Micromagnetic simulations of magnetoelastic heterostructures

Author: Javier Rodriguez AlvarezFacultat de Fısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.∗

Advisor: Arantxa Fraile Rodrıguez

Abstract: The system formed by a 10 nm thin film of Ni and a 10 µm-thick V2O3 layershows a reversible enhancement of the coercivity (∼300%) due to the structural phase transition(metal/insulator) of the V2O3layer. Performing micromagnetic simulations and comparing the re-sults with experimental data we have studied the coupling mechanism on the Ni/V2O3 interface. Wehave proven that this coupling is responsible for the coercivity enhancement and that the magneti-sation domains in the Ni do not follow exactly the metal/insulator domains on the V2O3. Finally,we have determined that the results of the simulations are in agreement with a phase coexistencein V2O3.

I. INTRODUCTION

Since the last part of the XX century, the world hasexperienced an unprecedented technological revolution.Throughout this process, the development of materialswith emergent properties and novel functionalities hasbeen responsible for many major scientific breakthroughs[1].

In the case of magnetic materials, a novel interestingapproach is being able to control the magnetic propertiesof a material by other means rather than the applicationof a magnetic field; e.g. light, pressure or an electricfield. The manipulation of magnetic properties withoutthe application of a magnetic field opens the door for acompletely new range of devices and possibilities [2].

In this regard, we address this issue by studyingthe properties of the heterostructure formed by a thinlayer of Ni (10 nm) deposited onto another layer of 100nm-thick vanadium oxide (V2O3). V2O3 undergoes astructural phase transition (SPT), from a rhombohedralstructure at the high temperature phase, to a monoclinicphase at low temperature. Since the volume of thelow temperature phase is 1% bigger than the hightemperature phase, the SPT induces a strain in the Nilayer across the interface with the V2O3. This interficialstrain produces a lattice deformation on the Ni layerwhich, due to its magnetoelastic characteristics, inducesthe formation of a magnetic anisotropy direction. Thiscrystallographic structure change in V2O3 results ina remarkable and reversible modification of the Nimagnetic properties such as a large enhancement ofcoercivity in a narrow temperature range across the SPT[2][3].

However, a complete microscopic study of the magne-toelastic coupling in Ni/V2O3 is still lacking. Experi-mentally, this problem has been addressed at the groupof Magnetic Nanomaterials, UB, by imaging the Ni mag-

∗Electronic address: javier.rodriguez.alvarez95@gmail.com

netic domains as a function of temperature across theV2O3 SPT. This TFG represents a new approach to theproblem, micromagnetic simulations.

The goal of this TFG is to use micromagnetic simula-tions to deepen our understanding on this topic.

II. SAMPLES AND EXPERIMENTALCHARACTERISATION

Bilayers of 5nm Al/10nm Ni/100 nm V2O3 wereprepared using magnetron sputtering deposition byour collegues at USCD, as described elsewhere [2].[3](Fig.1).The Ni layer is polycrystalline, smooth (rough-ness of about 2 nm) and metallic. Macroscopic magneticmeasurements show an increase of about 300% of the co-ercive field at the low-temperature phase (Fig.2).

A microscopic, magnetic characterization of the sam-ple has been carried out by the advisor and coworkers bysynchrotron-based photoemission electron microscopy(PEEM) combined with X-ray magnetic circular dichro-ism (XMCD) [4]. The temperature-driven variationof the FM domain structure of Ni (10 nm-thick) wasmapped across the SPT of V2O3. It was observed thatthe initial ferromagnetic (FM) saturated state splitsupon crossing the SPT into a pattern of small domainswhich reverse by progressive rotation. The fraction ofsuch inverted domains monotonically grows across theSPT with the predominance of one of the coexistingstructural phases, in agreement with the evolution ofthe insulator/metal transition in the V2O3 thin film[5]. It was observed that the Ni domains form stripepatterns with length and periodicity consistent withthe nanotexture of the metal/insulating domain in theadjacent V2O3 film [4].

The main objective of this TFG is to gain further in-sight on the microscopic Ni/V2O3 coupling mechanismby performing micromagnetic simulations of our 10 nm-thick Ni layer under comparable experimental conditionsand geometry used in the PEEM experiments. Simula-tion has proven itself a very useful tool in order to de-

Micromagnetic simulations of magnetoelastic heterostructures Javier Rodriguez Alvarez

FIG. 1: Schematic representation of the samples used in theexperimental measurements.

termine which parameters are important in the systemand which of them are not. Furthermore, through simu-lation experiments we can measure the effect that has onthe system a change in some of the parameters that aredifficult to disentangle through the experimental mea-surements.

III. SIMULATIONS

The simulations were carried out using the ob-ject oriented micromagnetic computational framework(OOMMF), which is a project developed by the NationalInstitute of Standards and Technology (NIST)[6]. How-ever, some extra work has been done using python codein order to process the output data from the simulationsand to provide the input data, depending on the simula-tion experiment carried out at each point of the research.

The phase transition in V2O3 is carried out inhomoge-neously, following a very characteristic pattern (Fig.5c).Because of this pattern formation, the anisotropyinduced in the Ni layer is neither homogeneous norabsolutely random. Thus, the different magnetisationdomains that appear in the Ni, due to the inducedanisotropy, follow a similar pattern, as will be discussedlater.

The simulations performed during this work, and pre-sented in this TFG, are mostly of two types. On onehand, several hysteresis loops of the Ni layer have beenmeasured for each temperature aiming to understand thebehaviour of the coercivity of the system with tempera-ture. On the other hand, the magnetisation of the systemhas been homogeneously oriented in a certain directionand a constant magnetic field (200 Oe) has been appliedanti-parallel to this direction (Fig.2a) to study the rel-ative rotation of the system as a function of the tem-perature under this particular protocol matching that ofPEEM experiments.

A uniform exchange constant of 3.4 · 10−12 J/m hasbeen used. In addition, an easy axis of magnetisationwas introduced in the Ni layer in the x direction withan anisotropy constant of 2470 J/m3 [3]. OOMMF com-putes the state of the system dividing it in cells of con-trollable dimensions distributed in a mesh. For each cell,

the magnetisation is considered to be constant and theinteractions with its neighbours are taken into account,as well as other long range parameters. The evolutionof the system is controlled throughout iterations by theLandau-Lifshitz-Gilbert equation:

d ~M

dt= −|~γ| ~M ×Heff +

α

MS

(~M × d ~M

dt

)(1)

where γ is the Gilbert gyromagnetic ratio, Heff the effec-tive field and α the damping constant, the Hamiltonianof the system is:

H = HZeeman +Hanisotropy +Hexchange +Hdemag (2)

In this case a minimisation evolver has been chosen.This means that a stage in the simulation, such as e.g.a change in the applied magnetic field, is finished whenthe system reaches an equilibrium state. The equilibriumcondition is reached when, after a recursive integrationof eq.1, the energy does not significantly change, in otherwords, when the energy gets minimised for those partic-ular conditions.[7]

A. Evolution of coercive field vs temperature

The so called coercivity is the strength of the magneticfield necessary to reduce to zero the magnetisation ofthe studied material and, thus, it is a useful measureof the ”magnetic hardness” of a material. In order todetermine the coercive field of the Ni, hysteresis loopsas seen in (Fig.2b) were calculated for each temperatureconfiguration.

For the simulations, a square mesh of 10µm x 10µmx 10 nm with 100 x 100 x 10 nm cells has been set,representing the thin-film nature of the Ni layer. Inaddition to the Ni magnetic anisotropy, the effect of thestrain produced by the V2O3 needs also to be takeninto account as schematically shown in (Fig1). Withthis in mind, an additional anisotropy of 20958 J/m3 isintroduced in the Ni layer forming an angle of 45◦ withthe ”natural” anisotropy axis of the Ni in the regionsthat are directly onto the domains of V2O3 that havetransited to the low temperature phase. The valuesfor both anisotropy constants were estimated fromexperimental data.

The OOMMF program reads the colour map from theexperimental measurement of the V2O3, as will be shownlater, and assigns the anisotropy values and directions ac-cording to the previously mentioned criteria. The regionspresenting a white colour are assigned to the emergentred phase. The boundaries between the two phases inthe V2O3 across the SPT (white colour) are assigned tothe emergent red phase.

Due the limited experimental data available to feed thesimulations (only five images corresponding to the con-

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Micromagnetic simulations of magnetoelastic heterostructures Javier Rodriguez Alvarez

FIG. 2: a) Magnetic measurements of the coercivity of thesystem versus the temperature. b)A typical hysteresis loopsused to calculate the coercivity of the system. c)Simulatedcoercivity vs temperature.

figuration of the V2O3) the data extracted from the sim-ulation shows that the change in coercivity is observed(Fig.2c) and the change in the rotation angle of the mag-netisation, when an external field is applied (Fig.4b), fol-lows the same trend as in the experimental data (Fig.4a).This indicates that assuming that the strain produced bythe phase coexistence in the V2O3 layer can explain thechange in the magnetic properties of the Ni film.

Apart from the original image measured using PEEM,additional images were taken into account in order togain additional statistical information. These extra im-ages were made cutting the original image in 9 squaresof the same size, rotating each piece randomly and com-posing an image of the original size from the pieces. Allthis process was carried out using a python script, and itwas motivated by the impossibility of making new mea-surements of the real system. The result showed that thecoercivity remains the same for all the created patternsstudied. This indicate the lack of long range correlationsin the formation of magnetic domains and that the onlyrelevant feature, for the coercivity enhancement, is thewidth and the shape of the metal/insulator domains inthe V2O3.

After all the process, the coercivity (HC) data for 5temperatures was successfully measured with an associ-ated statistical uncertainty that proved itself too low tobe noticed. The results are summarised in (Fig.2).

B. Rotation of the magnetisation under anexternal field

The above mentioned system, created with the sim-ulation of the hysteresis loops in mind, presented itselfas a valid tool for simulating other experiments with the

Ni/V2O3 heterostructure. In this case, we will be study-ing the rotation of the magnetisation as a function of thetemperature of the system under certain conditions. Thefollowing description corresponds to the simulation pro-cess; however, an analogous experiment was performedby the Nanomagnetic Materials group at Universitat deBarcelona. The data from these measurements are ingood agreement with the data extracted from the simu-lation, which supports the idea that the simulations wereusing reasonable parameters.

FIG. 3: Scheme illustrating the external field (red), the initialmagnetisation (blue) and the anisotropy directions (yellowand green).

FIG. 4: a)Experimental average rotation of the Ni spins ver-sus temperature across the SPT. b)Simulated average rota-tion of the spins. c)Simulated average rotation of the spinstaking as input the distribution of the magnetisation domainsmeasured in the Ni. d),e),f) Experimental measurements ofthe Ni magnetic domains using PEEM.

For this case, the initial magnetisation of the system

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Micromagnetic simulations of magnetoelastic heterostructures Javier Rodriguez Alvarez

FIG. 5: a)Fraction of spins versus the rotation angle with respect the initial magnetisaton. b)Equilibrium configuration of thesystem. Blue and red regions indicate magnetisation in the direction of the applied field and the high temperature anisotropy.c)Image of the metal/insulator domains in V2O3 using scanning near-field optical microscope (SNOM). Blue and red regionsstand for low (insulator) and high (metal) temperature phases across the SPT.

was set as the saturation magnetisation in a directionthat forms an angle of 45◦ with the ”x” axis (Fig.3). Aconstant magnetic field of 200 Oe was applied uniformlyin the system in the opposite direction to the initial mag-netisation (Fig.2a). Due to the effect of the magneticfield the magnetisation of the system rotated until thewhole mesh reached the stationary state. The behaviourof the system was greatly influenced by the competinganisotropies introduced to simulate the proximity of theV2O3. In particular, the magnetisation exhibited differ-ent configurations for each temperature.

In the simulated system, the Ni thin film was dividedinto 100x100x10 nm cells and the magnetisation of thesystem was computed for each of the 10000 cells formingthe mesh.The data of the average rotation angle of thespins with respect to their initial direction was calculatedindividually using a python script.

Evidence of phase coexistence has also been found, aspresented for T=176K in (Fig.5a,b). The distribution ofangles cover a wide range of values (from 10 to 60 degrees)and the image of the whole system contains regions ofhigh and low temperature magnetisation.

Apart from being able to accurately reproduce the con-ditions of previous measurements, additional aspects ofthe influence of the V2O3 layer in the behaviour of theNi have been studied through simulation. In order tosee with higher temperature resolution how the mag-netic domains in the Ni are imprinted by the differentstructural V2O3 configurations across the SPT, the samekind of simulations were carried out using as input forthe anisotropy distribution the experimental PEEM im-ages of the Ni magnetic domains (Fig.4d-f) instead ofthe near-field optical microscopy of the metal/insulatordomains in V2O3 (Fig.5c).

As can be seen in (Fig.4b-c), for high and low tempera-tures, the spins are oriented in the high temperature easyaxis and the direction of the field respectively. This be-haviour is due to the fact that the applied field is stronger

than the low temperature anisotropy but weaker than thehigh temperature anisotropy. The narrow nature of thepeaks and the global image of the spins confirm that forhigh and low temperatures the Ni layer presents a coher-ent magnetisation. Note that the tail in the right side ofthe angle distribution for T=179 is explained by consid-ering the remaining low temperature domains.

IV. ORDER PARAMETER

The change in magnetisation of the Ni due to the V2O3

SPT can be regarded itself as a phase transition. Consid-ering this, let us define an order parameter φ. The orderparameter will take values from -1 to 1, being equal to1 for the high temperature magnetisation (direction of

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Micromagnetic simulations of magnetoelastic heterostructures Javier Rodriguez Alvarez

the high temperature anisotropy) and -1 for the low tem-perature magnetisation (in the direction of the appliedfield). Thus, the order parameter will be

φ =1

N

N∑ai − bi (3)

where ai and bi are the absolute values of the projectionof each spin in the high and low temperature directionsand N is the total number of spins.

The behaviour of φ with respect to the temperature(Fig.6) is in good agreement with the data in (Fig.4b),which indicates that is an appropriate order parameter.Moreover, it is compatible with the phase coexistencebecause the standard deviation σ reaches its maximumin the middle of the SPF (Fig.6b). in good agreementwith the wide distribution of metal/insulator domains inV2O3 shown in figure (Fig.5c).

FIG. 6: a)Average order parameter versus temperature acrossthe SPT. b)Standard deviation of the order parameter versustemperature.

V. CONCLUSIONS

• The simulations implemented with OOMMF havesuccessfully reproduced the important microscopic

features of the Ni/V2O3 heterostructure studiedacross the SPT. Our results indicate that the straininduced at the interface with V2O3 is responsiblefor the reversible enhancement of the coercive fieldof the Ni layer.

• The data of the rotation angle under the presenceof a magnetic field provided by the simulationsalso reveals that magnetisation in the Ni film doesnot follow exactly the pattern induced by theV2O3. When the distribution of the anisotropyfollowed the configuration of magnetic domainsin the Ni (Fig.4c) the results did not match theexperimental data as well as they did when theanisotropy was introduced following the V2O3

phase configuration. Even though the high andlow temperature phases were in agreement withthe experimental data, the behaviour across theSPT was not reproduced.

• The spread in the distribution of the Ni magneti-sation angles is very wide around the V2O3 SPTtemperature and sharp far away from the transition(Fig.5a), in agreement with the phase coexistencein V2O3.

Acknowledgments

I would like to express my sincere gratitude to Aran-txa Fraile Rodrıguez and Amılcar Labarta, for havingbrought me the opportunity to discover such an interest-ing field and for their help throughout this final degreeproject.

I would also like to thank my family and friends fortheir always kind advise and patience.

[1] Vogel, et al.”Optically reconfigurable magnetic mate-rials” Nature Physics 11, pages 487491 (2015); DOI:10.1038/NPHYS3325

[2] de la Venta, et al.”Control of magnetism across metal toinsulator transitions” Applied Physics Letters 102, 122404(2013); DOI: 10.1063/1.4798293

[3] de la Venta, et al.”Coercivity enhancement in V2O3/Nibilayers driven by nanoscale phase coexistence”, Ap-plied Physics Letters 104, 062410 (2014), DOI:10.1063/1.4865587

[4] Fraile Rodrıguez et al, in prep

[5] D.N. Basov, et al.”Nanotextured phase coexistence inthe correlated insulator V2O3”, Nature Physics 13 80-86(2017), DOI: 10.1038/nphys3882

[6] Donahue, Michael Joseph and Porter, Donald Gene(2016), ”OOMMF: Object Oriented MicroMagneticFramework” https://nanohub.org/resources/oommf(DOI:10.4231/D3XS5JJ23).

[7] The criteria used to determine that the energy change isnegligible can be specified by the user.

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