Micromagnetics of magnetisation reversal mechanism in Permalloy chain-of-sphere structure with magnetic vortices

Computational Materials Science 45 (2009) 240–246

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Micromagnetics of magnetisation reversal mechanism in Permalloy chain-of-sphere structure with magnetic vortices P. Barpanda * Department of Materials Science and Engineering, 607, Taylor Road, Busch Campus, Rutgers University, Piscataway, NJ 08854-8065, USA

a r t i c l e

i n f o

Article history: Received 23 April 2008 Received in revised form 10 September 2008 Accepted 15 September 2008 Available online 1 November 2008 Keywords: Micromagnetics Magnetic vortex Reversal mechanism Coercivity Permalloy

a b s t r a c t Magnetisation reversal mechanism in a chain of closely spaced Permalloy nanospheres is presented using 3D micromagnetic simulation. While smaller spheres (d  20 nm) containing single domain reverses via domain switching mechanism, the larger sphere (d  50 nm) possessing magnetic vortices changes the orientation through vortex creation and annihilation (VCA) mechanism. The effect of total number of spheres (comprising the chain structure) and incidence angle of externally applied field on the VCA mechanism is explained in detail. The variation in coercivity, remanence, exchange energy and vortex parameters are discussed for the Permalloy chain-of-sphere system with uniform sphere size. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction Over the years, magnetic nanostructures are increasingly getting significant scientific attention for their potential use in varieties of nanostructured devices, logic gates, MRAM and high-density magnetic recording [1,2]. In an effort to advance the high-density magnetic recording, the geometry of magnetic nanostructures has been successfully extended to rectangles, squares [3], circular/elliptical dots [4], circular/elliptical hollow rings [5], pentagons, nanocones [6], truncated structures using ferromagnetic materials like Fe, Co, Ni, Permalloy and other ferromagnetic alloys. Apart from these isolated elements of different geometric shapes, research has been focussed on thin films and magnetic nanowires [7]. One such magnetic nanostructure is the ferromagnetic chainof-sphere structure, which was originally proposed by Jacobs and Bean [8] in an effort to explain the magnetisation reversal behaviour of elongated spherical particles. Though the chain-of-sphere structure is less studied, it forms an interesting system of strongly interacting uniform ferromagnetic particles, which can serve as a suitable model for understanding the magnetic phenomena of elongated structures like magnetic nanowires. The magnetic properties of ferromagnetic nanostructures are mainly controlled by the shape, dimensions and material parameters. Depending on the dimension of nanospheres, the chain-of-sphere structure may * Corresponding author. Tel.: +1 732 986 3945; fax: +1 732 932 6855. E-mail address: [email protected] 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.09.014

comprise of smaller sphere having single domain or larger sphere with bi-domain magnetic vortex [9,10]. The original model proposed by Jacobs and Bean deals with smaller spheres with single domains. On the contrary, the current study is focused on larger spheres possessing magnetic vortices. This type of structure has been observed by off-axis electron holography in chain of FexNi1x spherical particles prepared by vapour phase condensation route [11]. A thorough study of the ferromagnetic chain-of-sphere nanostructure is quite important to know the underlying magnetisation reversal process involving magnetic vortex that can help to gain insight for designing three-dimensional nanomagnetic devices. Chain-of-sphere (CoS) system can have different relaxed states from saturated domain to multi-domain, which essentially governs its magnetisation phenomena. In an earlier work, a magnetic phase diagram of domain state and magnetisation reversal mechanism has been briefly described for four alloy compositions of Fe–Ni system [9]. While single domain sphere involves magnetic domain switching with high coercivity, the chain possessing larger sphere with magnetic vortices reverses gradually involving vortex creation and annihilation (VCA) mechanism. If neighboring spheres are of different size, then a hybrid reversal occurs. In the current study, Permalloy chain-of-sphere structure with larger spheres possessing magnetic vortices are investigated, focusing on the magnetisation reversal mechanism. For the sake of simplicity, chain of uniform sphere size is considered throughout. Using micromagnetic simulation, the magnetisation reversal mechanism in chain structure is studied as a function of number of sphere (even

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or odd) and incidence direction of externally applied field. The variation in coercivity, remanence, exchange energy and vortex parameters (vortex diameter and pitch-of-helix) is discussed for the Permalloy chain-of-sphere system. 2. Computational simulation details In the field of magnetism, micromagnetic study [12–14] is widely used in conjunction with experimental observation of magnetic phenomena. In micromagnetic study, any ferromagnetic system is considered as an integration of large number of individual magnetic spins. The amplitude of the net magnetisation vector M(r) i.e. |M(r)| = Ms|m(r)| = Ms (saturation magnetisation) is constant, but the localized individual magnetic moments can be randomly oriented. The total Gibb’s free energy for a given magnetisation distribution M(r) is expressed as:

1 l Ms ðm  Hdem Þ 2 0

3. Results and discussion

 l0 M s ðm  Happ Þ where, Hdem and Happ are the demagnetizing field and externally applied field respectively, Aex is the exchange constant. The term g(m,uk) presents the magnetocrystalline anisotropy energy density, expressed as:

gðm; uk Þ ¼ K u ½1  ðm; uk Þ2  where, Ku = anisotropy constant and uk = unit vector parallel to easy axis. Minimizing the Gibb’s free energy, an equilibrium state is reached when m  Heff = 0 (Heff = effective magnetic field =  l1 ðDG=DMÞÞ. The micromagnetic equation is highly non-linear 0 and can be complemented by numerical simulations employing finite element method (FEM, [15]) or finite difference method (FDM, [16]). In the current fast-fourier transformation (FFT) based micromagnetic simulations [17], a finite difference method is employed based on Landau–Lifshitz–Gilbert (LLG) equation, expressed as:

ð1 þ a2 Þ

oM ¼ cðM  Heff Þ  ðac=M s ÞM  ðM  Heff Þ ot

where, a is damping parameter and c is gyromagnetic ratio of individual electron spin. The geometry of linear chain-of-sphere structure is shown in Fig. 1 (with chain axis along x-direction). The individual sphere is a summation of spatially divided cubic cells ðN x  N y  N z Þ of uniform magnetisation, with demagnetisation field computed to all

y

Single Domain

z

S1

S2

S3

S4

x

S5

3.1. Magnetisation reversal mechanisms involving vortices As per previous study, domain structure in magnetic materials largely depends on its size and composition. While smaller Permalloy spheres (d < 20 nm) are saturated, larger spheres (d > 30 nm) support magnetic vortices. Further, the domain in any sphere is strongly affected by the neighboring spheres and external field. Fig. 2 shows the magnetic domain phase diagram of Permalloy (Fe0.20Ni0.80), showing the size range of nanospheres possessing single domain and magnetic vortices. In the present investigation, magnetisation reversal of chain-of-sphere system with uniform sphere size of 50 nm (vortices) (point B in Fig. 2) was studied. Smaller sphere with single domain follows standard mechanisms of reversal like coherent rotation, symmetric/non-symmetric fanning, curling and buckling [8]. However, till date, there is no work on chain of larger sphere supporting vortex structure. The very presence of vortex state, in essence, triggers completely different type of magnetic reversal and hysteresis cycle. The whole reversal process shows an evolution, growth and propagation of

100

Diametre of neighbouring sphere (nm)

GðMÞ ¼ dV½Aex ðDmÞ2 þ gðm; uk Þ 

orders. In order to assure the accuracy of all micromagnetic simulation, the size of these cubic cells were kept smaller than the characteristic exchange length of Permalloy lex = ½2Aex =ðl0 M 2s Þ1=2 = 5.1 nm. Here, the equilibrium criterion was set when the maximum change in torque is below a given tolerance of m  Heff 6 104Ms A gyromagnetic frequency c of 17.6 MHz/Oe and a damping parameter a value of 1 was used throughout to accelerate the computation process while successfully capturing the equilibrium static configurations. The material parameters used for Permalloy are: saturation magnetisation (Ms) of 800 emu/cc, exchange stiffness (A) of 1.05 lerg/cm and uniaxial anisotropy (Ku) of 1000 lerg/cc. The study of magnetisation reversal mechanisms was conducted by applying external field from 4000  4000 Oe and by changing the incidence angle from 0° (parallel to chain axis) to 90° (normal to chain axis). Thermal fluctuations were not considered and the simulations were independent of anisotropy direction and initial magnetisation state.

Single Domain

Vortex

80

60

B 40

20

A

0

Magnetic Vortex Fig. 1. Schematic presentation of geometry of chain-of-sphere system of uniform sphere size. Smaller spheres (d = 20 nm) and larger spheres (d = 50 nm) have single domain and vortex, respectively. The chain-axis is parallel to the x-axis. The individual spheres are marked S1, S2, S3, S4 and S5. Here, the neighboring spheres have single-point contact.

0

20

40

60

80

100

8Diametre of centre sphere (nm) Fig. 2. A magnetic phase diagram showing the possible magnetic remanent states (single domain or vortex) in any sphere of chain structure depending upon its size and that of adjacent spheres. The current simulation focuses on point B (sphere size = d = 50 nm) supporting magnetic vortex state.

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0.5

S1

S2

S3

S4

S5

0 30 45 60 90

0

Decreasing Field

Magnetisation (MMs)

1

-0.5

-1 -3000

-2000

-1000

0

1000

2000

External field (Oe) Fig. 3. A graph showing the combined hysteresis loop for a chain-of-sphere (sphere size = 50 nm) system with external field (3000 to +3000 Oe) applied at different angle (0 to 90 degree) to the chain axis.

magnetic vortices at different field throughout the structure. This reversal process is termed as vortex creation and annihilation mechanism (VCA), which has been observed in circular dots [18]. A detail description of this VCA mechanism is presented here for chain of larger spheres with different orientation to external field. In order to study the reversal mechanism of a group of randomly aligned CoS structures, the effect of different angle of incidence to CoS can be investigated by using a combined hysteresis loop as shown in Fig. 3 for the current CoS structure. When the external field is applied along the chain axis (incidence angle  0°), it aligns each individual magnetic moment along the axis. Permalloy spheres of 50 nm diameter naturally support magnetic vortices. Thus, upon lowering the field, the saturated chain slowly starts relaxing into twisted magnetic vortices, which grow to become complete vortices with a vertical saturated domain in centre part surrounded by a horizontal circular swirling domain. Pair of oppositely swirling magnetic vortices starts forming from both ends of the chain. Gradually, the vortices penetrate towards the centre from both ends of the chain. The oppositely swirling vortices lead to the formation of an ‘inversion symmetry’ feature in the whole structure. This feature can be observed in elongated magnetic structures like cylindrical rods, large ellipsoid etc. While the central sphere remains unaffected, the left and right half of the structure possess equal magnetic domain structure aligned in opposite direction, thus locating the centre of inversion symmetry exactly at the physical centre of nanostructure. The vortex state remains in anticlockwise direction (right-handed) at the two left spheres and in clockwise direction (left-handed) at the two right spheres having different chirality. After the coercive field value is reached, the outer swirling domains in all spheres gradually switch to opposite direction forcing an instant domain switching of saturated domain in the centre sphere. It was observed the centre sphere never get a chance to develop magnetic vortex. But the reversal of adjacent spheres forces the centre sphere to reverse like a single domain. It is worth noting the magnetic vortices always enter the CoS structure from both ends simultaneously. A series of simulation snapshots (at different stage of hysteresis half cycle) are presented in Fig. 4 to display the magnetisation reversal involving VCA mechanism with parallel external field. If the external field is applied angularly to the chain axis, it modifies the magnetic reversal mechanism owing to the effect of the perpendicular field component on vortices. Similar to the case

Fig. 4. A series of snapshots showing the variation in domain states of each sphere in a chain of 5 uniform size spheres (d = 50 nm) during hysteresis cycle with the external field parallel (0 degree) to the chain-axis. The evolution and propagation of magnetic vortices and formation of inversion symmetry feature is depicted. For convenience, the central y–z cross-section of each sphere (in CoS) is shown to capture magnetic vortices during reversal process.

of parallel field, upon reducing field from maxima, a set of oppositely swirling vortices appear at both ends of the chain. However, due to the lateral component of the field, these magnetic vortices are off-centred from the beginning. Later the opposite-handed vortices propagate towards the centre of chain. But in this case, the inversion symmetry is broken at a very early stage of reversal. The normal component of external field interacts with the vortices pointing in y-direction giving larger exchange energy. Consequently, the magnetic vortices in the end-spheres are off-centred. Here, complete development of vortices occurs only in the endspheres. With further continuation of reversal, the vortices in centre spheres undergo domain switching leaving behind twisted off-centred shallow vortices in end-spheres that gradually disappear. This reversal mechanism holds good for any field direction between 0°–90°. Nevertheless, the vortex core domains in the end-spheres get aligned along the incidence field direction. Fig. 5 captures the snapshots of chain-of-sphere structure just before magnetisation reversal for external field applied at 0° < h< 90°. When the external field is normal to the chain structure, it significantly modifies the reversal mechanism. Here, upon decreasing the field from maxima, off-centred vortices appear at both ends of chain. With reduction of field, these end sphere vortices become centred forming an inversion symmetry feature (capturing three saturated spheres in between). With further continuation of reversal process, the end spheres get saturated along field, forcing domain switching in all three centre spheres without any formation of vortices. This reversal mechanism is depicted in Fig. 6 with a series of simulation snapshots. By and large, higher angle of incidence (of external field) favours early destruction of inversion symmetry feature and lesser degree of magnetic vortices growth. 3.2. Inversion symmetry in chain-of-sphere system As pointed out in earlier section, magnetisation reversal process in CoS structure having larger spheres yields an inversion symmetry feature, where each half of the CoS structure have oppositely

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S1

S2

S3

S4

S5

Anti-clockwise rotation N=2

Clockwise rotation

Decreasing Field

N=4

N=6

N=8 Fig. 7. Domain states (of centre layer) of each sphere during the magnetisation reversal process showing ‘inversion symmetry feature’ in chains having even numbered spheres (2, 4, 6, 8) of uniform size (d = 50 nm). While the vortices in left half are right-handed, they are left-handed in the right-half of the structure. Inversion symmetry feature is retained intact independent of number of sphere comprising the chain structure.

Fig. 5. A series of micromagnetic simulation snapshots showing various domain state of each sphere in a chain of five uniform size spheres (d = 50 nm) during hysteresis cycle with the external field applied angularly (0° < h < 90° degree) to the chain-axis. The evolution of twisted magnetic vortices and early breaking of inversion symmetry feature is shown clearly. The central y–z cross-section of each sphere (in CoS) is shown to capture magnetic vortices during reversal process.

S2

S3

S4

S5

Decreasing Field

S1

Fig. 6. A series of simulation snapshots showing the variation in domain states of each sphere in a chain of five uniform size spheres (d = 50 nm) during hysteresis cycle with the external field normal (90 degree) to the chain-axis. While the end sphere involves vortex creation and annihilation, the middle spheres switches like single domain. Once again, the central y–z cross-section of each sphere is shown to capture magnetic vortices during reversal process.

swirling magnetic vortices. This feature was further investigated in longer CoS system with axial external field. Interestingly, the inversion symmetry feature was observed in CoS with aspect ratio (number of sphere) varying from 2 to 9. Fig. 7 shows the interme-

diate step during reversal process of CoS with even number of spheres (n = 2, 4, 6, and 8). While left half of the spheres have vortices swirling in anticlockwise direction, the other half possesses vortices moving in clockwise direction. This creates an inversion symmetry with the centre of symmetry aligned exactly at the centre of CoS structure (marked as dashed line in Fig. 7). On another note, CoS structure with odd number of spheres (n = 3, 5, 7, and 9) was studied. It also results in formation of inversion symmetry feature during magnetisation reversal process. Micromagnetic snapshots of CoS with odd number of spheres are shown in Fig. 8 during an intermediate step. From both ends of the chain, magnetic vortices penetrate into the CoS structure. As shown in Fig. 8, the left half of the spheres has anticlockwise vortices while the right half of the spheres has clockwise vortices. Due to odd number of spheres, it leaves the centre sphere saturated. Here again, the centre of inversion symmetry is aligned exactly at the physical centre of CoS structure (as shown as dashed line). The appearance of ‘inversion symmetry’ feature is independent of the number of spheres and possible in very long chains (e.g. nanowires) too. However, irrespective of the length of CoS, the magnetic vortices always enter the structure from both ends of the nanostructure and penetrate towards the centre of chain structure. 3.3. Coercivity and remanence Coercivity and remanence of any magnetic system can be explained on the basis of magnetisation reversal mechanism and underlying domain structures. Fig. 9a presents the coercivity for varying angle of external field for CoS systems with even (n = 4) and odd (n = 5) number of spheres. When the field is parallel to the CoS axis (0°), it has the highest coercivity value. However, the coercivity in CoS (having vortices) is relatively smaller than CoS with saturated spheres as treated by Jacobs–Bean [8]. Magnetic vortices in CoS commence vortex creation/annihilation process that follow a gradual domain reversal resulting in lower coercivity. When the CoS is oriented angularly to external field, the coercivity decreases and interestingly follows the same pattern as long chain of saturated spheres [8]. In case of parallel field, magnetic vortices form and penetrate throughout the CoS along the field, which involves very high exchange energy and hence coercivity. But, for angular field, the normal component to chain axis favours easy magnetisation reversal with less degree of vortices formation as described earlier. It involves less exchange energy

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Anti-clockwise rotation

N=3

Clockwise rotation

N=5

N=7

for both even-numbered and odd-numbered CoS tried in this investigation. The variation of squareness (remanence/saturated magnetisation) of CoS with even and odd number spheres is shown in Fig. 9b. The squareness value gradually decreases from maximum to zero by varying the field orientation from 0° (parallel) to 90° (normal). As the reversal becomes increasingly easier supporting in and out of plane vortices, so the remanence value continuously decreases to zero. This holds true for both even and odd numbered CoS systems. 3.4. Exchange energy

N=9 Fig. 8. Domain states (of centre layer) of each sphere during the magnetisation reversal process showing ‘inversion symmetry feature’ in chains having odd numbered spheres (3, 5, 7, 9) of uniform size (d = 50 nm). There is inversion symmetry feature in all chain structure irrespective of number of sphere. In all cases, the centre sphere is saturated, separating equal number of magnetic vortices on each side swirling in opposite direction.

1400

Coercivity (Oe)

1200

sphere size= 50nm

n=5

1000 800 600

3.5. Vortex parameters (vortex core diameter/pitch-of-helix)

n=4

400 200 0 0

20

40

60

80

100

Incidence field angle (degree)

b

1 sphere size= 50nm

Squarenwss (M/Ms)

n=5

In case of chain-of-sphere system with larger spheres supporting magnetic vortices, magnetisation reversal involves creation, growth and annihilation of magnetic vortices in individual spheres. These vortices are sensitive to any surrounding magnetic field (external magnetic field or magnetic field of adjacent spheres). Thus, during magnetisation reversal, the individual magnetic vortex undergoes continuous change. This can be quantified by two vortex parameters; namely vortex core diameter (VCD i.e. dimension of central saturated domain) and the vortex pitch-of-helix (i.e. average angle of the swirling moments with y–z plane).

0.8 n=4

2500

0.6

0.4

0.2

0

0

20

40

60

80

100

Incidence field angle (degree) Fig. 9. The variation of (a) coercivity and (b) remanence in chain-of-sphere structure with the angle between external field and chain axis. It shows two CoS systems containing 4 and 5 uniform sized spheres (d = 50 nm), respectively.

Exchange entry (10-13 emu)

a

The variation of exchange energy of CoS system during the hysteresis cycle is shown in Fig. 10 for varying incidence angle of external field. The exchange energy is lowest for saturated state (aligned magnetic moments), whereas it increases when neighboring magnetic moments are angularly oriented (e.g. magnetic vortices). In the current case, exchange energy is lowest at higher fields affirming the saturated magnetic states. However, in the intermediate field range, the exchange energy increases due to the evolution, growth and propagation of magnetic vortices along the chain. When the external field is applied angularly (0° < h < 90°), the magnetic vortices do not penetrate to intermediate spheres completely and the reversal occurs quickly. Thus, angular field involves less amount of exchange energy. For normal external field (h = 90°), most of the spheres reverse directly without involvement of vortex structure, hence leading to very less exchange energy. These exchange energy profiles are consistent with the magnetisation reversal mechanisms and coercivity values of CoS nanostructure.

2000

0 deg

1500 30 deg 1000 45 deg 500

60 deg 90 deg

0 -2000

-1000

0

1000

2000

Applied field (Oe) (refer Section 3.4) and reversal at lower fields. When the external field is normal to CoS axis, it forms least number of vortices and exchange energy resulting in zero coercivity. This trend holds good

Fig. 10. Exchange energy plotted as a function of external field during hysteresis cycle for different angle of incidence. The CoS system contains five Permalloy spheres of uniform size (d = 50 nm).

P. Barpanda / Computational Materials Science 45 (2009) 240–246

Normalised vortex core diameter

3.5 3

Middle spheres (S2, S4)

Middle spheres (S2, S4)

2.5 2 1.5

End spheres (S1, S5)

End spheres (S1, S5)

1 -1500

-1000

-500

0

500

1000

1500

External field (Oe) Fig. 11. Micromagnetic simulation results showing the vortex core diameter (VCD) in the spheres comprising the chain structure during magnetisation reversal process. The VCD is normalised to exchange length of Permalloy.

245

throughout. This steady decrease in VCD continues till complete reversal around 1300 Oe. The same trend appears when positive field is applied to the CoS system. Fig. 11 does not show the VCD of centre sphere as it does not form any vortex (refer Section 3.1). For a relaxed magnetic vortex, the VCD is surrounded by inplane circular swirling magnetic moments. However, the presence of external field forces the vortices to form out-of-plane helix structure. The orientation of these helical vortices can be quantified as pitch-of-helix. Here, the pitch-of-helix is defined as the angle between out-of-plane magnetic moments and y–z plane measured at a radius of half of that of the sphere. For axial external field (0°), Fig. 12 presents the pitch-of-helix as a function of external field during hysteresis cycle. From 0 Oe, when the field decreases, a gradual formation and growth of vortices occur. The decreasing field forms more perfect vortices, hence continuously decreasing the pitch-of-helix. As the outer spheres involve early formation and growth of vortices in CoS, thus they have smaller pitch-of-helix than the middle spheres. This trend continues till the coercive field of 1300 Oe, when complete reversal occurs. Upon increasing the field, same trend continue but with opposite orientation of magnetic vortices (hence the negative angle). 4. Conclusions

The vortex core diameter is pivotal vortex parameter as it affects the strength of magnetic interaction between individual and neighboring spheres. In the present study, VCD is defined as the distance (from centre of sphere) over which the x-component of the magnetisation (Mx) decreases by 50% of its maximum value at the centre of sphere. In case of axial external field (0°), the variation of VCD during hysteresis cycle is illustrated in Fig. 11. The VCD is normalized to the exchange length of Permalloy (5.1 nm), as exchange length is key factor affecting the underlying magnetic phenomena. It is observed with decreasing the field from 0 Oe, the VCD gradually decreases, indicating the formation of more stable vortices. As the vortex becomes more stable, it possesses a very confined central saturated domain surrounded by swirling moments in majority of the sphere. As the vortices appear in the end spheres early during hysteresis, the end spheres possess more perfect vortices throughout the hysteresis cycle. Thus, VCD values are less for end spheres than intermediate spheres (Fig. 11)

The hysteresis behaviour of chain of larger spheres supporting vortex state has been studied for Permalloy CoS with uniform sphere size of 50 nm. A thorough explanation of vortex creation/ annihilation mechanism and inversion symmetry feature is presented using micromagnetic simulation. The presence of vortices involves a gradual reversal in larger spheres, thus leading to lower coercivity in the system. The formation and growth of magnetic vortices are extremely sensitive to external field orientation and neighbouring magnetic elements. Evolution of these vortices was examined by gauging the vortex core diameter and pitch-of-helix in CoS system. CoS nanostructure with magnetic vortices forms a very interesting system for micromagnetic study. A detailed investigation of magnetic vortices in Permalloy CoS combining experimental observation and analytical calculation will be communicated shortly. Acknowledgement The author would like to thank Dr. T. Kasama (Cambridge, UK), Prof. M.R. Scheinfein (Simon Fraser University, Canada) and Prof. R.E. Dunin–Borkowski (TU, Denmark) for fruitful discussion on chain-of-sphere model. The author gratefully appreciates the support of Dr. Anil Kaza (Intel, OR), Dr. P. George (UMDNJ, NJ), Dr. Sai Doddi (Rutgers, NJ) and Prof. G.G. Amatucci (ESRG-Rutgers, NJ) during this work.

Vortex roration plate angle (degree)

80 60

Middle spheres (S2, S4)

40 20

End spheres (S1, S5)

0

References

End spheres (S1, S5)

-20 -40 -60 -80 -1500

Middle spheres (S2, S4) -1000

-500

0

500

1000

1500

External field (Oe) Fig. 12. Micromagnetic simulation results showing the pitch of helix (angle of vortex rotation plane) in the spheres comprising the chain structure during magnetisation reversal. The angle in all cases is measured at a distance of half of the radius of sphere from its centre.

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