J. Phys. D: Appl. Phys.33 (2000) R1–R16. Printed in the UK PII: S0022-3727(00)04626-X

REVIEW ARTICLE

Property variation with shape inmagnetic nanoelements

R P Cowburn

Nanoscale Science Group, Department of Engineering, University of Cambridge,Trumpington Street, Cambridge CB2 1PZ, UK

Received 9 September 1999

Abstract. Nanometre scale magnetic particles (‘nanoelements’ or ‘nanomagnets’) form arich and rapidly growing new area in condensed matter physics, with many potentialapplications in data storage technology and magnetic field sensing. This paper reviews anextensive study into the influence of shape on the properties of nanomagnets in the size range35–500 nm. Elliptical, triangular, square, pentagonal and circular geometries have all beenconsidered. It is shown that the size, thickness and geometric shape of nanomagnets all play avital role in determining the magnetic properties. The shape, size and thickness of ananomagnet are shown to be linked to its magnetic properties by two distinct phenomena.The first is called configurational anisotropy and describes the role played by small deviationsfrom uniformity in the magnetization field within the nanostructures, which allow unexpectedhigher-order anisotropy terms to appear. These anisotropies can often dominate the magneticproperties. The second is the competition which exists between exchange energy andmagnetostatic energy. This competition determines whether the nanomagnets exhibit singledomain or incoherent magnetization and also controls the non-uniformities in magnetizationwhich lead to configurational anisotropy. Understanding the influence of shape opens the wayto designing new nanostructured magnetic materials where the magnetic properties can betailored to a particular application with a very high degree of precision.

1. Introduction

One of the most exciting recent developments in magnetismhas been the use of nanometre fabrication techniques to formnanometre scale magnets. These so-called nanomagnets, ornanoelements, possess by virtue of their extremely smallsize very different magnetic properties from their parentbulk material. Interest in the study of these fascinatingparticles has been driven undoubtedly by the success of thesemiconductor microelectronics industry, in four differentways. First, demand for magnetic hard disk data storagehas grown commensurately with the spread and increasein the capabilities of microcomputers. Hard disks storetheir data as sub-micrometre scale magnetic domains. Adetailed understanding of magnetism on the nanometrescale is therefore essential for developing new hard disks.Moreover, it is generally thought that the current growth inperformance of hard disks (60–100% per annum compoundincrease in storage density) can only be sustained into thefuture if the hard disk itself is replaced by a massive array ofmagnetic nanoelements [1]. Second, the art and science offabricating structures on the nanometre scale have benefitedfrom the exponential shrinking in size of the integrated circuittransistor [2], leading to enormous investment in opticaland electron beam lithography. Third, the mathematicalequations governing magnetic behaviour on the nanometrescale are highly nonlinear and can generally only be studied

theoretically using computer-intensive numerical algorithms[3] (although, see as an exception [4]). The recent increasein availability of low-cost, high-power processors has nowenabled nanomagnetism to become a widely studied branchof condensed matter theory. Fourth, the semiconductorindustry may itself in the future benefit from nanomagnetism.Many of the problems currently facing future reduction insize of integrated circuits, such as the diminishing numberof carriers present in a transistor gate and difficulties inremoving heat, may be solved by combining magneticelements into microchips [5]. In the first instance this willprobably be in the form of non-volatile computer memorycalled MRAM (magnetic random access memory) andprocess control microchips with built-in magnetic sensorsfor the automotive industry. The recent discovery [6] ofhow to inject spin polarized current from a ferromagnetinto a semiconductor now means that in the future wemay see fully hybrid magnetic–electronic devices whichuse the spin of the electrons as much as their charge toperform logic operations and processing. Nanomagnetsmay ultimately even provide a suitable environment forimplementing quantum computation [7].

A further, more fundamental motivation for studyingnanomagnets comes from the history of magnetics research.While physicists have long recognised that reducinga lengthscale is the natural way to study magnetism,experimental techniques only reached a sufficiently advanced

0022-3727/00/010001+16$30.00 © 2000 IOP Publishing Ltd R1

R P Cowburn

stage of development in the 1950s, enabling foundationalwork to be performed by Prutton and others [8]. Thearrival of molecular beam epitaxy in the laboratory duringthe 1980s allowed monolayer thickness magnetic films ofhigh structural quality to be routinely grown, leading to muchresearch during the 1980s and 1990s into two-dimensionalmagnetic systems [9]. A number of exciting new phenomenasuch as oscillatory exchange coupling [10], giant magneto-resistance [11] and perpendicular surface anisotropy [12]were discovered during this rich period, all associated in someway with the fact that the film thickness had been reduced tothe nanometre regime. The next thing to do was, naturally,to apply this principle to the other two dimensions. One-dimensional magnetism has thus been studied in magneticwires [13] and zero-dimensional magnetism, in which allthree dimensions are structured in the nanometre lengthscale,is currently one of the fastest growing fields in condensedmatter physics.

The most important property of a naturally occurringmagnetic element or alloy is its anisotropy. This refers tothe presence of preferred magnetization directions withinthe material and is, ultimately, responsible for determiningthe way in which a magnetic material behaves and thetechnological applications for which it is suitable. In aconventional magnetic material, anisotropy arises from theshape and symmetry of the electronic Fermi surface and so isintrinsic to the particular element or alloy and cannot easily betailored. In nanomagnets, however, the anisotropy dependsnot only on the band structure of the parent material, but alsoon theshapeof the nanomagnet. One of the most attractivefeatures of nanostructured magnetic materials, therefore, isthat their magnetic properties can be engineered by the choiceof the shape of the constituent nanomagnets.

The most widely studied shape of nanomagnet to datehas been rectangular, due largely to the applicability of suchstructures to MRAM and spin-valve magnetic field sensors[14]. A number of experimental [15] and theoretical [16]studies have been conducted. Several other studies haveinvestigated circular magnetic elements with either in-planeor out-of-plane magnetization [17], this time often with aneye to patterned media for ultra-high density hard disk datastorage. A smaller number of workers have looked at squarenanoelements [18]. None of these studies, however, hassystematically compared the influence of different shapes.In this paper we review an extensive comparative studywhich we have performed into the influence of shape andsize on the magnetic properties of nanostructures. We havemade small arrays of magnetic nanostructures on a siliconsubstrate using high-resolution electron beam lithography.The nanomagnets were in the size range 35–500 nm and in thethickness range 3–15 nm and had elliptical, triangular, square,pentagonal and circular geometries, which respectivelycorrespond to rotational symmetries of order 2, 3, 4, 5 and∞. The parent material was Supermalloy (Ni80Fe14Mo5X1

where X is other metals), which we chose because inbulk, it is almost isotropic and so any anisotropy in thenanomagnets must come from their shape. A high-sensitivitymagneto-optical method was then used to probe the magneticproperties of these nanomagnets. We show that the geometricshape of the nanomagnets plays a key role in determining their

Figure 1. SEM images of some of the nanomagnets fabricatedduring this study. The sizes are 500 nm (left-hand column) and<100 nm (right-hand column).

magnetic properties in two distinct ways. The first is througha new phenomenon called configurational anisotropy, inwhich very small deviations of the magnetization fromuniformity can couple to the shape of the particle. The secondway comes as a result of the shape of the particle stabilizingor destabilizing the single domain state.

This paper is structured as follows. Section 2 describesthe experimental and theoretical methods which we usedduring the course of our study. Section 3 then describes theproperties which we have observed in elliptical nanomagnets(section 3.1), triangular, square and pentagonal nanomagnets(section 3.2), isolated circular nanomagnets (section 3.3)and finally interacting circular nanomagnets arranged on arectangular lattice (section 3.4). Important common themesare discussed in section 4, followed by a conclusion insection 5.

2. Experimental and theoretical methods

2.1. Nanomagnet fabrication and characterization

All of the samples were made using high-resolution electronbeam lithography with a standard lift-off pattern transfer

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Property variation with shape in magnetic nanoelements

process [19]. Two layers of polymethylmethacrylate(PMMA), one of molecular weight 495 000 and one of weight950 000, were spun onto a single crystal silicon substrate.Arrays of different geometric shapes were then exposed ontothe sample in a Jeol 4000EX SEM/TEM operating at 100 kV,followed by 30 s of development in a 1:3 solution of methyliso-butyl ketone/iso-propyl alcohol. The array size wasbetween (5µm)2 and (10µm)2; the spacing between eachnanomagnet was usually at least equal to the diameter of thenanomagnet, and for the smallest structures was usually aslarge as three times the diameter. This ensured that there wasnegligible magnetostatic interaction between nanomagnets.A 3–15 nm thick layer of Ni80Fe14Mo5 (‘Supermalloy’)followed, in most cases by a 5 nm thick anti-oxidationcapping layer of gold were then deposited at a rate of0.08 nm s−1 by electron beam evaporation in an ultra-highvacuum chamber with base pressure 4× 10−9 mbar. Anunpatterned substrate was also present in the chamber to allowstructural and magnetic characterization of the unpatternedmagnetic film. Ultra-sonic assisted lift-off in acetone wasused to remove the magnetic film from the unexposed partsof the patterned sample.

Transmission electron microscopy (TEM) and crosssectional TEM showed the deposited Supermalloy to havea random polycrystalline microstructure with grains of size∼10 nm and a surface roughness of less than 0.5 nm.Scanning electron microscopy (SEM) was used to check thesize and shape of the nanomagnets. Figure 1 shows some ofthese SEM images.

In addition to this structural characterization, we alsoperformed magnetic characterization. Magneto-opticalmagnetometry was used to measure the coercivity (∼1 Oe)and the anisotropy field (4± 1 Oe, uniaxial in-plane)of the unpatterned film. A B–H looper was used tocheck its thickness and saturation magnetization (800±60 emu cm−3). Temperature dependent measurementsshowed the unpatterned films were still ferromagnetic at300◦C, which is not inconsistent with the expected Curietemperature of 400◦C and hence an exchange stiffness of∼10−6 erg cm−1 [20].

2.2. Nanomagnetometry

We have probed the room-temperature magnetic propertiesof the arrays of nanomagnets using a high-sensitivitymagneto-optical magnetometer [21], shown schematically infigure 2. The sample surface can be viewed under an opticalmicroscope, while a laser spot (diameter 5µm) is moved overthe surface until it is focused on top of one of the arrays ofnanomagnets. The reflected laser beam is then polarizationanalysed in order to access the longitudinal Kerr effect [22],which serves as a probe of the component of magnetizationlying in the optical plane of incidence. An electromagnetallows magnetic fields of up to 1000 Oe to be applied in theplane of the sample. It is necessary to measurearrays ofnanomagnets in order to obtain a sufficiently large signal.The high definition of the lithography means, however, thatall of the particles in the array are virtually identical to eachother and so the measured average properties for the arraycan also be interpreted as the individual properties of a singlenanomagnet.

Two different magnetic measurements were made usingthe magnetometer. The first involved sweeping a field appliedin the plane of the sample at 27 Hz and measuring themagneto-optical response. This allowed a hysteresis loop(normalized in amplitude to±1) of the nanomagnets to berecorded. The second measurement uses a newly developedtechnique called modulated field magneto-optical anisometry(MFMA) [23, 24] to probe the energy surface experiencedby the magnetization as a function of the in-plane direction.In this technique, a large and static magnetic field,H , isapplied in the sample planeperpendicularlyto the directionof magneto-optical sensitivity. A small transverse oscillatingfield, Ht , is then applied in the direction of the magneto-optical sensitivity, in order to cause the magnetization tooscillate aboutH . The measured response (referred to asthe transverse susceptibility, once normalized toHt ) can bewritten as

∂φ

∂Ht≡ χt =

(E′′(φ)Ms

+H

)−1

(1)

whereφ is the mean magnetization direction,E′′(φ) is thesecond derivative with respect toφ of the energy densitysurface andMs is the saturation magnetization.E′′(φ)/Ms

is shown in the appendix to have the same magnitude andsymmetry as the anisotropy field. Put simply, the reciprocalof the measured magneto-optical response, 1/χt , is thetotal internal field in the direction of the magnetization, i.e.the externally applied fieldH plus the effective anisotropyfield coming from the internal energy surface. A powerfulprobe of the energy surface of the nanomagnet can thus beobtained simply by measuring 1/χt as a function of directionandH .

2.3. Micromagnetic modelling

We have used the semi-classical formalism of micromagnet-ics [3] in order to provide a theoretical framework in whichto interpret the experimental data. Micromagnetics is a con-tinuum theory for which a numerical finite-element methodis used to find a magnetization vector field which minimizesa Hamiltonian incorporating three energy terms: magneto-static energy, exchange energy and Zeeman energy. Magne-tostatic energy arises from the stray magnetic fields whichemanate from any divergence in the magnetization. Mag-netization divergence can arise inside a nanomagnet wherethe magnetization field changes direction (‘volume charges’)or at the interfaces of the structure where the magnetizationis not parallel to the interface (‘surface charges’). The highenergetic cost of the magnetostatic surface charges meansthat magnetostatic energy can often be reduced by introduc-ing non-uniformity into the magnetization field. In the lessercases, this may be a little bending of the magnetization closeto the edges of a structure. In more dramatic cases, this mayresult in the formation of a superstructure within the magne-tization field, such as a vortex. These so-called incoherentmagnetization fields can reduce the net moment carried bythe nanomagnet to zero and so are technologically very im-portant. The calculation of magnetostatic energy is entirelyclassical. Exchange energy is, in contrast, a quantum me-chanical phenomenon and is ultimately responsible for ferro-magnetism. A positive exchange energy term arises wherever

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R P Cowburn

Laser

N.D

polarizer B.S

lamp+diffuser

sample andelectromagnet

λ/4analyzer

B.S

CCD

photo-diode

laser5µm square

500nmarray

400nmarray

Figure 2. A schematic of the magneto-optical magnetometer used for probing the magnetic properties of nanostructures. Inset is an imagerecorded by the charge-coupled device camera showing the focused laser spot, an isolated 5µm square magnet and two 5µm× 5µm arraysof sub-micrometre nanomagnets. B.S denotes the beam splitter and N.D denotes the neutral density filter.

there is a gradient in the magnetization field within a nanos-tructure. Consequently, exchange energy drives nanostruc-tures away from incoherent magnetization towards uniformmagnetization and, is in this sense, in competition with themagnetostatic energy term. The balance point in the compe-tition is ultimately determined by the size, shape and thick-ness of the nanostructure: large structures are dominated bymagnetostatics and small structures are dominated by ex-change. The final energy term is Zeeman energy which issimply the energy of the magnetization in an externally ap-plied magnetic field. Zeeman energy is always minimizedwhen the magnetization is aligned with the applied field. Amagnetocrystalline anisotropy term would usually also beincluded in such calculations. We have ignored it for thecalculations shown in this paper because it is one to twoorders of magnitude smaller in Supermalloy than the otherterms.

Figure 3 shows examples of equilibrium magnetizationvector fields calculated for square nanostructures. Figure 3(a)shows the vector field (also called a configuration) whichoccurs when one magnetizes the structure parallel to one ofits edges. This configuration is usually poetically namedthe ‘flower’ because of the way it flares out at the top andbottom, like the centre of a daffodil. Figure 3(b) shows theconfiguration associated with a square magnetized along itsdiagonal. We name this configuration the ‘leaf’ becauseof the way that it bows out in the centre and then nipstogether at the ends, like a plant leaf. We shall discuss thephysics of these and other configurations in more detail insection 3.2.

The numerical calculation of an equilibrium magnetiza-tion field can be repeated for each value of a varying appliedfield in order to simulate a hysteresis loop, or as a function ofnanomagnet orientation in order to predict anisotropy values.Full details of the numerical algorithm used to implement thisformalism can be found in [25].

(a)

(b)

Figure 3. A plan view of the calculated magnetization vectorfields in a square nanomagnet of edge length 100 nm and thickness20 nm when magnetized (a) parallel to an edge and (b) parallel to adiagonal.

3. Results

3.1. Elliptical nanomagnets

3.1.1. Experimental results. Figure 4 shows the hysteresisloops for ellipses of major axis 250 nm, minor axis 125 nmand thickness 10 nm obtained by sweeping an in-planeapplied field and recording the magneto-optical signal. Onesees clearly that the loop which was measured with theapplied field directed along themajor axis (±4◦) of thenanostructures displays high remanence and high coercivity,as is characteristic of easy-axis behaviour, whereas that takenalong theminor axis (±4◦) of the nanostructures displaysnear-zero remanence and coercivity, as is characteristic ofuniaxial hard-axis behaviour. Supermalloy itself is nearlymagnetically isotropic, and so this large difference between

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Property variation with shape in magnetic nanoelements

-1

0

1

-600 -400 -200 0 200 400 600

Field (Oe)

H

H

Figure 4. Hysteresis loops measured in two in-plane directions ofelliptical nanomagnets of major axis 250 nm and thickness 10 nm.

the major and minor axes must come from magnetostaticshape anisotropy. The hard-axis saturation field (310±15 Oefor the loop in figure 4) is, in such cases, always a goodmeasure of the effective anisotropy field, enabling us todetermine the effective uniaxial shape anisotropy field as afunction of lateral size and thickness for the ellipses. Theresults are shown in figure 5 for 1:2 aspect ratio ellipses. Forcompleteness we have also plotted in figure 5 the coercivefield as measured from the easy-axis loops. One immediatelysees that the uniaxial shape anisotropy and coercivity falloff with increasing ellipse size and increase with increasingthickness. This is usual magnetostatic behaviour, governedby the fact that the demagnetizing field is, to first order,independent of the absolute size of a magnetic element, butdepends only on the ratio of the thickness to the lateralsize.

3.1.2. Theoretical results. In order to modelmathematically the expected shape anisotropy we have usedthe micromagnetic simulation code described in section 2.This allows the magnetic energy of a nanomagnet to becalculated firstly when magnetized along its major axis andthen along its minor axis. The uniaxial shape anisotropyfield is then given by 21U/m where1U is the difference inthese two energies andm is the magnetic moment carried bya nanomagnet. Note that this method takes full account ofthe deviations from uniform magnetization which occur in anon-ellipsoidal body, such as a planar ellipse. We have usedup to 21 870 cubic finite-element cells, each of length 2.5 nmand values of 800 emu cm−3 for the saturation magnetizationand 1.05× 10−6 erg cm−1 for the exchange stiffness.

We have plotted in figure 6, using a full curve, the resultsfrom the micromagnetic calculations for nanostructures ofthickness 5 nm alongside the experimental data for thisthickness taken from figure 5. One sees that the agreementbetween the theoretical line and the experimental data ispoor: the measured anisotropy fields are in general a factorof two smaller than the theoretical values. This is surprisinggiven that shape anisotropy is a long established concept.In performing the micromagnetic modelling, however, wehave assumed the nanomagnets to be perfect ellipses. Itis well established that PMMA lithographic processes havea maximum spatial resolution of 5–10 nm [26], due partlyto the size and radius of gyration of the PMMA molecules

0100200300400500600700

H

(a)

0100200300400500600700

0 100 200 300 400 500 600Major axis (nm)

H

(b)

Figure 5. The experimentally measured (a) saturation field in thehard direction and (b) coercive field in the easy direction ofelliptical nanomagnets of different size. The nanomagnets were ofthickness 5 nm (◦), 10 nm (•) and 15 nm ( ). The curves are toguide the eye.

0

200

400

600

800

1000

0 100 200 300 400

Major axis (nm)

Figure 6. A comparison between experimentally (points) andtheoretically (curves) determined shape anisotropy fields forelliptical nanomagnets of thickness 5 nm. The three theoreticallines are calculated assuming an edge roughness of 0 nm (——),5 nm (· · · · · ·) and 10 nm (- - - -). The insets are ‘silhouettes’ oftwo of the modelled ellipses using the same meshing density asused in the calculations, with an edge roughness of 0 nm (upper)and 10 nm (lower).

[27]. Furthermore, the intrinsic grain size of the depositedSupermalloy (∼10 nm) may also limit the lithographic edgedefinition. The edge definition of the experimental ellipsescan thus be no better than approximately 10 nm. We havetherefore repeated the modelling, only this time assuminga sinusoidal edge profile of peak-to-peak amplitudeξ (nm)

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Figure 7. Hysteresis loops measured from differentnanostructures. For comparison (a) shows the loop from theunstructured Supermalloy. They-axis of each graph ismagnetization, normalized to the saturation value. The appliedfield in the schematic pictures of the nanostructures is assumed topoint up the page. The nanostructures had thicknesses of (a) 6 nm,(b) 5 nm, (c) 5 nm and (d) 3 nm.

and wavelength 2ξ (nm). Figure 6 shows the result of thiscalculation (assuming again a thickness of 5 nm) for twodifferent values ofξ . The insets in figure 6 are ‘silhouettes’of two of the modelled ellipses using the same meshingdensity as used in the calculations. The agreement betweenexperiment is now much improved. The edge roughnessclearly plays a vital role; the effective shape anisotropyof the ellipses, even as large as 500 nm, can be reducedby a factor of just under two simply by introducing a fewnanometres of roughness to the lateral interfaces. Figure 6shows the best agreement between the experiment andtheory for an edge roughness of between 5 and 10 nm,which is entirely consistent with the PMMA grain size.This is also consistent with recent numerical results fromGadboiset al [28] who found a significant reduction in theswitching fields of elongated magnetic memory elementsonce a small edge roughness was introduced into thecalculations.

Figure 8. Experimentally measured coercivities as a function ofnanomagnet size for different thicknesses (3 nm (◦); 5 nm (•)and 7.5 nm (ut)) and nanomagnet geometries: (a) triangle,(b) square and (c) pentagon.

3.2. Triangular, square and pentagonal nanomagnets

Figure 7 shows a selection of some of the hysteresisloops measured from nanomagnets of triangular, square andpentagonal shape [29]. One immediately sees that the loopsare very different from each other and from that obtainedfrom the conventional unstructured material. The propertiescovered range from those usually associated with moderately‘hard’ magnetic materials with switching fields of hundredsof oersteds (Figure 7(b)) down to those associated with very‘soft’ magnetic materials with a high relative permeabilitysuitable for field sensing (e.g. the pentagons of figure 7(d)with an effective relative permeability of 3000 and zerohysteresis). These changes in properties are a direct result ofvarying the thickness and, most importantly, the symmetryof the nanomagnets in the arrays.

In order to quantify the effects of shape, size andthickness more precisely, we have measured the three keymagnetic parameters, coercivity, susceptibility (4πMs timesthe zero-field gradient of the normalized hysteresis loop), andhysteresis (4πMs times the area of the normalized hysteresisloop), from the loops as a function of size, thickness andsymmetry order of the nanomagnets. The results are shown

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Property variation with shape in magnetic nanoelements

Figure 9. Experimentally measured susceptibilities as a functionof nanomagnet size for different thicknesses (3 nm (◦); 5 nm (•)and 7.5 nm (ut)) and nanomagnet geometries, see figure 8.

in figures 8–10, where, in order to be able to compare differentgeometries, we express the size of the nanomagnets by thesquare root of their area.

Despite the large volume of experimental data presentedacross these three figures, one can see certain commonfeatures. The coercivity (figure 8) and hysteresis (figure 10)data both show a rise or plateau as the nanomagnet size isreduced, followed by a sharp fall to zero. As the thicknessis increased, both coercivity and hysteresis increase and thefall to zero coercivity and hysteresis occurs at smaller sizes.The squares show much stronger peaks in these two data setsthan the triangles and pentagons.

The susceptibility data (figure 9) tracks the coercivityand hysteresis data, but in the opposite sense. One seesthat as the size decreases, susceptibility at first remainsconstant or falls slightly, but then rises sharply. The thinnernanomagnets then show an additional fall in susceptibility asthe size is further decreased. As the thickness is increased,susceptibility is reduced and the susceptibility peak occurs atsmaller sizes.

Coercivity, hysteresis and susceptibility are all deter-mined by anisotropy. The fact that one sees such strong

Figure 10. Experimentally measured hysteresis as a function ofnanomagnet size for different thicknesses (3 nm (◦); 5 nm (•)and 7.5 nm (ut)) and nanomagnet geometries, see figure 8.

variations in these three parameters as the size and thicknessvary, and hence a strong change in the shape of the hys-teresis loops of figure 7, suggests that the nanomagnets pos-sess a size-dependent anisotropy. We have therefore used themagneto-optical magnetometer in MFMA mode to measuredirectly the magnitude and symmetry of any anisotropy in thenanomagnets of thickness of 5 nm, and we present the resultsin figure 11. In these polar plots, the angle gives the in-planedirectionφ within the nanomagnet, the radius gives the ra-dius of the nanomagnet in that direction and the colour givesthe experimentally measured quantity∂Ht/∂φ (and hencethe anisotropy field—see equation (1)) for a nanomagnet ofthat size. Figure 11 shows experimental data from 22 dif-ferent arrays of nanomagnets (eight sizes of triangles, eightsizes of squares and six sizes of pentagons), each measured ineither 19 or 37 different directionsφ (0–180◦ in 10◦ steps fortriangles and squares and 0–180◦ in 5◦ steps for pentagons)making a total of 526 experimental measurements.

It is readily apparent from figure 11 that there are indeedstrong anisotropy fields present in all of the nanomagnetsstudied. The triangular nanomagnets exhibit anisotropy with6-fold symmetry, the square nanomagnets show a 4-fold

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(a)

(b)

(c)

500 Oe

400 Oe

300 Oe

200 Oe

Figure 11. The experimentally measured anisotropy field inside5 nm thick nanomagnets of (a) triangular, (b) square and(c) pentagonal symmetry. These are colour polar plots where thedirection gives the in-plane direction in the nanomagnet, the radiusgives the nanomagnet radius in that direction and the colour givesthe experimentally measured anisotropy field of a nanomagnet ofthat size. High field values correspond to easy anisotropy axes andlow values to hard anisotropy axes. The lateral scale of the figureis such that the square data runs from edge length 50 nm to500 nm. Data were recorded across a 180◦ range and then plottedtwice to fill 360◦.

symmetric anisotropy and the pentagonal nanomagnetspossess a remarkable 10-fold anisotropy. Frequencydoubling occurs in the triangular and pentagonal structuresbecause energy is always quadratic in the magnetization andso odd symmetry orders cannot be supported.

We applied a Fourier analysis to the plots of figure 11in order to obtain the magnitude of the anisotropy fields asa function of the nanomagnet size and symmetry, and showthe results in two different forms in figure 12. The appendixshows that the anisotropy energyU and the anisotropy fieldHa of any system are related by

U = 2MsVHa

n2(2)

where n is the symmetry order of the anisotropy andVis the volume of the particle (a single nanomagnet in this

0

50

100

150

0

50

100

150

0 100 200 300 400 500√ Area (nm)

(a)

(b)

Figure 12. The strength of the dominant anisotropy term shownin figure 11 expressed (a) as an anisotropy field and (b) as ananisotropy energy per nanomagnet for nanomagnets of differentsize and symmetry (triangles (4), squares (ut) and pentagons (�)).

Table 1. Experimentally determined values for the parametersCandl0 of equation (3).

C l0 Cn2

Shape n (nm−1) (nm) (nm−1)

Square 4 0.33± 0.03 59± 6 5.3± 0.5Triangle 6 0.16± 0.02 55± 3 5.9± 0.7Pentagon 10 0.05± 0.005 79± 20 5.0± 0.5

case). In figure 12(a) we plot the anisotropyfieldsdirectly, asreturned by the MFMA experiment, whereas in figure 12(b)we have plotted the anisotropyenergyof a single nanomagnet(in units ofkT wherek is the Boltzmann constant andT is298 K) using equation (2). Importantly, one sees in this figurethat whereas the anisotropyfieldsshow an initial rise withincreasing lateral size, which then either falls again (squaresand triangles) or forms a plateau (pentagons), the anisotropyenergycan be described approximately by the straight linerelationship

U

kT= C(√Area − l0) whenU > 0 (3)

where the experimentally determined parametersC andl0 are given in table 1. The table also shows that thequantityCn2 is found to be approximately constant across allnanostructures (at one thickness). This thus gives us a usefulphenomenological tool for rapidly assessing how magneticproperties are influenced by size and symmetry.

The anisotropy energy is particularly interesting becauseof a phenomenon called superparamagnetism [30], whichis the process by which anisotropy energy barriers can beovercome by thekT thermal energy fluctuations in nanometre

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Property variation with shape in magnetic nanoelements

scale magnets. One would thus expect the coercivityand hysteresis to fall to zero once the anisotropy energybecomes comparable to a fewkT (the precise prefactordepends upon the symmetry order of the anisotropy andthe timescale over which the experimental measurementsare made). The phase transition from ferromagnetism tosuperparamagnetism should also be accompanied by a peakin the susceptibility at the critical point of the phase transition.Conversely, once the anisotropy energy is larger than afew kT , the coercivity and hysteresis should approximatelyfollow the anisotropy field and the susceptibility shouldbe low. This is all consistent with the observations offigures 8–10 for 5 nm thick structures. The anisotropy field ofthe squares shows a peak in figure 12(a) as the element sizeis reduced and this peak is reflected directly in the squarecoercivity data (figure 8) and the hysteresis data (figure 10).The pentagon anisotropy field shows no peak, and this isalso reflected directly in the coercivity and hysteresis data.Finally, the triangle anisotropy field does show a peak justlike the square, but because the anisotropyenergiesarelower in the triangle (due to the high value ofn in equation(2)), thermal activation sets in at a larger size and preventsthe peak from being seen in the coercivity and hysteresisdata. The anisotropy energies (figure 12(b)) fall below afew kT at around 80 nm and this is approximately whereone sees a peak, or one side of a peak, in the 5 nm thicksusceptibility data (figure 9). It is interesting to note that thesusceptibility peak occurs at much larger lateral sizes andhas a much greater amplitude in the 3 nm thick structures.This is because the configurational anisotropy field is reducedbecause of the dependence of the demagnetizing field onthe ratio of the thickness to the lateral size. This reducedconfigurational anisotropy field then becomes an even morereduced anisotropy energy because of theV term in equation(2). Superparamagnetism in very thin structures thereforebegins at much larger lateral sizes. The high magneticsoftness which results could have important applications inmagnetic sensors.

An important question remains: why should thegeometric shapes studied in this section exhibit anyanisotropy at all? The demagnetizing field of any structure isdescribed by a second rank Cartesian tensor and so can onlyexhibit uniaxial (2-fold) symmetry. There is thereforenoshape anisotropypresent, at least in the conventional sense,in the plane of these non-elongated higher-order symmetrystructures. The answer comes from a phenomenon calledconfigurational anisotropy. First proposed by Schabes andBertram [31] during a theoretical study of magnetic cubes,we recently predicted [32] and observed experimentally[23] its key role in planar magnetic nanostructures, such asthose described in this paper. The demagnetizing field isonly uniform in an ellipsoidal body and so must be non-uniform in any finite planar structure. This can be seenin the micromagnetic simulations of planar squares shownin figure 3. The important point to realise is that the twoconfigurations shown in figure 3have different energies. Thisis entirely due to the deviations from uniform magnetizationwhich they exhibit. Consequently, although the energy ofa perfectly uniformly magnetized square is independent ofthe in-plane magnetization direction, as soon as one takes

Figure 13. Hysteresis loops measured from circular nanomagnetsof diameter (d) and thickness (t): (a)d = 300 nm,t = 10 nm and(b) d = 100 nm,t = 10 nm. The schematic annotation shows themagnetization within a circular nanomagnet, assuming a fieldoriented up the page.

into account the non-uniformity of the magnetization whichmust accompany a given magnetization direction, energydifferences do arise. An anisotropy thus appears with asymmetry which is related to that of the geometric shape ofthe nanostructure. This is called configurational anisotropybecause it comes from the differences in energy of thedifferent configurations (flower and leaf in the case of asquare) which arise as the magnetization direction is varied.

3.3. Circular nanomagnets

Circular nanomagnets are potentially very attractive formany technological applications. Their circular form meansthat they lack both shape anisotropy and configurationalanisotropy. If they are made from an intrinsically isotropicmaterial, it should therefore be possible to change theirmagnetization direction by even very weak applied magneticfields. They could thus form the heart of an extremelysensitive magnetic field sensor. More fundamentally, one ofthe most important theorems in nanomagnetism is Brown’sfundamental theorem [33] which states that, because ofa competition between magnetostatic energy and quantummechanical exchange energy, magnetic domain formationshould be entirely suppressed in very small (∼10−8 m)magnetic particles, causing nanomagnets to behave as singlegiant spins. Experimental data on the bounds of validity ofBrown’s theorem in well controlled systems are currently ingreat demand.

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R P Cowburn

0 5 10 15 200

100

200

300

400

500

Thickness (nm)

Figure 14. An experimentally determined phase diagram: vortex(◦) and single domain (•). The solid line shows a lower bound tothe theoretical phase boundary between the vortex state (above theboundary) and the single domain state (below the boundary).

3.3.1. Experimental results. We have measured hysteresisloops from circular nanomagnets as a function of diameter(50–500 nm) and thickness (6–15 nm) [34]. We find thatthe hysteresis loops thus obtained fall into one of twoclasses. Figure 13 shows a representative hysteresis loopfrom both of these classes with schematic annotation. Thefirst class, which we call the ‘vortex phase’ is typified by the300 nm/10 nm loop (figure 13(a)) as follows. As the appliedfield is reduced from minus saturation, the nanomagnetsretain full moment, until a critical field slightly belowzero at which point nearly all magnetization is lost. Themagnetization then progressively reappears as the field isincreased from zero, until positive saturation is achieved.The sudden loss of magnetization close to zero field is verycharacteristic of the formation of a flux closing configuration;the simplest of these is a vortex in which the magnetizationvector remains parallel to the nearest edge at all pointsin the circular nanomagnet. In large structures, this statelowers the system energy by reducing stray fields and hencelowering magnetostatic energy. Increasing the field thendeforms the vortex by pushing its core away from the centreof the nanomagnet, until it becomes unstable and the vortexis eventually annihilated [35], although not until a field ofseveral hundred oersted has been reached. This vortex phaseleads to magnetic properties which are dramatically differentfrom those which would occur if the magnetization simplyrotated under the action of a weak field.

The second class of loop, which we call the singledomain phase, is typified by the 100 nm/10 nm loop(figure 13(b)). These loops retain a high remanence (∼ 80%)and switch at a very low field (∼ 5 Oe). This is characteristicof single domain behaviour: all of the nanomagnets withinthe array retain all of their magnetization to form an arrayof giant spins, and magnetization reversal occurs by eachgiant spin rotating coherently [36]. The absence of shapeanisotropy (except that due to any ellipticity in the nominallycircular shape) and configurational anisotropy then meansthat the only anisotropy opposing the coherent rotation is the

0 100 200 300 400 5000

1

Field (Oe)

P Q

P

Q

R

R

Nor

mal

ised

mag

netis

atio

n

Figure 15. A theoretically determined hysteresis half loop for acircular nanomagnet of diameter 300 nm and thickness 10 nm (andso should be compared with figure 13(a)). The broken curve partsof the loop indicate a metastable state. The calculatedmagnetization vector fields are shown for three points on the loopP, Q and R, assuming a field oriented up the page.

weak in-plane anisotropy intrinsic to the Permalloy family.Hysteresis loops of this class thus have a saturation field ofmerely a few oersted. We found that the remanence vanishedif the field was applied parallel to the uniaxial hard axisinstead of the easy axis, as would be expected for such areversal mechanism.

We have classified all of our experimental data fromcircular nanomagnets in terms of vortex or single domainbehaviour and have plotted the result in figure 14.

3.3.2. Theoretical results. In order to verify that theclass of behaviour typified by figure 13(a) is indeed due tovortex formation we performed micromagnetic calculationsto simulate one half of the loop, in which 5656 cubic finite-element cells, each of length 5 nm, were used.

Figure 15 shows the half-loop which we have calculatedfor a circular nanomagnet of diameter 300 nm and thickness10 nm (i.e. the same size as that measured in figure 13(a)).The precise mechanism by which a vortex is first nucleatedis highly complex and is beyond the scope of this study. Wetherefore assume that a vortex is present under zero field.

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Property variation with shape in magnetic nanoelements

Figure 16. SEM images of rectangular lattices of circularnanomagnets. Each nanomagnet is 60 nm in diameter and has ay-direction lattice parameter of 180 nm. Thex-direction latticeparameter is (a) 180 nm, (b) 110 nm and (c) 90 nm.

One sees that as the applied field is increased from that point,the calculated loop traces out an almost identical path to thatobtained experimentally in figure 13(a). Both show a smalldegree of curvature followed by an abrupt annihilation eventat a field of several hundred oersted. Figure 15 also shows thecalculated magnetization vector fields within a single circularnanomagnet at three different points on the hysteresis loop.These agree well with the schematic representations shownin figure 13(a).

We have marked parts of the theoretical half-loop offigure 15 by a broken curve. These correspond to metastableregions, i.e. those for which the magnetization configurationleads to a local minimum in the free energy, but not a global

-1

0

1

10 0 1010 0 10

X=180nm X=180nm

-1

0

1

10 0 1010 0 10

X=120nm X=120nm

-1

0

1

10 0 1010 0 10

X=100nm X=100nm

-1

0

1

10 0 10 0

X=90nm X=90nm

-150 0 150-1

0

1

-150 0 150-1

0

1

-150 0 150-1

0

1

Field Field

X X

(a)

(b)

(c)

(d)

-150 0 150-1

0

-10 10 -10 100 0

Field (Oe) Field (Oe)

Figure 17. Experimentally measured hysteresis loops fordifferent lattice spacings and applied field directions. All loopswere measured within the field range±150 Oe, the large panelsshow high-magnification views around zero field; the insets showthe full measured loop. The vertical axis of all loops ismagnetization normalized by the saturation value.

minimum. Whether a thermally activated transition fromthe metastable state to the thermodynamic ground state isto occur depends upon the temperature and the timescaleover which the hysteresis loop is swept out. The theoreticalsimulation of figure 15 does not allow for thermally activatedtransitions, which accounts for the difference between theexperimental and theoretical vortex annihilation fields of283 Oe and 423 Oe respectively.

As the lateral size and thickness of the nanomagnets isdecreased, our calculations show that the range of appliedfields for which the vortex state is metastable increases untilthe vortex can never nucleate and so the reversal mechanismmust be replaced by the Stoner–Wohlfarth coherent rotationshown in figure 13(b). We have marked onto the experimentalphase diagram of figure 14 the calculated phase boundarybelow which vortex nucleation is impossible. One sees thatthe experimental data agree very well with the theoreticalline: no vortex nucleation was observed below it. We stressthat the theoretical line isnot a prediction for the transition

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R P Cowburn

from vortex to single domain behaviour, but is rather a lowerlimit to that boundary. Experiment and theory are thus inagreement: in order to observe the useful single domain state,circular nanomagnets must be either very thin or laterallysmall.

3.4. Interacting nanomagnets

So far in this section we have discussed the influence of theshape of nanomagnets on their magnetic properties. All of theexperiments performed in our study and many of the potentialtechnological applications of nanomagnets do not use a singlenanomagnet, but rather a large ensemble of magnets arrangedon some lattice. If the lattice spacing is sufficiently small(expressly not the case in the work described so far), themagnetic field emanating from one nanomagnet can influenceits neighbours. In this case, one must also consider thegeometry of thelatticeas well as the shape of the motif.

In order to demonstrate and investigate the phenomenonof magnetostatic interactions, we have made a number ofarrays of 60 nm diameter circular nanomagnets arranged ona rectangular lattice [37]. They-direction lattice period waskept constant at 180 nm (i.e. three times the diameter of thenanomagnets) whereas thex-direction lattice period varied indifferent arrays from 180 nm down to as small as 80 nm (i.e.leaving only 20 nm between neighbouring edges). Figure 16shows SEM images of some of the lattices.

Figure 17 shows hysteresis loops obtained from thedifferent lattices for the cases of the field applied alongthe latticex- and y-directions. One sees that when thenanomagnets are widely separated (e.g. figure 17(a)) thehysteresis loops have a characteristic ‘S’ shape, are fullyclosed (i.e. zero area inside the loop) and are virtuallyidentical in the two measurement directions. As thex-axisspacing is reduced, however, the loops show a significantchange in their central region, thex-axis loop openingup while the y-axis loop becomes more sheared (e.g.figures 17(c) and 17(d)).

Each nanomagnet, being 60 nm in diameter and 7 nmthick, is small enough to be in the single domain state(cf figure 14) and can therefore be represented, to a goodapproximation, as a point magnetic dipole located at thenanomagnet centre. The magnetic field emanating from sucha magnetic dipole falls off with the cube of the distance fromit. The largest lattice period (X = 180 nm—see figures 16(a)and 17(a)) causes the nanomagnets to be spaced by threetimes their own diameter, which is a sufficiently large distancefor magnetostatic interactions between nanomagnets to berelatively weak. The measured average property of the latticeis, thus, approximately the same as the individual propertyof an isolated nanomagnet, as in all of the other experimentsreported so far in this paper. In this case, the weak intrinsicuniaxial anisotropy of Supermalloy is unable to stabilizethe zero-field magnetization against thermal fluctuations (incontrast to the larger single domain particles measured infigure 13(c)), leading to a time-averaged remanence of zero,and hence the closed, superparamagnetic hysteresis loops offigure 17(a).

As the separation between the nanomagnets is nowreduced, as shown in figures 16(b) and 16(c), the

0

20

40

60

80

100

0

1000

2000

3000

4000

5000

(a)

0

20

40

60

80

100

0

500

1000

1500

2000

60 80 100 120 140 160 180 200

x-period (nm)(b)

Figure 18. Remanence (open circles) and susceptibility (fullcircles) measured as a function of lattice spacing for the fieldapplied along (a) the latticex-direction and (b) the latticey-direction.

magnetostatic coupling between nanomagnets (especially,but not exclusively, between nearest neighbours) becomesstronger, to the point at which it can overcome the thermalfluctuations. When this occurs, the spins essentially remainparallel and locked together in thex-direction even underzero applied field, leading to increased remanence in thex-direction loops of figures 17(c) and 17(d).

Magnetostatic coupling is an anisotropic coupling, withan energy minimum (easy axis) when the dipoles are alignedwith the line joining their centres (i.e. thex-direction inthis experiment) and an energy maximum (hard axis) whenaligned perpendicular to this line (i.e. they-direction).Consequently, whereas the remanence rises in the loopsmeasured in thex-direction, the loops measured in they-direction become increasingly sheared as a uniaxial harddirection appears.

We showed earlier in this paper how the shape of ananomagnet can induce configurational anisotropy whichcan overcome thermal fluctuations and thus lead to abruptchanges in the susceptibility, remanence and coercivity. Inthe case of interacting particles, the anisotropy induced by therectangular lattice can give the same effect. Figure 18 showsthe susceptibility (labelledχT this time) and remanencemeasured experimentally in the latticex- andy-directionsas a function of the latticex-direction spacing. One seesin Figure 18(a) a peak in the susceptibility atX = 100 nmcoinciding with the onset of remanence, in direct analogywith the peak in susceptibility in figure 9 coinciding with therise of coercivity (figure 8) and hysteresis (figure 10).

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Property variation with shape in magnetic nanoelements

As expected, no peak in susceptibility is observablein the y-direction of the lattice (figure 18(b)) becausethis is now the ‘hard’ magnetization direction and sonever displays remanence. The effect of magnetostaticinteractions is nevertheless still very much in evidence withthe susceptibility falling with decreasing separation as thenanomagnets become increasingly coupled. The fact thatthex- andy-direction susceptibilities do not become exactlyequal in the limit ofX = Y = 180 nm is due to the presenceof weak uniaxial anisotropy in the Supermalloy and otherexperimental artefacts.

4. Discussion

The experimental and theoretical results presented abovehave highlighted the two important ways in which ananomagnet’s shape influences its magnetic properties.The first is the extent to which the shape introducesanisotropy. This anisotropy then determines all ofthe macroscopic magnetic parameters such as coercivity,remanence, hysteresis and susceptibility. Unsurprisingly,elongated shapes such as ellipses and rectangles introducea uniaxial shape anisotropy. Shape anisotropy is aninteraction between the mean magnetization direction andthe form of the nanomagnet, i.e. it occurs even in perfectlyuniformly magnetized structures, such as ellipsoids. Shapeanisotropy is in this sense a phenomenon associatedwith the zeroth spatial order of the magnetization field.More surprisingly, non-elongated shapes which possess adefinite rotational symmetry, such as triangles, squares,pentagons, etc, also introduce an anisotropy related totheir size and symmetry, called configurational anisotropy.Configurational anisotropy is an interaction between thedeviations of the magnetization from uniformity and the formof the nanomagnet. It is therefore a phenomenon associatedwith first and higher spatial orders of the magnetization field.

The second way in which a nanomagnet’s shapeinfluences its magnetic properties is the extent to which itstabilizes a single domain state, by preventing incoherentmagnetization fields such as vortices or domain patterns.Size alone can do this, as was demonstrated in the circularnanomagnets, which were found to be single domain for sizesless than approximately 100 nm. Form can also play animportant role: all of the elliptical, triangular, square andpentagonal nanomagnets were found to be single domain,even at sizes as large as 500 nm. There are two reasons forthis. The first is associated with the sharp angles betweenedges in shapes such as triangles. These increase the energyof vortices because of the sharp turn, and hence the increasedexchange energy, which occurs if the magnetization is toremain parallel to the edges. The second reason is moreapplicable to elongated structures and concerns anisotropy.Anisotropy generally encourages the single domain state overvortex formation because, in the former case, the systemcan lower its energy by aligning all of the magnetizationwith an anisotropy easy direction. Incoherent magnetizationdistributions usually have a proportion of the magnetizationin the anisotropy hard direction. Thus, although we havepresented the influence of shape on magnetic properties asbeing comprised of two separate issues, i.e. anisotropy and

Figure 19. An experimentally measured hysteresis loop fromsquare nanomagnets of edge length 400 nm and thickness 10 nm.The schematic annotation shows the magnetization field assumingthe positive applied magnetic field to be pointing up the page.

stabilizing the single domain state, the two are actually linkedbecause anisotropy itself stabilizes the single domain state atremanence.

So far we have discussed the stability of the singledomain state at remanence. This, however, is not theonly case of interest. Incoherent magnetization structuresas a vehicle for achieving magnetization reversal are alsoimportant, i.e. a nanomagnet can be single domain underzero field, but then go via a vortex or similar state asthe magnetization changes direction under the action of anapplied field. We have not discussed the microscopic detailsof switching mechanisms in this paper except to postulatethat the very small circular nanomagnets switch by coherentrotation. The reason for our silence on this subject isthat hysteresis loops are not a good tool for investigatingthe microscopic details of magnetization reversal. Directimaging, such as Lorentz microscopy [38] or spin-polarizedSEM [39] are really required for this. Nevertheless, onecan infer a certain amount about the reversal mechanismfrom hysteresis loops. Figure 19 shows a hysteresis loopmeasured from 10 nm thick squares of edge length 400 nm.As described earlier, the loop shows almost full remanence,even though a circle of such dimensions would have mostcertainly collapsed into a vortex at remanence. There isthen the usual abrupt coercive transition as the magnetizationbegins to change direction. This jump does not, however,achieve full switching, but rather leads into a more gentleslope which eventually takes the magnet into saturation. Ourexplanation for this is that the initial abrupt transition was notto a fully reversed single domain state but to some incoherentdistribution such as the ‘U’ shaped buckle shown in theschematic annotation of figure 19. The gentle slope thenfollows the further distortion and eventual annihilation of thebuckle. Thus, even though geometric shaping had suppressedincoherent magnetizationat remanence, domains are stillinvolved in the reversal process at higher fields. Defects in thenanomagnets will certainly play some role in the nucleationand annihilation of the buckle structure, although furtherexperiments are required to establish their precise role.

R13

R P Cowburn

Exchange energy and magnetostatic energy wereintroduced in section 2 as ‘natural enemies’: magnetostaticenergy is minimized by non-uniform magnetization,exchange energy is minimized by uniform magnetization.The simplest application of this competition principle canbe seen in figure 14 where small, circular structures aredominated by exchange energy and so show the singledomain state and large circular structures are dominated bymagnetostatic energy and so show the vortex state. We can,however, go further than this and say that the exchange-magnetostatic competition is also evident in the phenomenonof configurational anisotropy. As we have already stated,configurational anisotropy is a property of non-uniformmagnetization fields. The degree of the non-uniformityand hence the strength of the configurational anisotropyare, however, dictated by the exchange-magnetostaticcompetition. This can be seen in figure 12(b) where theconfigurational anisotropy energy is seen to depend uponthe nanomagnet’s size by a straight line relationship. Thestraight line does not, however, pass through zero, but rathercuts thex-axis at around 60 nm (l0 in table 1). This is the sizeat which exchange begins to dominate the structure and sosuppresses non-uniformity in the magnetization field, leadingto a collapse in the configurational anisotropy. Thel0 offset,and hence the exchange-magnetostatic competition, can evenbe shown to be responsible for all of the peaks at around100–150 nm in the experimental data of figures 8–10 and 12.Suppose that the configurational anisotropy energy varied ina perfectly linear fashion with the square root of the area(i.e. l0 = 0 in equation (3)). Substituting equation (3) intoequation (2) would then give

Ha = CkT n2

2Mst√A

(4)

where t is the thickness of the nanomagnet, i.e. theconfigurational anisotropy field would vary with thereciprocal of the nanomagnet size, showing a divergence asthat size is reduced to zero instead of the rise and subsequentfall which one sees in figure 12. The high magneticsoftness which we have observed in small nanomagnets ofdefinite geometric shape is therefore another example of thedominance of exchange over magnetostatic energy in verysmall structures, and is in this sense the same phenomenonas that which causes the single domain phase in figure 14.

We concluded section 3 with a description ofmagnetostatically interacting circular nanomagnets. Weshowed that the shape of thelattice could be imposedon the nanomagnets’ properties through magnetostaticinteractions. In that case, we showed how uniaxial propertiesdeveloped. This is analogous to shape anisotropy, i.e. auniaxial magnetic property arising from a difference in theexperimental lengthscales of two orthogonal directions. Justas perfectly uniformly magnetized particles can only exhibitshape-induced uniaxial anisotropy, and no higher symmetryorders, in principle only a rectangular lattice can induceanisotropy through interactions: a square or hexagonallattice of interacting, uniformly magnetized particles shouldbe isotropic. Nevertheless, Mathieuet al [40] havereported experimentally afour-fold anisotropy in a sampleof interacting circular magnetic dots arranged on a square

lattice. They explained this as being due to deviationsfrom the perfect uniformly magnetized state of each particle.This is very interesting because it is the interacting analogyto configurational anisotropy: a higher-order anisotropyappears because of first and higher spatial order terms inthe magnetization field. In this case, however, the non-uniformity couples to the geometry of thelattice insteadof to the geometry of themotif as occurs in configurationalanisotropy.

5. Conclusion

We have presented a review of an extensive study whichwe have performed into the influence of shape and size onthe magnetic properties of nanostructures. We have shownthat shape plays an essential role in determining magneticproperties, on the one hand by inducing anisotropy andon the other hand by stabilizing/destabilizing the singledomain state. We have shown the key role played bysmall deviations from uniformity in the magnetization fieldwithin the nanostructures. These small deviations allowthe shape of the nanomagnets themselves and the shapeof the lattice on which they are arranged to couple intothe magnetic energy surface, causing unexpected higher-order anisotropy terms to appear which can dominate themagnetic properties. Equally important is the competitionwhich exists between the quantum mechanical exchangeenergy and the classical magnetostatic energy. This notonly determines whether a nanostructure will exhibit a singledomain state, but also controls the magnetization deviationswhich allow the coupling between (non-elongated) shape andmagnetism. Understanding these things opens the way todesigning new nanostructured magnetic materials where themagnetic properties can be tailored to a particular applicationwith a very high degree of precision.

Acknowledgments

This work would not have been possible without the excellentassistance of my colleagues, Professor M E Welland,Dr A O Adeyeye and D K Koltsov. Much of the workdescribed in this paper was funded by the UK Royal Society.The author is supported by St John’s College, Cambridge.

Appendix

A.1. Proof that the anisotropy field has the samemagnitude and symmetry asE′′(φ)/Ms

Suppose that the anisotropic energy density of a magnetcan be described by the functionE(φ) where φ is themagnetization direction. The anisotropyfieldHa is definedas the magnetic field which would need to be applied tothe magnet in the directionφ in order to give the sameenergy profile as the functionE(φ) for small deviations of themagnetization aboutφ. Let the size of any deviations of themagnetization direction fromφ be described by the variableθ . In this case, the Zeeman energy expression gives

E = constant−MHa cosθ (5)

R14

Property variation with shape in magnetic nanoelements

and hence

dE

dθ= MHaθ in the limit of smallθ. (6)

Differentiating with respect toθ once more gives

d2E

dθ2= MHa. (7)

Sinceθ is simply the deviation inφ, the identity dθ = dφcan be made and hence

Ha = 1

Ms

E′′(φ). (8)

A.2. Proof that anisotropy energy and anisotropy fieldare related byU = 2MsV Ha/n

2

A magnet with an anisotropy of rotational symmetry ordern will have a dependence of energyU on the magnetizationdirectionφ, which can be described by

U(φ) = AnKn cos2nφ

2(9)

whereAn is a scaling constant andKn is the anisotropyconstant. Differentiating this expression twice with respectto φ gives

d2U(φ)

dφ2= −n

2AnKn

2cosnφ. (10)

The energydensity Ecan then be found by dividing the energyby the volume of the magnetV to give

d2E(φ)

dφ2= −n

2AnKn

2Vcosnφ. (11)

Substituting (11) into (8) then gives

Ha(φ) = −n2AnKn

2MsVcosnφ. (12)

Comparing the magnitudes of the trigonometric functions of(12) and (9) then gives the desired relationship

U = 2MsHaV

n2. (13)

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