Packing fraction dependence of the coercivity and the energy product in nanowire based permanent magnets

Packing fraction dependence of the coercivity and the energy product innanowire based permanent magnetsIoannis Panagiotopoulos, Weiqing Fang, Frédéric Ott, François Boué, Kahina Aït-Atmane et al. Citation: J. Appl. Phys. 114, 143902 (2013); doi: 10.1063/1.4824381 View online: http://dx.doi.org/10.1063/1.4824381 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v114/i14 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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Packing fraction dependence of the coercivity and the energy productin nanowire based permanent magnets

Ioannis Panagiotopoulos,1 Weiqing Fang,1 Fr�ed�eric Ott,1 Francois Bou�e,1

Kahina A€ıt-Atmane,2 Jean-Yves Piquemal,2 and Guillaume Viau3

1Laboratoire L�eon Brillouin CEA/CNRS UMR12, Centre d’Etudes de Saclay, 91191 Gif sur Yvette, France2ITODYS, Universit�e Paris 7-Denis Diderot, UMR CNRS 7086 2, Place Jussieu, 75251 Paris Cedex 05,France3Laboratoire de Physique et Chimie des Nano-Objets, INSA de Toulouse, UMR CNRS 5215,135 Av. De Rangueil, 31077 Toulouse Cedex 4, France

(Received 13 August 2013; accepted 20 September 2013; published online 7 October 2013)

The maximum magnetic performance which can be expected from elongated single domain particle

based permanent magnets is assessed. The results are derived using micromagnetic calculations to

model the behavior of large bundles of aligned nanowires. In particular, we discuss the cases of Co

and Fe nanowires and their packing fraction dependence of coercivity, which is the main limiting

factor. We show that it is, in principle, feasible to achieve BHmax values close to 300 kJ/m3 in Co

nanowires with a packing fraction p¼ 0.7 and close to 400 kJ/m3 at p¼ 0.85. The packing fraction

limitations are essentially non-existing due to the intrinsic magnetocrystalline anisotropy of Co. On

the other hand, if a low anisotropy material such as Fe could be produced in the form of fine, well

crystallized wires it could yield a BHmax close to 200 kJ/m3 at an optimum p¼ 0.7. As the

performance of iron nanowires is solely based on shape anisotropy it becomes coercivity limited

above p¼ 0.35. VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4824381]

I. INTRODUCTION

Today, the turn towards hybrid and electric cars, wind

turbines, and other green energy applications is expected to

substantially increase the demand for magnetic materials.1

At the same time, the limited supply in rare-earths (RE)

motivates the research for a new generation of high added

value RE-lean and/or even RE-free permanent magnets. One

possible route to permanent magnetism without relying on

the RE magnetocrystalline anisotropy is to achieve high

coercivity based on shape anisotropy. Actually, this is the

main mechanism in AlNiCo magnets, which can be thought

as consisting of CoFe needles in a NiAl non-magnetic ma-

trix.2 The same approach applies for any system of elongated

nanoparticles. Mono-crystalline and mono-disperse cobalt

nanowires synthesized by the polyol process exhibit remark-

able hard magnetic properties for a simple 3d metal with

coercivity values close to 600kA/m (0.75 T) at room temper-

ature.3 Half of the observed coercivity can be accounted for

by the magneto-crystalline anisotropy,4 while the remaining

part is due to the shape anisotropy contribution. However,

even the Co market is periodically exhibiting limited and

fluctuating supply. On the other hand, creating competitive

permanent magnets based solely on shape anisotropy might

not be feasible in practice. The energy product BHmax of a

magnetic material depends sensitively on the shape of the

hysteresis curve in the demagnetization quadrant. For the

best case of a completely rectangular M(H) loop, an intrinsic

coercivity HC > MR=2 is required to achieve the full poten-

tial of the material ðBHÞmax ¼ l0ðMR=2Þ2, where Mr is the

remanent magnetization. Otherwise, we are in the coercivity

limited case with ðBHÞmax ¼ l0ðMR � HCÞHC and the mag-

net has to be fabricated in a prolate shape in order to operate

at its optimum working point, defined by a low demagnetiz-

ing factor.2 If no coercivity limitations exist the BHmax scales

with the square of the remanent magnetization. For an as-

sembly of nanowires with packing fraction p and an average

misalignment angle w, the remanence is MR � pMScos w.

This along with the steep decrease of coercivity with mis-

alignment in Stoner-Wolhfarth (SW) particles makes

obvious that for shape anisotropy magnets, as for any magnet

in general, alignment is a key issue in achieving high per-

formance. Furthermore high packing fractions can only be

achieved in aligned nanowires. However, one has to take

into account the decrease of coercivity with packing fraction

originating from dipolar interactions and demagnetization

fields effects. If such limitations did not exist, then the recipe

for making a high performance magnet would simply be to

produce it at the highest achievable packing fraction. Since

the first efforts to create permanent magnets by compacting

elongated particles,5 it was recognized that there is a loss of

coercive field with packing fraction and that this leads to a

maximum achievable energy product at intermediate packing

fractions. After the recent renewal of interest in this route of

permanent magnet fabrication, Skomski and coworkers6

have carefully addressed the problem and based on simple

arguments have shown that for shape anisotropy, there is a

maximum energy product up to BHmax ¼ l0M2S=12 for

p¼ 2/3. Here, using micromagnetic modeling, we present

the dependence of the coercivity on the packing fraction and

achievable energy products in shape anisotropy magnets

based on Co nanowires. We compare these results with the

low magnetocrystalline anisotropy case of Fe. In practice,

the synthesis of elongated needle-like Co nanoparticles is

based on the anisotropic growth of the hexagonal crystal, a

recipe that cannot be applied to the cubic structure of Fe.

0021-8979/2013/114(14)/143902/6/$30.00 VC 2013 AIP Publishing LLC114, 143902-1

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Nevertheless Fe, being a high moment and low anisotropy

material, is an ideal system to test the effect of packing frac-

tion in a system in which the coercive field is mainly due to

shape anisotropy.

II. MICROMAGNETIC SIMULATION PROCESS

The micromagnetic calculations have been performed

with the Nmag package.7 The damping constant was set to

0.8 to achieve fast damping and reach convergence quickly.

The convergence criterion for each field step was that the

magnetization should move slower than 10�/ns (globally or

on average for all spins). The nanowire dimensions (length

L¼ 100 nm and diameter D¼ 14.5 nm) were chosen to

match those observed in recent high coercivity, polyol-

synthesized, mono-disperse cobalt nanowires.3 The bulk

magnetic parameters have been used. Namely, MS¼ 1.4

MA/m, KU¼ 520 kJ/m3, A¼ 28pJ/m for cobalt and

MS¼ 1.7MA/m, K1(cubic)¼ 48 kJ/m3, A¼ 21pJ/m for iron.

According to these values, the exchange lengths are 3.4 nm

and 2.4 nm for Co and Fe, respectively. A mesh length 7 nm

was used in order to be able to include a large number of par-

ticles (>400) within the simulation cell. For such large sam-

ples, the memory usage was 30 Gb. When using a mesh

smaller than the exchange length (2.4 nm), the calculated

switching fields are only increased by 1.3% and 6% Co and

Fe, respectively (as calculated for an isolated wire). We con-

sidered that increasing the system size was more important

to produce realistic results than aiming for (pseudo)-accurate

switching field values which are experimentally anyway

very sensitive to any defect in the system.

The external field was applied at 5� with respect to the

nanowires’ axes. In order to be able to simulate aligned

nanowire aggregates with a systematic variation of packing

fractions (ranging from zero up to close to unity) in a con-

sistent manner, 209 nanowires have been placed in regular

arrays of 19 rows arranged in an hexagonal 2D-lattice, each

row consisting of 11 nanowires stacked along their lengths.

For some packing fractions, larger aggregates consisting of

407 nanowires in 37 rows of 11 wires have been also simu-

lated. The numbers 19 and 37 are simply neighbors up to dis-

tances 2a and 3a, respectively from a central point in a

hexagonal lattice with a lattice constant a¼D þ d, where D

is the nanowires diameter and d is the distance between the

nanowires. The maximum packing fraction for a 2D hexago-

nal lattice (limit of d¼ 0) is pmax ¼ p2ffiffi3p . Different volume

packing fractions can obtained by adjusting d and the dis-

tance along the wire length dx according to p ¼ pmaxD

Dþd

� �2

LLþdx

h i. In order to consistently and uniquely define d and dx

for a given p, one could choose either to keep the bundle as-

pect ratio constant or simply set dx equal to d. We have cho-

sen the later in order to avoid large distances between

consecutive wires in a row at low packing fractions. A ran-

dom displacement of each row along the bundle axis by dL,

with �25 nm � dL � þ25nm was used to produce more re-

alistic bundles of nanowires, as it has been shown that such

configuration leads to a strong effect of coercivity (HC) loss

due to local stray fields.8 In order to validate the coercivity

values obtained in elongated bundles, especially for the case

of low anisotropy Fe, we have also performed calculations in

samples with aspect ratios close to unity consisting of 374

nanowires in 187 rows (neighbors up to 7a) of two wires.

The simulation of a single nanowire with the same character-

istics is considered as the p¼ 0 point.

The aspect ratio 19 � 11 bundle of nanowires as a whole

AR ¼ 11 � ðLþ dxÞ=ð5Dþ 4dÞ, ranged from AR¼ 6.8 to 15

for p¼ 0.05 to 0.87, respectively. Approximating their

demagnetizing factors by those of ellipsoids with the same

aspect ratio, these aspect ratios yield N¼ 0.04 to

N¼ 0.0015. Cylinders9 and prisms10 with the same aspect

ratios present equally low values. Based on these values, the

maximum demagnetizing field is found to be 0.009MS (for

p¼ 0.87), which is low compared to the HC derived from our

simulations for the isolated nanowires (HC¼ 0.54MS for Co

and HC¼ 0.21MS for Fe). Thus, the simulation samples can

be considered to reproduce well the conditions of closed

magnetic circuit measurements used for the characterization

of permanent magnet materials,2 in which the demagnetiza-

tion factor is practically zero. Nmag offers the possibility to

include the effects of the demagnetizing field by specifying

an appropriate arrangement of copies of the primary simula-

tion cell.11 However, we found that, in this specific case, it

should be used with caution as creating very long samples in

this way tends to create square loops with HC close to that of

a single nanowire. So we chose to simply work with high

AR prolate simulation cells without introducing further elon-

gation through extra copies. Examples of the magnetic con-

figurations close to the nucleation and coercivity of the 407

Fe nanowire bundle are shown in Fig. 1. The snapshots of

Fig. 1 represent static magnetic configurations after conver-

gence has been achieved for each field step. If the

in-between magnetic states are monitored during the relaxa-

tion process, it is revealed that the reversal starts by nuclea-

tion at both wire edges and propagation of two domain walls

towards the center. Of course, in bundles of wires nucleation

begins by one of the two edges, whichever is favored by the

interactions. For isolated wires, the loops qualitatively

resemble those of the SW model (for ellipsoids with similar

aspect ratio) and the nucleation field shows an angular de-

pendence similar to that of the SW model, though the values

differ. Thus, these particle assemblies show macroscopically

a magnetic response that can be mistaken for SW particles.

The reversal of the bundle as a whole proceeds by nucle-

ation and propagation along the length of the bundle in a

domino fashion. This can be understood to a first approxima-

tion, by the fact that dipolar interactions tend to align the

magnetic moments of nanowires stacked along their lengths

and, on the contrary, antiparallel when stacked along their

diameters. Thus, dipolar interactions seem to promote rever-

sal in elongated regions since after a region has flipped, it

will tend to destabilize the lengthwise neighboring ones and

stabilize the lateral ones. Considering that this bundle has an

aspect ratio of 10, we can estimate an average demagnetizing

field pNMS¼ 0.013MS¼ 22 kA/m. This low value cannot

explain the huge difference of the nucleation field

HN¼�140kA/m compared to the one of an isolated

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nanowire HN¼�335 kA/m. Thus, for these elongated struc-

tures, nucleation is instigated by local fields and cannot be

directly ascribed to macroscopic demagnetizing factors. Due

to the presence of highly inhomogeneous magnetic configu-

rations and collective reversal, it becomes questionable if

local effective fields can be reconciled by a mean field

approach of the type Heff¼ -NM with a fixed Neff for all the

magnetic configurations. This further poses a question con-

cerning the applicability of the derivation of the response to

the demagnetizing fields under different magnetic circuits

with different demagnetization factors using the zero demag-

netizing field measurement as a starting point. In a way, it is

the opposite process of the demagnetization correction for

which, a typical counter-example is the over-skewed

(S-shaped) loops6 that are obtained if one applies the demag-

netization correction by N¼�1 to a magnetic film sample.

It is known that the use of average homogeneous demagnet-

izing factors is limited to macroscopic samples in which the

details of the local magnetization M(r) can be ignored, while

the inhomogeneous magnetic states which exist in nanostruc-

tured materials create strong deviations.2,6,12 Since this

method has been empirically found to work in macroscopic

objects, addressing this question becomes more compelling,

in the case, where the effect of dipolar fields from a finite

number of nanowires is used to simulate macroscopic

objects. If the limited simulation cell represents in a correct

way the response of a macroscopic magnetic material, the

approximation Heff¼ -NM should give meaningful results.

To address this point, we have compared loops obtained in

bundles with the same p but different aspect ratios. A good

example is shown in Fig. 2 trying to reconcile the magnetiza-

tion curves of two bundles with p¼ 0.67 and AR¼ 1 and

AR¼ 10, respectively by a linear demagnetization factor

correction.

The nucleation field is HN¼�100 kA/m for the

AR¼ 10 bundle and HN¼þ150 kA/m for the AR¼ 1. This

difference of DHN¼ 250 kA/m corresponds to an effective

demagnetizing factor Neff¼DHN/pMS¼ 0.22. Indeed by try-

ing different Neff¼ 0.1 to 0.3 Neff¼ 0.2 seems to be the cor-

rect value to make the curve of the AR¼ 10 bundle coincide

with that of AR¼ 1. The coincidence is restricted in the sec-

ond quadrant (M> 0, H< 0) of the loop, while in the third

quadrant (M< 0, H< 0) sever deviations occur. These would

result in the well-known over-skewing if we were trying to

do the opposite, i.e., to obtain the AR¼ 10 curve by a

demagnetization correction of the AR¼ 1 curve. To fit the

data of the third quadrant, lower N should be used but strong

deviations would still exist. There are two interesting points:

(i) though the HC values seem to differ, the estimation of

BHmax (comparing between AR¼ 1 and Heff corrected

AR¼ 10) is not seriously affected as it is based on the shape

of the loop on the second quadrant away from coercivity (see

discussion below). (ii) The differences of the HN of different

AR bundles seems to give results that can be reconciled with

a choice of an appropriate Neff, in contrast to the comparison

between an elongated bundle and the isolated wire, discussed

in Fig. 1.

An example of demagnetization curves and BHmax cal-

culation is shown in Fig. 3 for the same bundle of 407 Fe

wires with p¼ 0.67. In comparison, the hysteresis curve of

an isolated nanowire scaled to magnetization pMS

FIG. 2. Magnetization curves of two bundles with p¼ 0.67 and AR¼ 1 (2 �187 nanowires, solid circles) and AR¼ 10 (11 � 37 nanowires, diamonds).

The continuous curves are the (H, M) data of AR¼ 10 transformed to

(HþNM,M) for different N.

FIG. 1. Magnetic configurations of an

aggregate of 407 iron nanowires (37

rows of 11 nanowires) from nucleation

(top left) to coercivity (bottom right).

The numbers next to each configura-

tion denote the applied field H in kA/m

and the normalized total magnetization

m¼M/MS. The colors represent the

component of the magnetization vector

along the nanowire axis (red for

m¼þ1 to blue for m¼�1).

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(representing the hypothesis that the wires could be packed

without any HC loss). A first observation is that a coercivity

HC>Mr/2¼ 569 kA/m (i.e., much higher than that of an iso-

lated nanowire HC¼ 335 kA/m) is required to take full

advantage of the BR¼ l0MR¼ 1.43T remanence. Thus, for

Fe wires at this packing fraction, the performance is anyway

HC limited. The BHmax of the Fe wire bundle is estimated to

196 kJ/m3 (circle). The energy product that corresponds to

HC¼ 335 kA/m is BHmax¼ 315 kJ/m3 (indicated by a star in

Fig. 3). Even with HC¼ 220 kA/m, if the loop of the Fe bun-

dle was square, we would get BHmax¼ 255 kJ/m3 (yellow tri-

angle in Fig. 3). In other words not only HC but the

nucleation field HN (the field value at which M departs from

MR) and the loop shape are important for the energy product.

The corresponding example of demagnetization curves

and BHmax calculation for a bundle of 209 Co nanowires

with p¼ 0.7 is shown in Fig. 4. The HC and HN safely exceed

the value of 488 kA/m required to take full advantage the

remanence BR¼ 1.22T.

III. PACKING FRACTION DEPENDENCE OF THECOERCIVITY AND THE ENERGY PRODUCT

As mentioned, in the case of magnets consisting of

packed particles the saturation magnetization is simply pro-

portional to pMS, where MS is the magnetization of the pure

magnetic phase. The main concern is the limitation imposed

through the HCðpÞ and HNðpÞ dependence. There is a scarcity

of experimental work on the HCðpÞ and especially in the case

of high packing fractions, as the published work is mainly

related to recording media.13,14 Thus, the main contributions

on the effect of packing fraction on elongated particles

remain that of Luborksy5 on Co-Fe and Morrish and Yu15 on

iron oxides. These contributions refer to materials that are

not nanostructured and thus deviations from the SW homo-

geneous rotation are not surprising.

Empirically, it has been found that the data can be

described by a relationship of the form

HCðpÞ ¼ Að1� pÞ þ B: (1)

Intuitively, the first term that has been proposed by N�eel,16

can be attributed to the shape anisotropy contribution and the

second to the magneto-crystalline anisotropy. Of course this

kind of assignment can only be approximate. One could

argue that for diameters large enough to favor curling modes

of reversal (for which, in principle, there is no magnetostatic

energy contribution), the HC should be independent of p as

opposed to coherent rotation. The values of A that can be

estimated from the data in literature5,15 are more than 3 times

lower than what is expected from the SW model (A¼MS/2)

indicating that a large part of the shape anisotropy is already

lost due to the large diameter of the particles and does not

contribute to the p dependence. The reported HC dependence

of elongated FeCo particles in Ref. 5 extrapolates to zero

before p¼ 0.8. On the other hand, nickel nanowires grown in

nanoporous alumina templates show very weak coercivity

dependence on packing fraction,17 despite the fact that this

variation is achieved by simultaneous increase of their radii,

which should cause further loss of HC. Interestingly, devia-

tions from the linear relationship have been observed close

to p¼ 0, which seem to follow an exponential decay law,13

but these are not relevant to energy product optimization

considered here. A complication of the experimental studies

in low p comes from the tendency of magnetic particles to

form clusters, which locally increases packing fraction.

The simple relation of Eq. (1) suggests that there are no

BHmax coercivity limitations up to a packing fraction of pc <AþB

AþMS=2and thus, no coercivity limitation at all, for materials

with high enough magnetocrystalline anisotropy to ensure

B > MS=2. Assuming A ¼ MS=2, B ¼ HK ¼ 2K=l0MS; we

get for Co pc ¼ 0:92, while for iron pc ¼ 0:52.

Above pc the optimum packing fraction would be

popt ¼ ðAþBÞAðAþMS=2ÞðAþMSÞ . For needle-like particles of negligible

FIG. 3. Magnetization curves M(H) (open blue circles), B(H) (open red dia-

monds) in the quadrant H< 0, M> 0, and BHmax (solid black circles) for a

bundle of 407 Fe nanowires with p¼ 0.67. The solid lines represent the

same curves for an isolated wire. The dashed line shows the energy product

for the non Hc limited case. The maximum energy product for the Fe wires

bundle is 196 kJ/m3 (red circle). The energy product that would correspond

to the same bundle, if there was not any HC loss due to packing is denoted

by a star. The energy product for a square loop with the same HC (i.e., in the

absence of nucleation effects at HN before HC) is denoted by a triangle.

FIG. 4. Magnetization curves M(H) (open blue circles), B(H) (open red

squares) in the quadrant H< 0, M> 0, and BHmax (solid black circles) for a

bundle of 209 Co nanowires with p¼ 0.70. The solid lines represent the

same curves for an isolated nanowire. The maximum energy product

obtained for H ¼ �MR=2 is denoted by a star.

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magneto-crystalline anisotropy, B¼ 0 and A ¼ MS=2 give

popt ¼ 2=3. For non-zero B, popt ¼ 23

1þ BMS=2

� �which shows

that for any material with B > MS=4, popt> 1 so that the ma-

terial should, in principle, be made as densely packed as pos-

sible. For instance, with the Co bulk parameters, popt > 1,

while in contrast for Fe popt ¼ 0:702.

The results of Nmag simulations for cobalt and iron bun-

dles are summarized in Fig. 5. The coercive field HC and the

nucleation field HN are plotted as a function of p. Their val-

ues are in agreement with those reported in literature: for

D< 15 nm iron nanowires HC¼ 280–320 kA/m at 5 K and

HC¼ 190–220 kA/m at 300 K,12 while for cobalt with

D¼ 6 nm HC � 1000 kA/m at 5 K and �600 kA/m at 300 K

have been have been reported.18 HC and HN show a linear

dependence on p. However (as summarized in Table I),

when fitted to Eq. (1) the values of the parameters A and B

differ considerably from the A ¼ MS=2, B ¼ HK , even if cor-

rected by a factor of aw ¼ 0:766 for the 5� misalignment

according to the well-known expression of the angular de-

pendence of the switching field for SW particles (page 187

of Ref. 2 for instance). Of course such a difference is not sur-

prising as the reversal mechanism differs from the simple ho-

mogeneous rotation SW mechanism. Qualitatively the

results show that, as intuitively expected, low anisotropy

leads to a low B. The nucleation field of Fe nanowires is the

only one that has a dependence close to that predicted by

N�eel’s simple prediction (B¼ 0).

The HC¼MR/2 limit can be estimated by the intersec-

tion of the each set of data with the corresponding pMS/2

line. According to the previous discussion, the intersection

with the HN is also meaningful as nucleation also lowers the

energy product. Above p¼ 0.35 for iron, BHmax becomes

coercivity limited and the magnet has to be supported by an

elongated shape. For instance, at the maximum p¼ 0.7, the

working point must be defined by a low demagnetizing fac-

tor N¼ 0.16 (load line B¼�5.25l0M), compared to N¼ 0.5

(B¼�l0M) in the non-Hc limited case. For Co, this limit is

above p¼ 0.8. As the shape anisotropy related coercivity

seems to persist even at high packing fractions, the realiza-

tion of permanent magnet materials with high anisotropy

materials seems, in principle, feasible.

In Fig. 6, the calculated BHmax values for Co and Fe

nanowires as a function of the packing density are summar-

ized and compared with the ones obtained by the simple pre-

diction that the loops are square and have a remanence

MR ¼ pMScos w and coercivity

HCðpÞ ¼ awMS

2ð1� pÞ þ HK

� �: (2)

For Co, the values are really close to those predicted for this

simple model as coercivity limitations appear at very high

packing fractions. In other words, the loops remain square with-

out any significant nucleation up to fields greater than pMS/2.

For Fe, the deviations are greater but the maximum is still close

to p¼ 2/3 as the magneto-crystalline anisotropy is low.

IV. CONCLUSIONS

Micromagnetic simulations show that it is, in principle,

feasible to achieve BHmax values close to 300 kJ/m3 in Co

nanowires with a packing fraction p¼ 0.7 and close to

400 kJ/m3 at p¼ 0.85. The packing fraction limitations are

essentially non-existing due to the intrinsic magnetocrystal-

line anisotropy of Co. On the other hand, if a low anisotropy

material such as Fe could be produced in the form of fine,

well crystallized wires it could yield a BHmax close to

FIG. 5. Coercivity (HC, solid symbols) and nucleation field (HN, semi-filled

symbols) for cobalt (red diamonds) and iron (black circles) nanowires bun-

dles determined by micromagnetic simulations. The intersections with the

pMS/2 straight lines show the critical packing fractions above which the

BHmax is HC limited.

TABLE I. Comparison of the constants A and B derived by fitting the Nmag

simulation data to Eq. (1) to the shape and magneto-crystalline anisotropy

fields. All values are in kA/m.

Material Field A (calc.) aw � ðMS=2Þ B (calc.) awHK

Co HC 169 453 582 536

HN 263 471

Fe HC 142 651 196 34

HN 262 69

FIG. 6. Energy product as a function of packing fraction for cobalt (red dia-

monds) and iron (black circles) nanowires. The continuous lines show the

calculations under the assumption of square loops with a coercivity given by

Eq. (2).

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200 kJ/m3 at an optimum p¼ 0.7. This value is coercivity

limited and corresponds to a low demagnetization factor

(N¼ 0.15) working point (on the load line B¼�5.25l0M).

As the performance of iron nanowires is solely based on

shape anisotropy it becomes coercivity limited above

p¼ 0.35. If we want to extend the results to other low anisot-

ropy high magnetic moment materials, we can state that this

value amounts to �l0M2S=18 when compared to the upper

theoretical limit l0M2S=12 derived in Ref. 6.

ACKNOWLEDGMENTS

This work was supported from the European

Commission FP7 for the REFREEPERMAG (No. EU

NMP3-SL-2012-280670) project.

1O. Gutfleisch, M. A. Willard, E. Br€uck, C. H. Chen, S. G. Sankar, and J. P.

Liu, Adv. Mater. 23, 821 (2011).2R. Skomski and J. Coey, Permanent Magnetism (Institute of Physics

Publishing, Bristol and Philadelphia, 1999).3Y. Soumare, C. Garcia, T. Maurer, G. Chaboussant, F. Ott, F. Fi�evet, J.-Y.

Piquemal, and G. Viau, Adv. Funct. Mater. 19, 1971 (2009).

4T. Maurer, F. Ott, G. Chaboussant, Y. Soumare, J.-Y. Piquemal, and G.

Viau, Appl. Phys. Lett. 91, 172501 (2007).5F. E. Luborsky, J. Appl. Phys. 32, 171S (1961).6R. Skomski, Y. Liu, J. E. Shield, G. C. Hadjipanayis, and D. J. Sellmyer,

J. Appl. Phys. 107, 09A739 (2010).7T. Fischbacher, M. Franchin, G. Bordignon, and H. Fangohr, IEEE Trans.

Magn. 43, 2896 (2007).8T. Maurer, F. Zighem, W. Fang, F. Ott, G. Chaboussant, Y. Soumare, K.

A. Atmane, J.-Y. Piquemal, and G. Viau, J. Appl. Phys. 110, 123924

(2011).9D.-X. Chen, E. Pardo, and A. Sanchez, J. Magn. Magn. Mater. 306, 135

(2006).10D.-X. Chen, E. Pardo, and A. Sanchez, IEEE Trans. Magn. 38, 1742

(2002).11H. Fangohr, G. Bordignon, M. Franchin, A. Knittel, P. A. J. de Groot, and

T. Fischbacher, J. Appl. Phys. 105, 07D529 (2009).12D. J. Sellmyer, M. Zheng, and R. Skomski, J. Phys.: Condens. Matter 13,

R433 (2001).13S. Umeki, H. Sugihara, Y. Taketomi, and Y. Imaoka, IEEE Trans. Magn.

17, 3014 (1981).14C.-R. Chang and J.-P. Shyu, J. Appl. Phys. 73, 6659 (1993).15A. H. Morrish and S. P. Yu, J. Appl. Phys. 26, 1049 (1955).16L. N�eel, C. R. Acad. Sci. 224, 1550 (1947).17S. D. Col, M. Darques, O. Fruchart, and L. Cagnon, Appl. Phys. Lett. 98,

112501 (2011).18K. Soulantica, F. Wetz, J. Maynadi�e, A. Falqui, R. P. Tan et al., Appl.

Phys. Lett. 95, 152504 (2009).

143902-6 Panagiotopoulos et al. J. Appl. Phys. 114, 143902 (2013)

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