studies [43,63,66].However, the reduction in remanent magnetization with increasing nanowire T was attributed to the formation of multidomain structures or moment rotations within the magnetic structure due to the efect of the magnetostatic anisotropy and demagnetizing feld, which increases with increasing nanowire thickness.Now, the drop in the remanent magnetization with increasing nanowires angle can be described as follows: when the nanostructure's long axis is parallel to the external magnetic feld, H β , the remanence ratio, as predicted, reveals the maximum value in all of the thicknesses of wires explored here.Tis is due to the shape anisotropy because the magnetic spins are primarily directed along the easy axis of magnetization.
Upon increasing the wire's angle and removing this feld, the magnetic spins relax and are redirected again parallel to the nanowire's long axis.As a result, the remanent magnetization is decreasing in that direction with an increase in the nanowire angles.Tis behavior is more remarkable in relatively thick nanowire (150 nm) due to the reduction of the shape anisotropy with increasing nanowire T and the efect of the demagnetizing feld.
Returning to Figures 3(d)-3(e) and their insets, clearly, the loop shapes are complicated and exhibit various switching structures with increasing the wire angle, not as predicted by the curling model of domain reversal [13][14][15].Tis complication in the switching events might be due to the formation of complex multidomain structures upon increasing the wire's long axis angle with respect to the magnetic feld applied.
To understand the complexity behavior in the switching events of relatively thick wires, an example of micromagnetic moment distributions for the 150 nm thick nanowires obtained during the switching states at several angles is shown in Figure 5. Te color variation represents the magnitude along a certain direction, such as the x-y and x-z angles of the magnetic spin.Tese micromagnetic spin structures were obtained instantaneously from energy-minimized states at two diferent feld steps before and after the switching events and are shown as an axial portion over the center of such wires.Tese snapshots reveal the complexity in the confguration of magnetic spins throughout the switching, which may also indicate that the magnetization reversal in such relatively thick wires is not as simple as in the curlinglike behavior, as will be discussed in the subsequent investigations.
To investigate the magnetization reversal in detail in such nanowires, the switching felds were extracted from the hysteresis loops and plotted against the nanowire's long axis angle, as shown in Figure 6.Two regimes were recognized for nanowires with T ≤30 nm; strong reduction in the switching feld was observed with an increase in the wire angle up to 20 °. Between 20 °and 60 °, the switching feld is approximately fxed.Ten, the switching feld increases quickly at greater nanowire angles (∼60 °).A similar trend was noticed with nanowires of thickness 10 nm but with greater values of switching felds in all angles studied here (not shown in the fgure due to their large values of switching felds).In contrary, for nanostructures with T higher than 50 nm, it is increasing gradually up to ∼40 °, and then increasing quickly to the highest value.Tis result indicates that there is a transition in the magnetization behavior between nanowires of 30 nm and 50 nm thicknesses.Tis fnding is in excellent agreement with the other fndings investigated in the literature using other materials for ferromagnetic wires and the same and other characterization methods [35,36,45].
As an example, the critical thickness, t o , for the transition of transverse to vortex wall modes of Fe and Ni wires was established to be 20 nm and 40 nm, respectively, as calculated using micromagnetic modeling on cone crosssectioned Fe and Ni wires [45].Nonetheless, this is close to 34 ± 4 nm obtained experimentally from electrodeposited Ni nanowires using micro-SQUID and was attributed to the transition between uniform and nonuniform modes of reversal [35,36].
Te critical thickness of the nanowires considered here was calculated using the relations ( 6)- (10) presented in the methodology section and was found to be around 30 ± 5 nm, which is in full agreement with the fnding stated above and reported in the literature [35,36].
Finally, for all angles investigated here, there was a reduction in the switching felds with increasing nanowires T. Te reduction in the switching felds with increasing Journal of Nanotechnology 6 Journal of Nanotechnology nanowires T was analyzed in detail elsewhere [43,44], and it was attributed to the formation of multidomain structures or moment rotations within the sample due to the efect of the shape anisotropy and demagnetizing feld, which increases with increasing nanowire thickness.
According to the theoretical calculations and micromagnetic analysis performed in the literature, there are various distinct classical mechanisms that are able to describe the reversal processes in such nano-objects, including buckling and fanning models, as well as coherent rotation of the Stoner-Wohlfarth and incoherent rotation of curling models of domain reversal.Detailed theoretical descriptions of these classical models can be found elsewhere [28,29,[66][67][68][69][70][71].
Tus, to understand the mechanism responsible for the angular dependence of the nanowires investigated here, it should frst be noted that since buckling reversal is expected to occur when the nanowires T are comparable to the exchange length (reduced thickness around unity) [66,70], buckling may be excluded from the applicability on such nanowires.On the other hand, the aspect ratio and shape of such nanowires are far from being a chain of spheres, which would reverse as in the fanning model.Terefore, the angular dependence of the switching felds is likely to be the consequence of a coherent (thin wires ≤ 30 nm) and an incoherent rotational process (thick wires ≥ 50 nm) following the Stoner-Wohlfarth and curling models of domain reversal, respectively.Accordingly, all the switching   Journal of Nanotechnology felds extracted from the simulation hysteresis loops were matched with the theoretical calculations of the Stoner-Wohlfarth (for nanowires of T ≤ 30 nm) and curling models (for nanowires of T ≥ 50 nm) after normalizing the switching felds, H sw , to the minimum values using the equations ( 6)- (10) presented in the methodology section and are shown in Figure 7. Excellent agreement was seen at all angles of discussion, proposing that these classical models of domain reversal are a rational analytical illustration of the magnetization reversal in such permalloy wires, in spite of the micromagnetic spin confgurations and the loops shape, which were established earlier, demonstrating the complexity of the spin structure and the appearance of diferent switching events through the magnetization reversal in relatively thick permalloy nanowires.To understand this behavior in more details, similar investigations are required using wires with T between 100 nm and 300 nm.

Conclusions
Micromagnetic simulations were performed using the OOMMF platform to explore the magnetic properties of Ni 81 Fe 19 nanostructures with various T up to 150 nm and an identical stretch of 1 µm.Te highest (lowest) remanent M β was found when the external magnetic feld was applied parallel (normal) to the wire's long axis, demonstrating the dominance of shape anisotropy on the behavior.Tis result was in full agreement with the mathematical representations performed here.
Te shape of the hysteresis loops of relatively thick (150 nm) wire was complicated, displaying various switching events by increasing the wire's angle.
Te angular dependence revealed that nanowires ≤30 nm thicknesses behaved diferently from nanowires ≥50 nm.Tis result was in excellent agreement with the calculations performed here theoretically and with the results stated experimentally in the literature using other characterization methods and other compositions of ferromagnetic nanowires.Te angular dependence of the switching feld was also in full agreement with the classical models of reversal.Tis agreement indicates that these models are a rational and methodical demonstration of the magnetization reversal in such wires, although the micromagnetic spin structures of relatively thick nanowires showed the complexity of the magnetic moment structure before and after the reversal.

Figure 1 :
Figure 1: Permalloy wires were produced using rectangular sections divided into three-dimensional arrays of cubic cells.Te cell sizes are 2 × 2 × 2 nm 3 and 5 × 5 × 5 nm 3 for nanowires <50 nm and ≥50 nm, respectively.Te thickness/width marked in the fgure is denoted as T throughout this article.

Figure 3 :Figure 4 :
Figure 3: Normalized loops acquired from the simulations of Ni 81 Fe 19 wires with diferent T at various wire angles with respect to the feld applied, as specifed in the loop titles.Note the diference in the magnetic feld scale bars of the hysteresis loops.

Figure 5 :
Figure 5: Snapshots of magnetic moment distributions along the center of the permalloy wires of 150 nm thickness obtained from energyminimized states at two diferent feld steps during the switching feld and at various angles of simulations, as indicated in the fgures.Te color variation represents the x-y and x-z angles of the magnetic spin.

Figure 6 :Figure 7 :
Figure 6: Switching feld as a function of the angle of permalloy wires for diferent T of nanowires, as shown in the fgure.Te dashed lines provide a guide for the eye.