torque magnetometry analysis of magnetic anisotropy distribution in ni nanowire arrays

Phys. Status Solidi A 208, No. 3, 553–558 (2011) / DOI 10.1002/pssa.201026390 p s sa

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applications and materials science

Torque magnetometry analysis ofmagnetic anisotropy distribution in Ninanowire arrays

Victor Vega*,1, Victor M. Prida**,1, Jose Angel Garcıa1, and Manuel Vazquez2

1Dpto. Fısica, Universidad de Oviedo, Calvo Sotelo s/n, 33007 Oviedo, Asturias, Spain2 Instituto de Ciencia de Materiales de Madrid, CSIC, 28049 Madrid, Spain

Received 9 July 2010, revised 17 October 2010, accepted 19 October 2010

Published online 19 November 2010

Keywords anodic alumina templates, dipolar interaction, ferromagnetic nanowires, magnetic anisotropy

*Corresponding author: e-mail [email protected], Phone: þ34985103294, Fax: þ34985103324** e-mail [email protected]

Highly ordered arrays of Ni nanowires have been prepared by

pulsed electrochemical deposition into nanopores of anodic

alumina membranes (NAAMs) used as templates. They have

been experimentally characterized by magnetic torque meas-

urements and vibrating sample magnetometer (VSM) techni-

ques in order to determine the magnetic anisotropy of the

hexagonal array of nanowires. A detailed analysis of the

experimental data has been performed based on a phenomeno-

logical model taking into account the influence of the nanowire

shape anisotropy added to the dipolar magnetostatic interac-

tions among them. An overall agreement is obtained between

the simulations derived from the model and the experimental

magnetic torque anisotropy curves.

� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Since Masuda and Fukuda firstreported in the 1990s on the preparation of highly orderedarrays of self-assembled nanopores in anodic alumina by atwo-step anodization process [1], the synthesis of patternednanostructured systems using those porous templates hasattracted large interest. This is due to the advantages infeasibility of different nanostructures of relevance in a broadspectrum of research fields [2, 3] including basic research innanomagnetism through such a low-cost technique. Amongothers, potential technological applications include novelperpendicular magnetic storage media and self-assembledfunctional materials, where the tailoring of the magneticanisotropy plays an important role [3–7].

Magnetic nanowire arrays are thus electrochemicallygrown by electroplating filling of the self-assemblednanopores of anodic alumina membranes (NAAMs) usedas templates. Therefore, metallic nanowires are arrangedwith hexagonal symmetry reproducing the ordering of poresin the template [8].

Arrays of magnetic nanowires with high aspect ratio arecharacterized by a narrow size distribution at the nanoscale,well below the limits of conventional nanolithographytechniques [9, 10]. This allows one to achieve a higher

magnetic shape anisotropy that increases their magneticstability, prevents spontaneous switching of magnetizationby thermal fluctuations, and avoids the superparamagneticlimit restrictions in perpendicular magnetic recording media[3, 11–13].

On the other hand, magnetostatic interactions amongnanowires have been widely studied in the literature sincethey can play a determining role in the final magneticproperties of the nanowire arrays [14–23].

Several experimental techniques are conventionallyemployed to determine the magnetic properties of individualnanowires as magnetic force microscopy (MFM) or givenmagnetotransport studies [13, 24–26]. Other experimentaltechniques as ferromagnetic resonance (FMR) measure-ments, superconducting quantum interference device(SQUID), or vibrating samplemagnetometry (VSM)providecomplementary information on the array as a whole, aboutthe average effectivemagnetic anisotropy andmagnetostaticinteractions[5, 16, 17, 21–24, 27].

In this work we report on the determination of theanisotropic magnetic behavior of highly hexagonallyordered Nickel nanowire arrays fabricated by pulsedelectrodeposition into self-assembled NAAMs templates.


554 V. Vega et al.: Magnetic anisotropy distribution in Ni nanowire arraysp

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Figure 1 SEM cross-sectional view of a NAAM template. Thealumina nanopores are 35 nm in diameter and 105 nm of interporedistance. Inset: SEM cross-sectional view of another NAAM afterpores electroplating filling with Ni.

Figure 2 (online color at: www.pss-a.com) Magnetic torque ani-sotropy curves, experimentally measured (&), Fourier fit (solidline), and simulated curves (dashed lines) taking into account onlythe shape anisotropy ( ) and both, shape anisotropy and dipolarinteractions ( ).

The total magnetic anisotropy is experimentally evaluatedby the magnetic torque magnetometry (MTM) technique.The analysis of MTM curves was also correlated with VSMhysteresis loops, achieving a good agreement between bothdata. A simplified phenomenological model has beendeveloped based on the cylindrically shaped magneticanisotropy of nanowires. In addition, dipolar-like magneto-static interactions between nanowires are introduced tooptimize the fitting between the model and the experimentaldata. The use of magnetic torque measurements has not beenpreviously employed to quantify themagnetic interactions inthis kind of nanometric systems. Here, it is proven to be asuccessful tool to provide relevant information not onlyabout the magnitude of the magnetic anisotropy, but alsoabout its preferred orientation.

2 Experimental details NAAMs templates wereobtained through a two-step anodization process carriedout in a 0.3M oxalic acidic electrolyte by applying apotentiostatic 40Vdc voltage as it has been previouslyreported elsewhere [1, 28]. Afterwards, a progressivethinning of the barrier layer was performed in order toreduce the thickness of the alumina barrier layer to about8 nm, improving its electric conductivity and enabling thesubsequent homogeneous pulsed electroplating process [2].

Nickel nanowires were electrochemically grown in thealumina nanopores employing a Watts bath [27, 28]. Theelectroplating process lasted for 40min and an averagenanowires length of 2.85mm was estimated throughFaraday’s laws and current transients recorded during thenanowires growth.

Morphological characterization of both NAAMs and Ninanowire arrays was carried out by means of scanningelectron microscopy (SEM) and force microscopy (AFM/MFM). Magnetic torque measurements were carried out in ahomemade torque magnetometer based on the magnetictorque exerted by a ferromagnetic sample that is magnetizedto saturation by a uniform applied magnetic field, andemploying the same applied field to act as the counteractingtorque [29]. The experimental torque measurements havebeen performed at room temperature under an appliedmagnetic field of 9 kOe, by varying the angle u between thefield and the axis of the nanowires. The hysteresis loops wereobtained with a VSM under a maximum applied field of10 kOe.

3 Results and discussion3.1 Magnetic torque analysis Figure 1 shows a

SEM cross-sectional view of a nanoporous anodic aluminamembrane. Nanopores diameter and internanopores spacinghave been estimated to be 35 and 105 nm, respectively. Agood parallelism between the nanopores is achieved as canbe observed in Fig. 1, following the highly ordered andparallel aligned nanochannels along the alumina template.

The inset shows the cross-sectional SEM image ofanother NAAM after the Ni electroplating filling of thepores, resulting in the parallel arranged growth of Ni


nanowires, about 3mm long (with fluctuations in length lessthan 10%), in the channels of the NAAM template.

Magnetic torque measurements shown in Fig. 2 wereperformed by varying the angle u between the appliedmagnetic field (9 kOe in amplitude) and the longitudinal axisof the nanowires array. Experimental data werewell fitted bya Fourier series. As observed, the Fourier line fit displays atwofold symmetry that denotes the uniaxial character of themagnetic anisotropy. The fitted anisotropy constant gives avalue of 5.7� 105 erg/cm3, indicating also that the magne-tization easy axis lies parallel to the nanowires axis.Meanwhile, the plane of the sample substrate remains as ahard axis plane. The in-plane angular dependence of the

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Phys. Status Solidi A 208, No. 3 (2011) 555

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Figure 3 (online color at: www.pss-a.com) VSM room-temper-ature hysteresis loops under magnetic field applied parallel to thenanowires axis ( ) and perpendicular to the nanowires axis at threedifferent in-plane orientations forming 608 to each other, respec-tively (�, , ).

magnetic torque measurements has shown no relevantanisotropy (and hence these measurements are not pre-sented). Differences in the slopes of the measured magnetictorque curve that make the sinusoidal curve asymmetriccould be ascribed to a slight misalignment between theapplied magnetic field and magnetization direction [30, 31].

The magnetic torque results are consistent in theirqualitative behavior with the room temperature VSMhysteresis loops shown in Fig. 3. The parallel hysteresisloop shows a high remanence (mr¼Mr/Ms¼ 0.75) thatdenotes an easy magnetization direction close to the wiresaxes with a coercive field of 800Oe. In turn, the reducedremanence (mr¼ 0.25) of perpendicular hysteresis loopsindicates a magnetization hard plane. Coercivity takes avalue of 250Oe, and magnetization approaches saturationfor an applied field of aboutHappl� 2.5 kOe. The lack of anypreferred in-plane magnetization orientation is confirmed byrotating the sample plane at three different angles, making anangle of 608 to each other, obtaining similar results for thedifferent directions of the applied field.

3.2 Magnet ic anisotropy and dipolarinteractions modeling In order to explain the origin ofthe uniaxial magnetic anisotropy for the hexagonally orderedNi nanowires array, a simplified phenomenological modelhas been developed taking into account both shape anisotropyof the nanowires and magnetostatic dipolar interactionsamong them in a mean-field approach. In this particular case,the magnetocrystalline anisotropy is neglected in a firstapproximation since its contribution is expected to be muchsmaller than the shape anisotropy [13, 32].

A single nanowire can be approximated by a prolatespheroid with a long axis two orders, at least, longer than theshort one [11]. In this context, the prolate spheroid can betaken as infinitely long so the magnetic shape anisotropy

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constant can be given by ksh¼pM2s [17, 30], and the

demagnetizing field due to the nanowire’s cylindrical shapewould be given by:

Hsh ¼ 2pM: (1)

For the sake of simplicity, and as can be deduced fromthe hysteresis loops in Fig. 3, we assume that the nanowiresare magnetically nearly saturated along the direction of theapplied field, irrespective of that orientation, when it reachesa value above 2.5 kOe [30]. Considering the more generalcasewhen both the appliedmagnetic fieldH and therefore thesaturation magnetizationMs, make an angle uwith respect tothe nanowires axis (Fig. 4a), the demagnetizing field alongthe perpendicular direction of the nanowires axis, due to theshape anisotropy, can be written as:

Hsh ¼ �2pMs sinu ur ¼ �2pM? ur; (2)

with ur the unitary vector parallel to the radial direction ofthe nanowire (Fig. 4a).

Dipolar-like magnetostatic interactions among nano-wires give rise to an effective field sensed by the nanowiresthat adds to the applied magnetic field and their strengthdepends on the exact spatial configuration of all nanowires inthe array. Here, we will consider two particular configur-ations of the magnetically saturated sample.

In the first case, we assume that nanowires aremagnetized along their long axis (Fig. 4b).Magnetic chargesaccumulate at the top and bottom of the nanowires, and thatleads to a demagnetizing dipolar field, HDjj, in a given wirethat is antiparallel to the saturating applied magnetic field.Such demagnetizing field is given by [21]:

HDjj ¼ �4pPMs cosu ul ¼ �4pPMjj ul; (3)

with ul the unitary vector along the parallel direction to thenanowire axis and P the filling factor of the sample, which isgiven by P¼ (Dp/Dint)

2�p/(2H3) for the case of anhexagonal array of nanowires, where Dp is the nanowirediameter and Dint is the lattice parameter of the hexagonalarrangement.

In the case of an array of nanowires magnetizedperpendicularly to their long axes, the charges on thecylindrical wire surfaces (Fig. 4c) create a dipolar demagne-tizing field, HD?, which is perpendicular to the nanowire’saxes and depends on their diameter, Dp, and the interwirespacing, Dint, according to Ref. [21]:

HD? ¼ 2pPMs sinu ur ¼ 2pPM? ur: (4)

Considering Eqs. (2), (3), and (4), for the case when theexternal magnetic field is applied at an arbitrary directionmaking an angle, u, with respect to the wires axis, themagnetostatic energy density can be written as:

Ems ¼ M �H þ 1=2M �Hef ¼ Ms H

�pM2s ½ð1�PÞ sin2u þ 2P cos2u�; (5)



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Figure 4 (online color at: www.pss-a.com) (a) Scheme of themagnetization components for a single nanowire along its axis(Mjj) and over the radial direction of the nanowire (M?), being theapplied magnetic field and the magnetization making an arbitraryangle uwith respect to thewire’s axis. (b) and (c)Magneticmomentsconfiguration and surface magnetic charges for the hexagonallyordered array of the dipolar interacting nanowires when the externalfield is applied in the axial direction (ul) (b), or along the radialdirection (ur) (c), respectively.

where the first term on the right is the Zeeman contribution,andHef is the net magnetostatic field that takes into accountboth shape anisotropy and dipolar interactions among the Ninanowires.


The derivative of the magnetostatic energy with respectto u gives us an expression for the magnetic torque:

t � �@Ems=@u ¼ pM2s ð1�3PÞ sin2u (6)

where the factors pM2s and �3pM2

sP denote the contri-butions of shape anisotropy and dipolar interactions,respectively.

From Eq. (6), we can proceed to a comparison betweenthe results from the phenomenological model and exper-imental MTM measurements.

The dashed blue line in Fig. 2 depicts the torque curvecalculated throughEq. (6), taking into account only the shapeanisotropy, while the dashed red line also takes into accountthe contribution of the dipolar interaction among the Ninanowires. In all cases, the saturation magnetization valuefor the Ni nanowires has been taken as that for the bulk Ni(Ms¼ 485.56 emu/cm3) [33]. Starting from the magneticmodel but discarding the magnetostatic dipolar interactions,an anisotropy constant value of 7.4� 105 erg/cm3 has beenobtained,which is consistentwith previous calculations [34].Meanwhile, by considering both dipolar interactions andshape anisotropy contributions to the net anisotropy, aneffective anisotropy constant of 5.2� 105 erg/cm3 is thenobtained, which results in a better agreement with the valueobtained from MTM measurements (5.7� 105 erg/cm3).

The technique employed here is a reliable and directmethod that enables the quantitative determination of theeffective magnetic anisotropy in the arrays of interactingnanowires. For example,we could compare the present resultwith what is possible to obtain from VSM hysteresis loopsshown in Fig. 3. From a simple observation of the shape ofthe orthogonal hysteresis loops and their remanence values,we straightforwardly deduce an effective anisotropy, K,along the nanowire axis. Nevertheless, a proper estimation ofits quantitative strength is not so easy.We could estimate it asK¼ (½)MsHk, with Hk as the anisotropy field that can beroughly visualized in Fig. 3 like some value smaller than thefield to reach themagnetization saturation (about 2.5 kOe) byconsidering the field dependence of the susceptibility in theperpendicular loops.

Alternatively, the anisotropy field distribution can beobtained by numerical derivation of the descent fromsaturation to the remanent state of the orthogonal M(H)curve, as it has been reported in Ref. [22], giving a roughestimation for the anisotropy field of about Hk� 1.56 kOeand therefore an effective anisotropy value of aroundK� 3.9� 105 erg/cm3 can be deduced, which is of the sameorder of magnitude but below the proper estimation.

4 Conclusions In summary, the effective anisotropyconstant of a hexagonal array of interactingNi nanowires hasbeen determined by means of MTM measurements. Aphenomenological model describing the anisotropic mag-netic behavior of the nanowires array has been established bytaking into account the contributions of shape anisotropy andmagnetostatic dipolar interactions among nanowires.

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MTM measurements show the existence of a uniaxialanisotropywithmagnetization easy axis along the nanowiresaxis, giving a value of the effective anisotropy around5.7� 105 erg/cm3.

From comparison between MTMmeasurements and thesimulated model, we obtain a shape-dominated anisotropybehavior, together with a vanishing contribution of themagnetocrystalline anisotropy, having a value of around7.4� 105 erg/cm3. Only when the magnetostatic dipolarinteraction is also considered, which acts partially balancingthe shape anisotropy, is the total effective anisotropy valuereduced to 5.2� 105 erg/cm3, resulting in a better agreementbetween the simulated model and the measured torquemagnetic anisotropy curves. These direct magnetic torquemeasurements confirm thatmagnetostatic dipolar interactionalso plays an important role in determining the magneticbehavior of this kind of nanomagnetic entities, while themain contribution governing the total anisotropy is deter-mined by their shape.

In addition, as can be inferred from Eqs. (3) and (4), thedipolar interaction among ordered arrays of ferromagneticnanowires, and hence their effective magnetic anisotropy,can in fact be partially tailored by means of convenientlymodifying the geometrical parameters of the pores arrange-ment in the NAAM templates (pore diameter and/orinterpore distance) and therefore, the filling factor, P, ofthe sample. Such modification can be well done byemploying alternative anodization techniques such as hardanodization (HA), or nanoimprint (NI) processes, as well asby further modification of the as-produced nanoporoustemplates after submitting them to pore widening bychemical etching of the Al2O3, or pore narrowing throughthe atomic layer deposition (ALD) technique, creating aconformal coating over the NAAM surface. Furthermore, itis also possible to tune the effective magnetic anisotropy inarrays of ferromagnetic nanowires by considering multi-layered or barcode wires, made of alternating layers withmagnetic materials and nonmagnetic spacers [35].

Finally, it has been proven that the MTM techniqueappears as a feasible and direct method to account for themain features and magnitude of the effective magneticanisotropy of ferromagnetic materials on the nanoscale.

Acknowledgements FEDER and Spanish funding underMEC and FICyT research projects, numbers MAT2007-65420-C02-01, MAT2007-65097-C02-02, MAT2009-13108-C02-01,FC09-IB09-131, MAT2010-20798-C05-01, and MAT2010-20798-C05-04 are gratefully acknowledged. The scientificsupport from the University of Oviedo SCT’s, particularly fromthe Nanoporous Membranes Unit, is also acknowledged.

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torque magnetometry analysis of magnetic anisotropy distribution in ni nanowire arrays

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