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Local field loop measurements by magnetic force microscopy

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2014 J. Phys. D: Appl. Phys. 47 325003

(http://iopscience.iop.org/0022-3727/47/32/325003)

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Journal of Physics D: Applied Physics

J. Phys. D: Appl. Phys. 47 (2014) 325003 (15pp) doi:10.1088/0022-3727/47/32/325003

Local field loop measurements by magneticforce microscopy

Marco Coısson1, Gabriele Barrera1,2, Federica Celegato1,Emanuele Enrico1, Alessandra Manzin1, Elena S Olivetti1, Paola Tiberto1

and Franco Vinai1

1 INRIM, Electromagnetics Division, strada delle Cacce 91, 10135 Torino (TO), Italy2 Universita di Torino, Dipartimento di Chimica, via P. Giuria 9, 10125 Torino (TO), Italy

E-mail: m.coisson@inrim.it, g.barrera@inrim.it, f.celegato@inrim.it, e.enrico@inrim.it,a.manzin@inrim.it, e.olivetti@inrim.it, p.tiberto@inrim.it and f.vinai@inrim.it

Received 1 April 2014, revised 12 June 2014Accepted for publication 25 June 2014Published 18 July 2014

AbstractMagnetic force microscopy (MFM) is a valuable technique to investigate the reversalmechanisms of the magnetization in micrometric and sub-micrometric-patterned thin filmsthat cannot be studied by means of magneto-optical methods because of their limitedresolution. However, acquiring tens or hundreds of images consecutively at different appliedmagnetic fields is often impossible or impractical. Therefore, a field-dependent MFM-derivedtechnique is discussed and applied on square and circular dots of different materials (Ni80Fe20,Co67Fe4Si14.5B14.5, Fe78Si9B13) having sizes ranging from 800 nm to 20 µm. Experimentallocal hysteresis loops are obtained by properly analysing the phase signal of the MFM along aselected profile of the studied patterned structure, as a function of the applied magnetic field.Characteristic features of the magnetization process, such as vortex nucleation and expulsion,transition from C-state to saturated state or domain wall motion in Landau-like domainconfiguration are identified, and their evolution with the applied field is followed. Thenecessity to combine experimental and theoretical analyses is addressed by micromagneticsimulations on a model system (a Ni80Fe20 square dot with a lateral size of 800 nm),comparable to one of the studied samples. The agreement between experimental and simulatedMFM maps, at different applied fields, and hysteresis loops provides the necessary validationfor the technique. Additionally, the simulations have been proven to be necessary tounderstand the magnetization reversal processes occurring in the studied sub-micrometricstructures and to associate them with characteristic features of the hysteresis loops measuredwith the proposed technique.

Keywords: thin films, nanostructures, MFM, magnetization reversal, micromagneticsimulations

(Some figures may appear in colour only in the online journal)

1. Introduction

Nanostructured magnetic thin films have been extensivelyinvestigated in the last decade due to the development ofmagnetoelectronic devices for high-density and high speedinformation storage, MRAM and magnetologic [1–3]. In thiscontext, magnetic devices consisting of periodically patternedferromagnetic materials have also found considerable interestfor perpendicular hard disk media and magnetoelectronics.

In order to reach a complete understanding of themagnetic properties of nanopatterned thin films, themagnetic characterization techniques must follow the ongoingminiaturization of magnetic nanostructures to sizes of theorder of 100 nm. Generally, the study of magnetic domainpatterns is a powerful tool to analyse the local magneticbehaviour of magnetic thin films for magnetorecording devicesand more generally for spintronics [4]. In fact, in the fieldof nanomagnetism only few techniques can supply valuable

0022-3727/14/325003+15$33.00 1 © 2014 IOP Publishing Ltd Printed in the UK

J. Phys. D: Appl. Phys. 47 (2014) 325003 Marco Coısson et al

information about the behaviour of magnetic nanoscaledmaterials. In particular, one of the major challenges is theimaging of a single magnetic micro- or nano-structure. Severaltechniques for magnetic domain imaging are well consolidatedand available, such as Kerr microscopy [5, 6] and scanningelectron microscopy with spin polarization analysis (SEMPA)[7]. Lorentz and Faraday microscopy [8, 9] is also usedalthough transmission electron microscopy techniques can beused only for very thin samples. Some drawbacks are presentin each of them. Kerr microscopy usually cannot be used forthe investigation of new generation storage media with lateralbit dimensions significantly below 1 µm due to the spatialresolution being limited by diffraction, unless advanced signalanalysis techniques are used to optimize the signal-to-noiseratio and achieve resolutions of a few hundred nanometres[10]. Electron beam-based methods, with their higher lateralresolution, are better suited for this purpose. SEMPA is asurface technique and cannot provide the analysis below thefilm surface. Additionally, it is not widely available. Avalid alternative is represented by magnetic force microscopy(MFM), a technique based and derived from scanning probemicroscopy (SPM) [11, 12].

More than three decades have passed since thedevelopment of the first probe-based microscope, givingrise to scanning tunnelling microscopy (STM) and later toatomic force microscopy (AFM), allowing one to achieve sub-micrometric resolution as well as 3D images [13]. MFM isdirectly derived by exploiting a magnetic tip with an AFMand provided a new method for mapping the magnetic fielddistributions on a microscopic scale [14]. This techniqueis based on the magnetic force gradient acting between themagnetized sample surface and the magnetized tip. MFMsoon became one of the most powerful methods to studysurface magnetic properties with high resolution and requiringno sample preparation [15]. In the field of nanomagnetism,the MFM technique turns out to be characterized by thehighest spatial resolution and allows the study of magnetizationreversal [16], crucial for magnetorecording and magnetic logicas magnetic cellular automata [17].

However, the major drawbacks are that this techniquedoes not provide any quantitative values of magnetic momentintensity unless a detailed modelling of the tip and of itsinteraction with the sample is available [18–20]. Many studiesconcerning tip–sample interaction have been performed in thelast years to determine thin film properties on periodicallypatterned samples [21] or to take into account that the tipmay distort the magnetic structure by inducing reversible andirreversible changes in the sample magnetic state [22–24].Recently, works to calculate remanent magnetization using themagnetic contrast distribution functions have been presented[25]. More recently, a MFM probe made of a Fe carbonnanotube needle behaving as a monopole has been presented[26]. This approach allows one to establish a proportionalitybetween the first derivative of the sample stray field and theMFM signal [26, 27]. MFM is extensively employed to imagemagnetic domains of single magnetic nanostructures [28] ornanoscale phase separations in magnetic materials [29]. MFMdata are often complementary to the magnetic characterization

performed using standard high-sensitivity volumetric methodssuch as SQUID or VSM magnetometers.

Hysteresis loops of nanostructures in Ni80Fe20 thin films(i.e. single domain permalloy particles and rings) have beenreconstructed by counting the percentage of switched elementsimaged at magnetization remanence [30, 31]. The evaluationof hysteresis loops related to a single dot and its magnetizationreversal as a function of applied magnetic field by MFM playsa relevant role in nanomagnetism. Recently, a method basedon the integral value of the cantilever phase shift has beenproposed to measure the field-dependent magnetization ofsingle objects [32]. However, the single object magnetizationvalue is derived by integrating the magnetic signal over thewhole dot surface therefore limiting the resolution. Anothermethod to evaluate magnetic coercivity of single domainnanostructures (Co nanowires) is reported in [33]. In this case,the variation of MFM contrast as a function of applied field isacquired in a single line in a bistable structure.

In this work, a novel method derived from the single-pointtechnique [34] to image the hysteresis loop of an individualmagnetic element having more complex magnetic domainsarrangements (i.e. magnetic vortex) is proposed and testedin different systems (square and circular dots of Ni80Fe20,Fe78Si9B13, Co67Fe4Si14.5B14.5 having sizes ranging from800 nm to 20 µm). The hysteresis loop is calculated byacquiring the MFM contrast as a function of applied fieldin a single scan line, and by a subsequent application of asuitable data analysis process. The method has been validatedand the analysis of the experimental results supported bymicromagnetic simulations confirming the reliability of thetechnique together with hysteresis loop shape.

2. Samples preparation

Square dots with a lateral size of 800 nm and 2 µm havebeen prepared by first depositing a Ni80Fe20 (permalloy) thinfilm on a Si substrate with a 300 nm thick Si-oxide layer onits surface. The magnetic film is 30 nm thick and has beendeposited by means of rf sputtering. On top of the permalloylayer, a thin layer of Al, 10 nm thick, has been added inorder to ease the removal of the resist after completion ofthe lithographic process. Subsequently, a negative electronresist (ma-N2401) has been spinned on top of the metalliclayer, and electron beam lithography (EBL) has been used topattern the dots with the desired size. After the development ofthe electron-irradiated resist, an Ar+ plasma etching has beenused to remove the excess magnetic material and transfer thepattern to the permalloy film. Finally, the remaining resisthas been removed with a ma-D331 developer, which, beingNaOH based, dissolves the Al layer and leaves the permalloysub-micrometre structures exposed for MFM characterization.The dots are arranged in a square array configuration, withthe elements several tens of micrometres apart in orderto make magnetostatic interactions among them negligible.The saturation magnetization of a continuous permalloy filmhaving the same thickness and used for reference turned outto be ≈860 kA m−1 as measured by alternating gradient fieldmagnetometry after careful determination of sample volume;

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J. Phys. D: Appl. Phys. 47 (2014) 325003 Marco Coısson et al

90 nm

Figure 1. 3D AFM image of a 800 nm Ni80Fe20 dot. The dashed linerepresents the profile that is scanned repeatedly over time. In thepresent case, the lift step (pass 2) is performed at a height of 90 nm.

coercive fields of less than 1 Oe were measured on the referencecontinuous film.

Conversely, Fe78Si9B13 square dots, having a side of20 µm and a thickness of 250 nm, have been prepared byperforming an optical lithography on a Si3N4 substrate, usinga suitable mask. Subsequently, the magnetic material has beensputtered from a target made of amorphous ribbons of the samecomposition. The desired pattern has been finally obtained bylift-off. With a similar method, Co67Fe4Si14.5B14.5 circulardots, with a thickness of 100 nm and a diameter of 6 µm, andsputtered on glass, have been prepared.

AFM/MFM has been performed with a Bruker Multimode5 Nanoscope 8 microscope, equipped with a scanner witha horizontal excursion of 125 µm and a non-magnetic head.CoCr-coated HR-MESP tips have been used, with a coercivefield of ≈900 Oe and magnetized along the tip axis.

3. Experimental technique

A 3D AFM image of a permalloy dot with a lateral size of800 nm is shown in figure 1. The selected dot is representativeof the patterned array.

The evolution of the magnetic domain configuration inthe selected dot has initially been studied by means of aconventional technique consisting of an MFM under theapplication of an in-plane magnetic field, generated by anelectromagnet, with values up to 800 Oe (i.e. below thecoercive field of the MFM tip that has been used). MFMimages of the same dot are reported in figure 2 for selectedfield values.

When the dot is saturated, as in figure 2(a), the dotedges orthogonal to the magnetic field direction are detectedby the MFM tip as bright and dark bands, depending onthe magnetization direction (e.g. north pole on the left andsouth pole on the right of figure 2(a); the magnetization isschematically represented by the white arrows on each image).In fact, even if the magnetization is in plane, the poles at the two

dot edges are responsible for a stray magnetic field that expandsin the volume outside the dot, as schematically depicted infigure 3. The vertical component of the second derivativeof the fringing field with respect to the vertical direction z ismaximum close to the dot edges (red and blue shaded regionsin figure 3), and is detected by the MFM tip [35, 36], whichis vertically magnetized. The sign of the interaction of thetip close to the dot edges is opposite on the two sides (redand blue shaded regions), giving rise to opposite colours on anMFM image.

When the field is progressively reduced, as in figure 2(b),the bright and dark bands at the dot left and right edges remainvisible, but along the top edge they progressively expand,indicating that a magnetic vortex is about to nucleate. Indeed,a vortex structure is clearly visible at low field values, whichinclude the magnetic remanence, as shown in figures 2(c),(d) and (e). When the field is increased again along theopposite direction, a symmetrical configuration with respect tofigures 2(a), (b) and (c) is obtained, as shown in panels (e), (f )and (g). The vortex is firstly expelled, and then the magneticsaturation is reached, with the bright and dark bands at theedges of the dot having opposite colour, as the magnetizationis now reversed. The magnetization behaviour can be followedalong the opposite path, with the field increasing towards theremanence and then saturation along the initial direction. Inparticular, the magnetization follows a symmetrical behaviour.

All the constant-field MFM images, of which thosereported in figure 2 are a subset, have been obtained acquiringthe phase shift in pass 2, at a lift scan height of 90 nm. Inprinciple, the sample magnetic domain configuration can beaffected by the tip–sample interaction [31, 35]. Indeed, in pass1 the tip is in intermittent contact with the sample surface,which in turn is exposed to a non-negligible magnetic fieldgenerated by the tip magnetization. However, as this field isperpendicular to the sample surface, it has to act against theshape anisotropy of the dots, which opposes to the rotation ofthe magnetization perpendicularly to the film plane. Indeed,reversible motions of the vortex core as the tip is scanned inpass 1 cannot be ruled out; however, the absence of artefactsand sudden jumps in the phase-shift signal in all constant-field MFM images like those reported in figure 2 leans againstirreversible effects induced by the tip in pass 1, and even moreso in pass 2. This subject will be further discussed in thefollowing. A possible workaround, that can be applied oncertain microscopes but that does not guarantee a constantdistance of the tip from the sample surface, consists in a three-point fit of the substrate plane sufficiently far away from thestudied structure, then in a single-pass scan at a given distancefrom that plane. This approach has not been applied here.

The detailed study of the field-dependent magnetizationprocesses in sub-micrometric and nanoscale structures,requiring the acquisition of a large number of MFM images atconstant applied magnetic field, has serious disadvantages: thetechnique is very time consuming, in addition to sample driftand tip wearing during the numerous image acquisitions canseverely impact the possibility of obtaining all the necessarydata. Therefore, a different approach is envisaged.

The MFM-based setup that we have developed iscomposed of the usual microscope arrangement, with a

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J. Phys. D: Appl. Phys. 47 (2014) 325003 Marco Coısson et al

200 nm

H

(g)H = -600 Oe

(a)H = +600 Oe

(f)H = -450 Oe(e)H = -150 Oe(d)H = 0 Oe

(c)H = +150 Oe(b)H = +450 Oe

Figure 2. MFM images of a Ni80Fe20 dot with a size of 800 nm at selected applied field values. The field is applied in plane, along thehorizontal direction. The images represent in false colours the phase channel (retrace) in pass 2. The full colour scale is the same for (a),(b), (f ) and (g), and has been enhanced in (c), (d) and (e) for visualization purposes.

Figure 3. Schematic representation of the stray magnetic field fluxlines generated by an in-plane saturated dot. The red and bluearrows represent the vertical component of the magnetic field, andtheir size pictorially indicates its amplitude. The red and blueshaded areas indicate the regions where the interaction between thevertically magnetized MFM tip and the vertical component of thegradient of the field is significant.

controller dealing with all the input/output (I/O) and a PCfor human interface and data processing. In addition to thestandard configuration, the setup includes an electromagnetthat applies a magnetic field in the sample plane. A power

supply, controlled by the PC, generates the necessary current todrive the electromagnet. In our setup, a gap of approximately9 cm is required to accommodate the MFM head. Magnetpole caps of approximately 12 cm provide a magnetic fieldthat is homogeneous, in the middle of the gap, over a regionof approximately 1 cm3. This is enough for continuous filmshaving a lateral size of a few mm or sub-micrometric ornanometric structures. Without requiring water cooling, themagnet can sustain fields up to 1200 Oe, even if they oftenturn out to be higher than the field at which the magnetizationof the tip flips along the horizontal direction. The field readingis provided by a Hall probe connected to a gaussmeter. Theanalogue output of the gaussmeter is fed to an analogue input(input 1) of the microscope controller; in this way, the magneticfield value can be acquired and saved together with the othersignals of the MFM. Additionally, the ‘end of line’ (EOL)signal, a transistor-to-transistor-logic (TTL) generated by themicroscope controller after each line scan, is provided as aninput to the PC through an impulse counter.

The technique works as follows: a profile of the sub-micrometric structure under investigation is chosen, and theslow scan axis of the microscope is disabled. In thisway, the same profile is repeated over time, and the signals

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J. Phys. D: Appl. Phys. 47 (2014) 325003 Marco Coısson et al

heightpass 1, retrace

phasepass 2, retrace pass 2, retrace

dot width(800 nm)

(a) (b) (c)

0 750H (Oe)

scan line #

(d)

Figure 4. Ni80Fe20 dot with a size of 800 nm. (a) Height channel (pass 1, retrace) of the profile, repeated over time with the slow scan axisdisabled. (b) Phase channel (pass 2, retrace) sensitive to the magnetic interaction between tip and sample. (c) Input 1 channel (appliedmagnetic field value, pass 2, retrace). (d) Applied field values as a function of the scan line number, equivalent to a cross section of (c) alonga vertical line. The scan (time sequence) along the vertical direction is downwards.

acquired. The PC, through the EOL signal coming from thecontroller, can synchronize the magnetic field variations withthe beginning of a new scanning of the same line. As aconsequence, each time the same profile is measured undera different magnetic field, whose value is acquired by thecontroller and saved together with the other signals processedby the AFM. The magnetic field can be swept between arbitrarylimits, provided that the magnetization of the tip does notflip perpendicular to its axis. Arbitrary field histories can begenerated, although in this work we will limit to symmetricfield paths, as usually carried out in standard hysteresis loopmeasurements.

In contrast with conventional MFM, with this techniquea single ‘hysteresis loop’ acquisition can be made of as manyscan lines are acquired in an individual MFM image (usuallyseveral hundreds); as each line is scanned at a differentmagnetic field value, the sample is investigated at a largenumber of field equilibrium points, much larger than thenumber of full images (like those in figure 2) that can berealistically acquired. The speed improvement with respectto conventional field-dependent MFM is therefore achievedby obtaining the magnetic information not on the wholesub-micrometric structure, but only on the chosen profile.Therefore, issues related to tip wearing, although unavoidable,are considerably reduced, as a whole field history can beacquired in a single frame. The tip coating will eventuallyget damaged, but a new tip lifespan should allow the properacquisition of several ‘full frame’ MFM images as well asseveral ‘local hysteresis loops’ made of a few hundred appliedfield values without a significant performance loss. However,joining full-frame images as in figure 2 and ‘hysteresis loops’measured with this technique provides a rather comprehensiveunderstanding of the static magnetization processes in thestudied sub-micrometric structures. The acquired field-dependent images that represent these local (i.e. performed ona single profile) hysteresis loops require adequate processingin order to extract from them quantitative information, as willbe discussed in the following.

The proposed technique is derived from the single-pointapproach to measure hysteresis loops with MFM, that isdiscussed in [31, 34]. The single-point technique has theadvantage that no pass 1 is required, therefore the effects of theinteraction between the tip and the sample can be minimized.However, as the morphology is not acquired during hysteresisloop measurements, any effects due to sample drift cannotbe noticed and possibly corrected. Additionally, the single-point technique requires that the two-dimension domainconfiguration evolution of the studied structure is inferredfrom a zero-dimension measurement, as the tip is placed ina specific position and never moved during the measurement.In contrast, our technique requires pass 1, therefore limitingits applicability to the cases where severe effects of the tipon the sample domain structure are not observed, but givesaccess to one-dimension sample morphology and magneticdomain configuration, therefore providing a greater degree offlexibility in the analysis and interpretation of the results, aswill be discussed in the following, although only a limitedpossibility of compensating sample drifts will result.

According to figure 1, the dashed line represents the profileat which the slow scan axis is disabled, and that is scannedrepeatedly over time; a sketch of the MFM tip is also drawn,indicating the height (not in scale) at which it performs the lift(pass 2) step in the present case.

The result of the measurement procedure is reported infigure 4. In figure 4(a), the chosen dot profile is shown bymeans of the height channel. The dashed lines indicate the dotleft and right edges. It can be seen that the profile showsalmost no changes between adjacent scan lines, indicatingthat the sample is very stable during the measurements; thevery small drift does not significantly affect the results of themeasurements. This is achieved thanks to the fact that the AFMhead is non-magnetic and the applied magnetic field is uniformover the whole sample volume. In any case, the availabilityof the height profile of the studied structure for each appliedmagnetic field could allow, if necessary, to compensate forsample drifts, that could be not negligible especially in sub-micrometric patterns. However, it has to be noted that it is not

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1.20.80.40.0

0

200

400

position (µm)

phas

e sh

ift(1

0-3 d

eg)

Figure 5. Top: phase (pass 2, retrace) image representing themagnetic interaction between the tip and the sample, for differentapplied fields (scan lines). Bottom: profile along the yellow thickdashed line. The values in the shaded regions are used for the dataanalysis discussed in the text.

always possible to distinguish from drifts along the slow andthe fast scan axes; even though the height channel can helpin compensating for small sample movements along the fastscan axis, any drift along the (disabled) slow scan axis couldturn out undetected. Figure 4(b) shows the magnetic signal, asobtained by the phase channel during the lift scan (pass 2). Themagnetic signal clearly changes as a function of the scan linenumber, as will be discussed shortly. The applied magneticfield during each scan is shown in figure 4(c): bright coloursindicate a positive magnetic field (e.g. directed from right toleft on the dot), whereas dark colours indicate a negative one(e.g. directed from left to right on the dot). Its actual valuesare shown in figure 4(d), which is equivalent to a cross sectionof panel (c) along a vertical line.

Figure 4(c) shows that the applied magnetic field isconstant during each scanned line (each line in the imagehas uniform colour). This is achieved thanks to thesynchronization of the magnetic field steps with the beginningof a new line acquisition, as discussed earlier. Additionally,it can be seen that the first scanned lines and approximatelythe last third of the acquisition have been performed atconstant, zero field. While this has almost no influence onthe morphological part of the image (height channel), a clearlyvisible correlation with the magnetic signal, representing theinteraction between the tip and the sample, is demonstrated infigure 4(b), which is magnified in figure 5.

In correspondence of saturating positive fields, the phasechannel shows bright and dark regions at the left and rightdot edges respectively. This is in agreement with thedomain configurations discussed in figure 2. As the field isprogressively reduced (i.e. moving downwards in the images),

these bright and dark regions at the dot edges expand, as alreadydiscussed in figure 2, indicating that the domain configurationevolves from saturation to C-state. On further reducing themagnetic field, a sudden loss of the magnetic contrast marksan abrupt rearrangement of the domain configuration of thedot. This irreversible jump, in agreement with the simulationsdescribed in the following section, will be attributed to thenucleation of a magnetic vortex. It is an effect induced by theprogressive variation of the applied magnetic field; althoughwe cannot rule out the possibility that the exact value atwhich the vortex nucleates is slightly affected by the tip–sample interaction especially in pass 1, the same considerationsdiscussed for figure 2 apply. Therefore, we attribute thisirreversible jump of the phase signal in figure 4(b) and figure 5to the applied magnetic field and not to a remagnetizationof the sample induced by the tip. When the field is furtherchanged, the vortex moves and is eventually expelled. Afterthe vortex expulsion, the bright and dark regions at the dotedges swap their position (dark to the left and bright to theright), in agreement with the reversal of the magnetic fieldvalue, until the maximum negative field is applied. Thiscorresponds to the line that can be easily identified by thecusp in figure 4(d) at approximately one third of the scannedlines. From that moment, the field starts increasing again,and a symmetrical evolution of the domain configuration isobserved (which can be followed in figure 5 as well wherethe magnetization features are indicated with symbols), withformation of C-state, nucleation (•), motion and expulsion ofthe vortex (◦), until the positive saturation is reached again(�) at approximately two thirds of the scanned lines. In thisstate, the magnetic configuration of the dot is the same as atthe beginning of the loop (figure 4(b)), as expected. As themagnetic field has cycled through a whole loop, it is switchedoff and the dot reverts to the magnetic domain configurationat its remanence, that is acquired with no further evolutionthrough the end of the frame.

As a counter proof that the detected signal actually comesfrom the interaction between the magnetic sample and theMFM tip, and is not an artefact induced by the applicationof the external magnetic field, a similar procedure has beenapplied on a sample consisting of a 200 nm thick step of copper.As expected, a phase (pass 2) image that is featureless andremains constant (unchanged) as the applied magnetic field isswept between its extrema is obtained.

An example of a phase (pass 2, retrace) scanned lineis shown in figure 5, which proposes again figure 4(b), butcropped to the only relevant part. A profile taken along theyellow thick dashed line is also shown. In order to plot an actualhysteresis loop from the acquired data, a suitable analysis ofthe images needs to be performed. For each line of figure 5,the profile is extracted. The height channel (figure 4(a)) givesthe precise position of the dot edges, indicated in figure 5 bythe thin, dashed blue lines. For each line, a certain number ofvalues (pixels) is chosen around the position of each edge (10pixels outside and 10 pixels inside the dot, in the present case):the shaded area in figure 5 approximately indicates the twoportions of the curve that are taken into account. The averagevalue of the phase is computed for each line at both edges;

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0 300–300–600 –300–600

–300–600–300–600

600

H (Oe)

2 Hz

/ z2

(a.u

.)

experimental

0 300 600

H (Oe)

2Hz /

z2 (a.u.)

simulated

0 300 600

H (Oe)

2 Hz

/ z2

(a.u

.)

0 300 600

H (Oe)

2Hz /

z2 (a.u.)

(a) (b)

(c) (d)

30 nmfrom

dot side

150 nmfrom

dot side

1st branch

2nd branch

Figure 6. MFM (‘phase contrast’) local hysteresis loops of a Ni80Fe20 square dot, with a size of 800 nm. (a) Experimental loop with a phasecontrast measured close to the dot edges. (b) Corresponding simulated loop. (c) Same as (a) but with the phase contrast measured inside at≈150 nm from each edge. (d) Corresponding simulated loop.

then, the two average values are subtracted, to calculate a sortof ‘phase contrast’ representing the horizontal component ofthe magnetization that is responsible for the field that expandsin the volume outside the dot and whose z-component of thesecond derivative is detected by the MFM tip. As an example,the phase profile reported in figure 5 along the dashed line(bottom panel), which is close to the negative saturation, resultsin a negative phase contrast. The same analysis performedalong a line close to the positive saturation (e.g. along a line inthe top part of figure 5) would result in a profile with a reversedsign, and therefore in a positive phase contrast.

This ‘phase contrast’ value is then plotted as a function ofthe magnetic field that was applied when that line was scanned,as measured by the input 1 channel shown in figure 4(c). Theresult of this procedure, performed on all scanned lines, isshown in figure 6(a) for Ni80Fe20 square dots having a sizeof 800 nm.

Abrupt changes in the ‘magnetization’ at specific fieldvalues can be seen in the loop of figure 6(a): they coincidewith the abrupt changes of colour observed in figure 4(b)

and their interpretation will be discussed later with the helpof micromagnetic simulations. The first (from positive tonegative saturation) and second (from negative to positivesaturation) branches of the loop are plotted in blue and red,respectively: in this case, the first branch is higher than thesecond, but this is not necessarily the case when other profilescloser to the dot centre are investigated. However, the areaenclosed by the cycle is not representative of the dissipatedenergy, as the curves of figure 6 do not relate directly to themagnetization of the dot, but rather to the vertical componentof the second derivative with respect to z of the magneticfield that is generated by the magnetization of the dot, as withany MFM image. The most important magnetization reversalfeatures are marked in the return branch of the loop plottedin figure 6(a) with the same symbols that are also shownin figure 5. In particular, a full circle indicates the vortexnucleation, a hollow circle the vortex expulsion, and a downtriangle the jump from the C-state to saturation. Moreover,depending on the chosen profile and the details of the phase-analysis procedure, different loops or crossings of the branchescan be obtained, as will be discussed in sections 4 and 5.

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J. Phys. D: Appl. Phys. 47 (2014) 325003 Marco Coısson et al

Finally, it has to be noted that the local hysteresis loops shownin figure 6 are quantitative for what concerns the field axis,whereas there is no direct quantification of the magnetizationof the dot unless a proper calibration of the response of theMFM tip is performed. Nonetheless, the loop shape can beeasily observed and discussed.

4. Validation

The new measurement approach is validated by comparingthe experimental data (hysteresis loop and MFM images atdifferent equilibrium points) to the results of micromagneticsimulations performed on the square dot with 800 nm lateralsize. In particular, the spatial distribution of the magnetizationvector �M versus applied field value is calculated by solvingthe Landau–Lifshitz–Gilbert (LLG) equation:

∂ �M∂t

= − γ

1 + α2�M ×

[�Heff +

α

MS

(�M × �Heff

)](1)

where γ is the absolute value of the gyromagnetic ratio, α is thedamping coefficient and MS is the saturation magnetization.The effective field �Heff is here the sum of magnetostatic,exchange and applied fields, having considered negligible themagnetocrystalline anisotropy of permalloy. The parameterMS is set at 860 kA m−1 in agreement with the experimentallymeasured value on the reference continuous film, while theexchange constant is fixed to 13 pJ m−1.

The LLG equation is solved once having discretizedthe magnetic sub-micrometre structure into hexahedra withaverage size comparable to the exchange length, assumingthat magnetization and effective field vectors are uniform ineach mesh element. For the time integration and the efficientcomputation of static hysteresis loops, the used micromagneticsolver, described in detail in [37], adopts a norm-conservingformalism based on Cayley transform and a second-order Heunscheme [38, 39].

The calculated static hysteresis loop of the 800 nm sizedot is shown in figure 7, considering an applied field directedparallel to one of the dot edges. To reproduce the experimentalprocedure, the simulation starts from a zero-field condition.Then, the sample is nearly saturated by increasing the fieldup to 800 Oe. Next, the static hysteresis loop is calculated bygradually modifying the applied field in 12.566 Oe (1 kA m−1)steps, for each of them the time evolution of �M is computeduntil the equilibrium state, which is assumed to be reachedwhen the maximum local value of the torque | �M × �Heff |/M2

S

is lower than a fixed threshold. Damping coefficient, time stepand threshold values are chosen on the basis of the analysisreported in [40] to guarantee both energy minimum reachingand computational efficiency increase.

Figure 7 shows the typical hysteresis loop of a dotwith symmetric Landau vortex state at remanence. Whendecreasing the field from saturation the magnetizationconfiguration evolves towards a C-state [41, 42] (jump (a)in figure 7) with a successive compression of the domainmagnetized in the direction of the applied field (jump (b) inthe figure). At very low fields, the system is characterized

by an irreversible jump (labelled (c) in figure 7) from vortexnucleation at one of the dot sides to centred vortex state. Then,there is a reversible deformation of the Landau vortex statewith the progressive motion of the vortex towards the oppositeside. The final irreversible jump (marked as (d)) is due tothe expulsion of the vortex and leads again to a C-state withthe magnetization in the larger domain oriented parallel to thenegative applied field.

The MFM images have been numerically reconstructedassuming that the magnetization in the tip coating layer isuniformly distributed along the tip vertical axis and it isnot affected by the stray field due to the sample. Second,we assume that the magnetization configuration in the dotis not influenced by the stray field produced by the tip.This assumption is supported by the discussions provided forfigures 2 and 4, that do not show an experimental evidenceof a significant remagnetization of the sample due to thefield generated by the tip, but can be relaxed if requiredby the experimental conditions and studied samples. Byconsidering the previous hypotheses and by representingthe tip as a magnetic dipole, the MFM images have beenreproduced computing the spatial distribution of the second-order derivative of the z-component of the dot stray field∂2Hdot,z/∂z2 [35, 36]. This quantity, which is proportional tothe gradient of the force acting on the tip, has been calculated ona planar grid of points located at a distance z0 of 90 nm from thesample surface. In particular, the second-order derivative hasbeen approximated as Hdot,z(z0+�z)+Hdot,z(z0−�z)−2Hdot,z(z0)

�z2 , where�z = 2 nm. Specifically, the dot stray field in correspondenceof the generic point �r = �r(x, y, z) on one of the three planesparallel to the sample surface has been obtained from

�Hdot (�r) = 1

4π

T∑e=1

∫∂�e

�M (�re) · �ne

�r − �re

‖�r − �re‖3ds, (2)

where T is the number of hexahedra in which the dot has beendiscretized and ∂�e is the surface of the eth hexahedron withnormal unit vector �ne.

An example of a colour map of the second-order derivativeobtained from the spatial distribution of the magnetizationof the dot at an applied field value of −250 Oe is shown infigure 8(a). This map does not take into account the averagingeffect due to the finite size of the MFM tip. Therefore, eachpoint of each simulated MFM image has been averaged withits nearest neighbours up to a distance of 35 nm (the tip radiusaccording to the specifications of the used HR-MESP tip). Thetransformed maps reproduce more closely the experimentalones. The result of the averaging process is shown in figure 8(b)for the simulated image reported in (a).

It is now possible to visualize these simulated MFMimages at a few different applied field values, to identify thedifferent magnetization processes occurring in the dot. Theresult is shown in figure 9, which can be directly compared withthe experimental images in figure 2. The qualitative agreementis excellent.

Moreover, at each field value for which the MFM maphas been calculated, the same profile can be chosen and thecorresponding line can be extracted. By putting these lines

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J. Phys. D: Appl. Phys. 47 (2014) 325003 Marco Coısson et al

412.5 Oe

0 Oe

-300 Oe-287.5 Oe

25 Oe

87.5 Oe100 Oe

400 Oe

0 500 1000

0.0

0.5

1.0

H (Oe)

M (

arb.

uni

ts)

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

Figure 7. Loop of a Ni80Fe20 square dot, with a lateral side of 800 nm and a thickness of 30 nm, simulated with a micromagnetic codesolving the LLG equation. Colour maps of the simulated magnetization spatial distribution are reported in correspondence of the loopirreversible jumps, marked from (a) to (d), for the equilibrium points immediately before and immediately after the jump. The colour scaleidentifies the magnetization vector angle (in degrees) with respect to the x-axis.

(a) (b)

Figure 8. (a) Map of the second-order derivative of the dot stray field calculated from the spatial distribution of the simulated dot at a heightof 90 nm for an applied field of −250 Oe. (b) Effect on the image shown in (a) of the finite size of the MFM tip, having a radius of ≈35 nm.

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J. Phys. D: Appl. Phys. 47 (2014) 325003 Marco Coısson et al

(a)H = +600 Oe H = +400 Oe H = +65 Oe

H = +15 Oe H = 0 Oe H = -65 Oe

H = -250 Oe H = -400 Oe H = -600 Oe

(b) (c)

(d) (e) (f)

(g) (h) (i)

400 nmH

Figure 9. Simulated MFM images of a square Ni80Fe20 dot (size 800 nm) at selected applied field values. The field is applied in plane, alongthe horizontal direction. The dashed squares identify the dot edges.

in sequence, an image similar to the experimental phase shiftas a function of field (figure 5) can be obtained, as shown infigure 10 for a profile lying at 100 nm from the bottom edgeof the simulated dot. In addition to the excellent qualitativeagreement, the irreversible jumps of the dot magnetizationcan be clearly identified in figure 10, corresponding tothe transitions from saturation to C-state and to the vortexnucleation and expulsion, and identified with the same symbolsused in figures 5 and 6. As the effect of the tip magnetizationon the dot magnetic domain configuration is not taken intoaccount in the simulations, the irreversible jumps visible infigure 10 and therefore also in figure 5 cannot be attributedto the tip–sample interaction but are indeed induced by theapplied magnetic field.

Therefore, for each field value, a simulated MFM map isnow available, which can be processed with the same algorithmthat has been described previously, as if it were an experimentalMFM image. Therefore, an hysteresis loop has been calculatedfrom the second-order derivative simulated maps by analysing

the colour contrast, as described in figure 5, assuming that aprofile is scanned at ≈100 nm from one edge of the dot; thecorresponding result is reported in figure 6(b).

The calculated loop clearly marks a few irreversibletransitions in the magnetization process occurring in thedot, that correspond to the abrupt colour changes visible atspecific lines in figure 10. The transition marked with thetriangle pointing up (�) in the second loop branch reported infigure 6(b) corresponds to the transition from the saturated stateto the C-state. Further decrease in the applied field leads to thenucleation of the vortex, identified by the transition labelledwith a full circle (•). Then, the vortex expulsion and theformation of the C-state are marked by the hollow circle (◦)and the triangle pointing down (�), respectively. The sameirreversible jumps can be identified in the experimental loopshown in figure 6(a), although the formation of the C-statefrom saturation is probably hidden in the measurement noise.

Although the experimental and simulated fields at whichthe various irreversible features of the magnetization process

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J. Phys. D: Appl. Phys. 47 (2014) 325003 Marco Coısson et al

0.0 1.0

H (kOe)

line #

Figure 10. Left: simulated phase-shift image as a function of theapplied magnetic field along a profile located at 100 nm from thebottom edge of the simulated dot. Right: magnetic field value foreach line of the image.

take place do not perfectly match due to the difficulty ofproperly taking into account in the calculations all the detailsand imperfections of the real sample, the detailed analysisof the simulated MFM images and of the correspondingcalculated ‘phase contrast’ hysteresis loop enables theidentification and understanding of the magnetization reversalprocesses occurring in the studied dots, investigated bymeans of the described technique. The excellent qualitativeagreement of figures 6(a) and (b) and of figures 5 and 10satisfactorily validates the proposed methodology.

5. Distance from dot edges

To further validate the technique, the same data reported infigure 4 that produced the loop of figure 6(a) are analysedby shifting the areas where the phase contrast is calculated≈150 nm from the dot side, towards its centre. Therefore,with respect to figure 5, the shaded areas are not locatedon the positive and negative peaks of the phase, but atan arbitrary (symmetrical) position inside the dot. Theresulting experimental hysteresis loop is shown in figure 6(c).The corresponding simulated loop, calculated with the sameparameters used for figure 6(b) but with a distance from thedot edges of ≈150 nm, is shown in figure 6(d). The qualitativeagreement between the experimental and simulated loops isexcellent. Additionally, the significant role played by thechoice of the regions between which the phase contrast isdetected has been put into evidence in the determination ofthe final loop shape.

The measurement technique has then been applied onlarger systems, i.e. a Ni80Fe20 dot having a size of 2 µm anda Co67Fe4Si14.5B14.5 circular dot having a diameter of ≈6 µm.The obtained ‘phase contrast’ hysteresis loops are shown infigure 11. On these larger systems, the two loop branches(blue and red curves) cross, giving the loops a different shapewith respect to the 800 nm sized Ni80Fe20 dots. However,on both systems MFM images acquired at several magnetic

field values indicate that the magnetization reversal process iscomparable to the one described in figure 2 for the 800 nmNi80Fe20 dots. As an example, MFM images at the magneticremanence displaying a vortex are shown in figure 11 for bothNi80Fe20 and Co67Fe4Si14.5B14.5 dots. Therefore, we assumedthat, although only qualitatively, the same calculations ofthe 800 nm Ni80Fe20 dot can be used for comparison: wecannot expect good quantitative agreement between simulationand experiment in figure 11 as the simulated sample doesnot match in size or even in shape and composition theexperimental ones, although a qualitative comparison is stillmeaningful since the magnetization processes remain the sameas discussed before. Simulated loops having a shape closelyresembling that of the experimental ones could be obtained,as demonstrated by figure 11. However, in order to make thetwo loop branches cross, it has been necessary to place theregions between which the phase contrast has been calculatedat slightly different distance with respect to the dot edges. Infact, although experimentally the phase contrast is calculatedat approximately the same distance from both left and rightedges of the dots, for the loops reported either in figure 6 andin figure 11, dots having a larger size are subject to larger errorsin the precise determination of the region over which the phasecontrast is calculated. Conversely, simulations allow for theirprecise control. In figure 11, the distance from the left and rightedges of the dot of the two regions where the phase contrastis calculated is not symmetrical, but differs by only 15 nm, adistance that is smaller than the MFM tip radius. Therefore, thetechnique can be applied also to microstructures, although theresulting loops are more sensitive to the exact placement of theshaded areas of figure 5. On sub-micrometric structures, theexperimental results are much less subject to these fluctuations.

6. Domain wall position as a function of applied field

The amount of information that can be extracted from the field-dependent MFM images analysed in the previous sections tomeasure local hysteresis loops can be larger than discussed sofar, if the sample provides sufficient details on its local domainconfiguration, thanks to the fact that we are acquiring the phaseshift data as a function of the magnetic field along a full scanline and not on a single, fixed point. An example is the MFMimage, acquired at the magnetic remanence, of a Fe78Si9B13

dot with a thickness of 250 nm and a size of 20 µm, shownin figure 12(a). In this case, a Landau domain configurationis clearly visible, with the domain walls separating the fourtriangular regions in which the dot is divided.

The procedure previously discussed for measuring localhysteresis loops with the field-dependent MFM is appliedto this sample, and the magnetic phase signal is shown infigure 12(b); the dependence of each scanned line with theapplied magnetic field is shown in figure 12(c).

Since the domain walls separating the four Landau regionsare clearly visible, it is possible to plot the intercept betweenthe scanned line and the domain walls as a function of theapplied magnetic field. The result is shown in figure 13(a).At high negative applied fields the walls are more far apart(≈12 µm), indicating that the magnetization of the domain

11

J. Phys. D: Appl. Phys. 47 (2014) 325003 Marco Coısson et al

Co67Fe4Si14.5B14.5 circular dot(diameter 6 µm)

Ni80Fe20 square dot(size 2 µm)

0 300 600

H (Oe)

2 Hz

/ z2

(a.u

.) experimental

0 300 600

H (Oe)

2 Hz

/ z2

(a.u

.) simulated

H (Oe)0 250 500

2Hz /

z2 (a.u.)

experimental

1st branch

2nd branch

0 300 600

H (Oe)

simulated 2Hz /

z2 (a.u.)

500 nm 2 µm

Figure 11. MFM (‘phase contrast’) local hysteresis loops on a Ni80Fe20 square dot (size 2 µm) and on a Co67Fe4Si14.5B14.5 circular dot(diameter of 6 µm). Top row: experimental loops. Middle row: corresponding simulated loops approximated with the data calculated on theNi80Fe20 square dot having a size of 800 nm. Bottom row: MFM images acquired at magnetic remanence for the measured dots.

wall in the bottom half of the dot is aligned parallel to negativefields, hence the domain increases its volume. On increasingthe applied field, the domain shrinks and the two walls comecloser (≈4 µm apart for an applied field of +300 Oe). Themaximum applied field has been chosen to be not sufficientto reverse the magnetization of the domains that are aligned

antiparallel to the field, therefore the magnetic saturationis not reached and the Landau domain configuration is notdestroyed. However, the domain wall motion is not completelyreversible, as some hysteresis is observed in figure 13(a) forboth walls. If the ‘phase contrast’ loop is determined followingthe procedure described in the previous sections, the resulting

12

J. Phys. D: Appl. Phys. 47 (2014) 325003 Marco Coısson et al

0 300

H (Oe)

scan line #

5 µm

phasepass 2, retrace

5 µm

(a)(b) (c)

Figure 12. (a) MFM image at remanence of a Fe78Si9B13 dot with a thickness of 250 nm and a size of 20 µm. (b) Field-dependent phasesignal along the profile shown with the dashed line in (a). (c) Applied field values as a function of the scan line number.

hysteresis curve is reported in figure 13(b). As expected, thisloop does not show irreversible jumps, since the maximumapplied field is not sufficient to destroy the Landau domainconfiguration, however, it is characterized by hysteresis, thatmay not necessarily be the same discussed in figure 13(a),since the portions of the field-dependent phase image thatare analysed to build the two loops plotted in figure 13 aredifferent.

7. Conclusions

The investigation of the reversal mechanisms of themagnetization in micrometric and sub-micrometric structuresrequires special techniques that are sensitive to the nanometricscale and allow a detailed map of the magnetic domain patterns.MFM can be exploited at different applied magnetic fieldvalues; however, the technique is very time consuming and it isnot possible to acquire tens or hundreds of images at differentfield values because of the wearing out of the tip.

In this paper, we have modified the field-dependent MFMtechnique by disabling the slow scan axis and repeating theacquisition of a single profile of a magnetic element severalhundreds of times; by properly synchronizing the applied fieldsteps with the scanning of each line by the MFM, a detailedimage of the magnetization variation as a function of the fieldcan be acquired, at a local scale. Suitable data processinghas allowed one to plot local hysteresis loops measured onindividual Ni80Fe20 square dots (size of 800 nm and 2 µm),Fe78Si9B13 square dots (size 20 µm) and Co67Fe4Si14.5B14.5

circular dots (size 6 µm) and identify local reversible andirreversible features in the magnetization reversal processes.

The thickness of the investigated sub-micrometricstructures does not have, in principle, a significant effect onthe measurement technique, provided that the scanner of themicroscope is able to cope with the height of the steps that thetip encounters during the scanning of the profile. However,the actual domain configuration of a given sub-micrometricor manometric structure may be significantly affected by itsthickness [43], therefore making this technique quite useful

for experimentally investigating such effects. Similarly, thematerial composition and therefore its magnetic properties canconsiderably impact on the performance of the microscopeand therefore of the technique. In particular, the magnetictip coating has to be chosen with care in order to minimizeboth the effect of the applied field on the alignment of thetip magnetization, and the effect of the tip stray field on thedomain configuration of the sample. Therefore, investigatinghard magnetic samples such as Nd–Fe–B or hard Fe-oxidescan be challenging because large magnetic fields need to beapplied in order to reverse the sample magnetization. Providedthat the applied field is sufficiently homogeneous in order toneglect drifts of the sample or of the microscope, special caremust be taken when the field flips the magnetization of thetip from perpendicular to parallel to the sample plane. In thiscase, MFM is still able to detect a magnetic contrast, althoughits interpretation needs to be revised. Additionally, a properqualification of the tip magnetic properties has to be carried outto deconvolve them from the MFM signal and reconstruct thefield evolution of the magnetization of the sample. Approachessimilar to those used in [20, 32] should therefore beexplored.

The experimental technique has been validated by meansof a micromagnetic simulation of the magnetization processin a Ni80Fe20 square dot, with a size of 800 nm and athickness of 30 nm, comparable to one of the experimentalsamples. The calculated equilibrium configurations of themagnetization at all applied field values have been transformedinto equivalent MFM maps that have been analysed withthe same technique proposed for the experimental images.The simulated ‘phase contrast’ loop is able to identify themain magnetization processes occurring in the dot (vortexnucleation and expulsion, transition from C-state to saturatedone), and has an excellent qualitative agreement with theexperimental one, thus providing the necessary validation ofthe technique.

The proposed experimental approach can therefore besuccessfully applied for the study of the magnetization reversalprocesses in arbitrary micrometric and sub-micrometric

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J. Phys. D: Appl. Phys. 47 (2014) 325003 Marco Coısson et al

0 150 300

H (Oe)

M (

arb.

uni

ts)

FeSiB dotsize 20 µm

(b)

0 150 3000

2

4

6

8

10

12

14

16

18

H (Oe)

wal

l pos

ition

(µm

)

(a)

Figure 13. (a) Local hysteresis loop as measured with thefield-dependent MFM on a Fe78Si9B13 dot with a thickness of200 nm and a size of 20 µm. (b) Position of the intercept betweenthe scanned profile (figure 12(a)) and the domain walls as a functionof the applied magnetic field.

structures through local hysteresis loops measurements withfield-dependent MFM. Even though the main features of themagnetization reversal process can be identified by properlyanalysing the field dependence of the phase channel of theMFM along the selected profile, micromagnetic simulationsor another locally resolved knowledge of the domain structurewill most likely be needed to fully understand and describe theprocesses occurring in a hysteresis loop. However, throughthe combination of the proposed technique and micromagneticmodelling, a deep and complete understanding of the fieldevolution of the magnetization in sub-micrometric structuresand patterns can be successfully achieved.

Acknowledgments

Sample preparation and patterning have been performed atNanoFacility Piemonte, INRIM, a laboratory supported byCompagnia di San Paolo.

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