Magnetic islands modelled by a phase-field-crystal approachgrant/Papers/EPB2018.pdf · approach to study microstructure formation occurring at atomic length scales and di↵usive

Eur. Phys. J. B (2018) 91: 55https://doi.org/10.1140/epjb/e2018-80543-9 THE EUROPEAN

PHYSICAL JOURNAL BRegular Article

Magnetic islands modelled by a phase-field-crystal approachNiloufar Faghihi1,a, Simiso Mkhonta2, Ken R. Elder3, and Martin Grant1

1 Physics Department, Rutherford Building, 3600 rue University, McGill University, Montréal, Québec H3A 2T8,Canada

2 Department of Physics, University of Swaziland, Private Bag 4, Kwaluseni M201, Swaziland3 Department of Physics, Oakland University, Rochester, MI 48309, USA

Received 26 September 2017 / Received in final form 9 December 2017Published online 26 March 2018 – c� EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2018

Abstract. Using a minimal model based on the phase-field-crystal formalism, we study the coupling betweenthe density and magnetization in ferromagnetic solids. Analytical calculations for the square phase in twodimensions are presented and the small deformation properties of the system are examined. Furthermore,numerical simulations are conducted to study the influence of an external magnetic field on various phasetransitions, the anisotropic properties of the free energy functional, and the scaling behaviour of the growthof the magnetic domains in a crystalline solid. It is shown that the energy of the system can depend onthe direction of the magnetic moments, with respect to the crystalline direction. Furthermore, the growthof the magnetic domains in a crystalline solid is studied and is shown that the growth of domains is inagreement with expected behaviour.

1 Introduction

Predicting the properties of real materials can be di�-cult, as such materials are produced by various processingmethods. These lead to topological defects and multiplegrains that control mechanical, thermal, electrical andmagnetic responses. Important phenomena such as plas-ticity, hysteresis, work hardening and glassy relaxation arestrongly a↵ected by these microstructures. Understand-ing the formation of these complex microstructures ischallenging due to the non-equilibrium nature and theirmultiple length scales.

Processes on mesoscopic length scales and di↵usivetime scales, occurring in microstructure evolution, can bemodeled using continuous phase fields [1–3], but detailson atomic length scales are averaged out. A successfulapproach to study microstructure formation occurringat atomic length scales and di↵usive time scales makesuse of phase-field-crystal (PFC) method [4,5]. The PFCapproach is an extension of the phase-field methodsto produce structures that are periodic in space. Assuch, features such as elasticity, dislocations, anisotropy,grain boundaries and polycrystalline structures naturallyemerge from PFC free energy functionals [6]. The phase-field-crystal formalism encompasses elastic and plasticdeformations in crystals and has been used to studyphenomena such as grain growth, epitaxial growth, mate-rial hardness, crack propagation, grain boundary melting,glass formation, amongst other phenomena [7]. It hasbeen shown that the PFC formalism can be crudely

a e-mail: [email protected]

derived from a classical density functional theory (DFT)of freezing [8,9] and incorporates the essential physics offreezing [10].

In its simplest formulation, the PFC free energy isdominated by a single set of modes of the same length. Theone-mode model truncates the number-density expansionin the Fourier space to one set of reciprocal lattice vec-tors. Higher modes are assumed to have much smalleramplitudes at high enough temperatures. This simplefree energy gives rise to the following equilibrium statesin three dimensions: liquid, bcc, hcp, and stripe [11].Recently, variants of the PFC free energy have been devel-oped to address more complex crystal symmetries andtheir coexistence with the bulk liquid. One such approachintroduces a class of multipeaked two-point correlationfunctions in the free energy functional [12–14]. Otherapproaches have been formulated by coupling di↵erentsets of crystal density waves corresponding to di↵erentreciprocal lattice vectors [15–17]. Using this mechanism,it has been possible to produce di↵erent Bravais latticesand other structures, including honeycomb and kagomeordering in two dimensions [17].

The PFC method has been exploited to study therelationship between magnetic and elastic properties inferromagnetic solids. The magnetic PFC model introducedin references [18,19] contains a new coupling term whichconnects the free energy of the phase-field-crystal model tothe Ginzburg-Landau free energy of ferromagnetic phasetransition. The phase diagram of the model was calcu-lated and the magneto-elastic e↵ects and the e↵ect of thegrain boundaries on the magnetic hardness were stud-ied. It was shown that the grain boundaries facilitated

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the formation of the magnetic domains and decreased themagnetic hardness. It was noted that in order to incor-porate anisotropy into the model – i.e., the dependenceof magnetic properties on the crystallographic directions– higher degrees of coupling of the magnetization to thegradient of the density were required. In reference [20],the magnetic coupling was extended to higher degrees ofthe coupling. In addition to anisotropy, that work extendsthe magnetic PFC model to incorporate polarization andconcentration fields.

In this paper, we introduce a model system where themagnetization regions are coupled to atomic sites. Thesemagnetized regions which are co-located with the localphase-field crystallinity are called magnetic islands in thispaper. This is unlike the previous work [18–20] wherethis coupling was not considered and the magnetizationlength scale spanned over several atomic length scales.In this work, the system can exhibit arrays of magneticislands that are arranged into a crystalline structure thatis mediated by the density field and thus the magneti-zation length scale conforms to the atomic length scale.Our work is partly motivated by experiments on artifi-cial spine-ice systems [21,22]; these systems are arrays offerromagnetic islands on the nanometer scale which arefabricated lithographically [23,24].

2 Model

Our free energy is a combination of the two-mode phase-field-crystal formalism [6,15,17,25] for the crystal growthand the classical Ginzburg-Landau formalism for ferro-magnetic ordering, FGL [26]. A coupling term connects thedensity and the magnetization, such that natural prop-erties of the magnetic crystals, e.g. magneto-crystallineanisotropy and magnetostriction e↵ects, are included.The free energy terms below are written in dimen-sionless units. The two-mode PFC free energy is givenby

FPFC =

Zdr

⇢

2

⇥ �q20 +r2

�2+ b0

⇤⇥ �q21 +r2

�2+ b1

⇤

+r

2 2 +

4

4

�, (1)

where (r) is the particle number density di↵erence atpoint r defined to be (r) = (⇢(r) � ⇢̄)/⇢̄, and ⇢̄ is thereference density, taken to be the density of the liq-uid at coexistence [3]. The parameter r is related tothe temperature of the system and the parameters bicontrol the relative stability of di↵erent modes. For thetwo-mode-PFC model, in order to obtain stable hexag-onal and square phases, q1/q0 =

p2. This is based on

the fact that to get a square phase we need to couplethe h10i and h11i reciprocal lattice vectors. The gradi-ent terms in this free-energy functional are related tothe multi-peak pair correlation functions of the classicaldensity functional theory of freezing where the peaks arelocated at wave numbers qi. The Ginzburg-Landau free

energy is

FGL =

Zdr

⇢W

20

2|rm|2 + rc

|m|2

2

+�|m|4

4�m ·H

�, (2)

where W0 is the exchange correlation length and rc and �are the phenomenological constants that control the bulkbehaviour of magnetization. The constant rc is relatedto the Curie temperature at which the ferromagneticphase transition occurs, and H is the external magneticfield. Note that all of the bold-lettered symbols in theequations represent two-dimensional vectors.

The coupling term is

FC =

Zdr

n�↵m2 � ✏

2(m ·r )2

o. (3)

The first term is minimized when the magnetization haspeaks on atomic sites and becomes negligible betweenthose sites. This term therefore induces magnetic islandsat atomic sites where > 0. The second term gives rise tomagneto-elastic coupling, and hence the magnetostrictione↵ect [18–20].

To minimize the free energy, we solve the dynamicalequations of motions based on the dissipative dynamics.We exploit conserved dynamics for the density n and non-conserved dynamics for mx and my:

@n

@t= r2

✓�F

�n

◆, (4)

@mi

@t= � �F

�mi, (5)

where i = x, y. We use a pseudospectral algorithm tosolve the equations of motion with periodic boundaryconditions in two dimensions.

3 Results

3.1 Phase diagram

It is known that the application of a magnetic field canbe used to control a material’s microstructure. To studythe influence of an external magnetic field on the phasediagram of our system, we ran simulations at di↵erenttemperatures r and mean densities. Parameters of thefree energy were chosen to be (b0, b1,W0, rc, �,↵, ✏) =(0, 0.1, 0.5,�0.1, 1, 1, 0.1). The parameters are chosen tobe representative, so that a wide range of behaviors can beobserved, and, in particular cases, for numerical stability.

We chose these values for b0 and b1 for simplicity. Theparameter W0 controls the magnetic domain wall width.This value for W0 is chosen to be comparable to the lat-tice constants q0 and q1. The parameter rc must havea negative value for the magnetic domains to form. Theparameter � is related to the saturation magnetization andthe magnetic susceptibility of the system. Knowing the

Eur. Phys. J. B (2018) 91: 55 Page 3 of 7

Fig. 1. Phase diagram in terms of the temperature and meandensity for H = 0 (a) and H = 0.2x̂ (b).

experimental values of the saturation magnetization andmagnetic susceptibility for a specific system, one can inprinciple calculate �. The parameters ↵ and ✏ are couplingcoe�cients. We chose ↵ in such a way that the magneti-zation is conveniently localized to the atomic sites. Thevalue of ✏ is related to the magnetostriction coe�cient.

We found that one of the following phases was stabledepending on the values chosen for r and the mean den-sity: uniform, hexagonal, square or striped. We started oursimulation from an initial condition consisting of a slabof hexagonal phase surrounded by two square crystals.In this way we tracked the competition between hexago-nal and square ordering, and determined which phase wasmore stable. We ran simulations for t = 105 time-stepsand calculated the free energy of the system to determinewhen the system reached its stable state.

Fig. 2. (a) The initial configuration of the system. (b) The con-figuration of the system after 105 time-steps when the externalmagnetic field H = 0. (c) The configuration of the systema short time after an external magnetic field of H = 0.2x̂ isapplied to the configuration (b). (d) The configuration of thesystem after the whole system is filled by the square crystalunder the application of an external magnetic field.

In the presence of an external magnetic field, we foundthat the equilibrium phase boundaries shifted. At thetemperatures and densities where the uniform phase wasstable, the square phase was now stable. Phase diagramsof the system are presented in Figures 1a and 1b forthe cases where the external magnetic field H = 0 andH = 0.2x̂, respectively. Comparing the two phase dia-grams, the phase shifts are apparent. Note that the phasediagrams are asymmetric with respect to the density. Thereason is the existing of the odd order term of �↵m2 in the free energy density (Eq. (3)). Since the phase dia-gram is asymmetric, m2 6= 0 which means that the phaseboundaries in Figures 1a and 1b are well below the Curieline. It should be mentioned that in principle it is possi-ble to calculate the position of the Curie line in the phasediagram, using the methods described in [18,19].

It follows that our model recovers the feature that anexternal magnetic field can give rise to a phase transfor-mation in crystalline solids [27,28]. In fact, di↵erent phasesof the material have di↵erent magnetic susceptibilities(and dielectric constants). Thus the magnetic or electricfields can modify the Gibbs free energy and thereby thecircumstances under which the phases are stable [29].

We now consider how the magnetic field can induce aphase transition. We ran simulations at r = �0.3, andmean density of �0.4, for H = 0, and started from a smallsquare seed at the centre of the simulation box. The sys-tem equilibrated to the uniform state after 105 time-steps(Fig. 2b). We then applied an external magnetic field ofH = 0.2x̂ and observed that the square phase started togrow from the centre of the box (Figs. 2c and 2d). The

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configurations of the system at di↵erent time-steps of thesimulation are shown in Figure 2.

3.2 Magneto-elastic calculations

The change in the dimensions of a ferromagnetic mate-rial in the presence of an external magnetic field is calledmagnetostriction. To study magnetostriction, we considera small deformation in the density field in the presence ofa uniform magnetization. For simplicity we will considera square lattice with b0 = 0 and q0 = 1. We first expandthe density in terms of the Fourier modes for a square lat-tice to determine the equilibrium lattice constant. For asquare lattice the density takes on the form,

(r) = 0 +8X

i=1

aieiq·r

, (6)

with q1 = h1, 0iqe, q2 = �q1, q3 = h0, 1iqe, q4 = �q3,q5 = h1, 1iqe, q6 = �q5, q7 = h�1, 1iqe, q8 = �q7. Weconsider the case where modes with the same wavenumberhave the same amplitudes ai = A1 for i = 1, . . . , 4 andai = A2 for i = 5, . . . , 8. In the limit that b1 goes to zerothe free energy is minimized by qe = 1. Assuming qe =1 + � and expanding in � it is possible to show that thefree energy is minimized by

� = � b1A22

A21(1 + b1) +A

22(4 + 5b1)

, (7)

to leading order in �.To obtain the strain energy in the system we con-

sider a small deformation, u(x, y) in which in terms ofamplitudes can be written (using qe = 1 + � and Eq. (7))

aj = �eiqj ·u, (8)

where as before � = A1 for j = 1 . . . 4 and A2 for j =5 . . . 8. Substitution into the free energy and expanding tolowest order gives (ignoring terms independent of u)

felas = �q2✏(A21 + 2A22)(Uxxm2x + Uyym2y + 2Uxymxmy)

+C11

2

⇥U

2xx + U

2yy

⇤+ C12UxxUyy + 2C44U

2xy

+2K44W2xy, (9)

where felas is the free energy per unit area and the elasticmoduli are,

C11 = 8(A21(1 + b1) +A

22(2 + 3b1)

C12 = 16A22(1 + b1)

C44 = 4(4 + 5b1)A22

K44 = 4A22b1. (10)

The Uij and Wxy are the standard and asymmetric straintensors respectively, i.e.,

Uij =1

2

✓@ui

@xj+@uj

@xi

◆(11)

and

Wxy =1

2

✓@ui

@xj� @uj@xi

◆. (12)

These results are in agreement with the results ofWu et al. [15] in that the shear modulus vanishes if onlyA1 is included in the analysis. It should be noted that Wxygoes to zero in the limit b1 = 0.

The magnetostriction e↵ect is obtained by minimizingfelas with respect to Uij and Wxy which gives,

Uxx = ✏q2 (A

2 + 2B2)(C11m2x � C12m2y)C

211 � C212

Uyy = ✏q2 (A

2 + 2B2)(C11m2y � C12m2x)C

211 � C212

Uxy = ✏q2 (A

2 + 2B2)mxmy2C44

(13)

and Wxy = 0. The expressions in equation (13) imply thatthere is a direct coupling between strain and magnetiza-tion: an external magnetic field can be used to changethe value and direction of magnetization vector, m, and,therefore, the field can deform a ferromagnetic sample toa specific shape. This is known as the magnetostrictione↵ect.

3.3 Anisotropy

In this section we consider whether the magnetic prop-erties of the system depend on the crystalline direction.We develop a method to examine whether the energy ofa perfect crystal depends on the direction of the magneti-zation. The simulation box was constructed by preparinga single unit cell containing one atom. Then we placed amagnetic moment on the single atom by turning on thecoupling term of f = �↵m2 .

Magnetic anisotropy can be incorporated into the modelby adding even powers of the term (1/k)(m · rn)k, inagreement with the symmetries of the system. It has beenshown theoretically that to obtain D-fold symmetry kshould be at least of order D [20]. This is a necessary butnot su�cent condition to have D-fold anisotropy. It turnsout that we require a quartic anisotropy for square symme-try through the terms (✏2/2)(m ·rn)2 + (✏4/4)(m ·rn)4,and a six-fold symmetry of a hexagonal lattice requires anadditional sixth-order term.

To confirm that this model is capable of producingmagnetic solids in which the properties of the magneticsystem depend on the crystallographic directions, we runsimulations and we set only the initial direction of themagnetization. We update both density and magnetiza-tion and observe that the direction of the magnetization

Eur. Phys. J. B (2018) 91: 55 Page 5 of 7

Fig. 3. Manifestation of anisotropy in the energy of a unitcell. (a) The energy of a square unit cell as the angle of themagnetization with respect to x axis is modified from 0� to360�. (b) The energy versus angle plot for a hexagonal unitcell.

remains fixed during the simulation. We set ↵ = 1 and forthe square lattice we set (✏2, ✏4, ✏6) = (10, 107, 0) and forthe hexagonal lattice we set (✏2, ✏4, ✏6) = (1, 102, 108). Weran the simulation for N = 105 time steps and calculatedthe energy of the system. Then we rotated the directionof the magnetic moment by �✓ = 10� and calculated theenergy. We repeated this for di↵erent angles of the mag-netic moment with respect to the x axis. The energiesof the system as a function of the angles are depicted inFigure 3a for the square unit cell in Figure 3b for thehexagonal unit cell. It can be seen that for the square thepeaks of the free energy happen at 45�, 135�, 225�, 315�

which is corresponding to the symmetries of the squarephase. For the hexagonal phase, the peaks in the energyhappen at 0�, 60�, 120�, 180�, 240�, 300�, 360� which isagain in agreement with the symmetries of the hexagonalphase.

Fig. 4. The configuration of the magnetization when there isno coupling between the density and the magnetization (a),and when we turn on the coupling term by setting ↵ = 1 (b).

The values of ✏i, i = 2, 4, 6 were chosen based on thefollowing: as the magnetization vector rotates, the energydi↵erences for di↵erent angles are very small; therefore, weincrease the values of ✏i to have more pronounced energydi↵erences. On the other hand, if the values of ✏i are toolarge, the localization of the magnetization on the atomicsites as well as the direction of the magnetization withrespect to the x axis deviate from the correct configura-tion. Thus the values of ✏i were chosen so that we haveas large an energy di↵erence as possible and the rightphysical configuration for the magnetization vectors.

Page 6 of 7 Eur. Phys. J. B (2018) 91: 55

Fig. 5. The growth of the magnetic domain in a crystal ofhexagonal symmetry. A domain of magnetization m = 0.1x̂(red) is located at the centre of the simulation box in (a). (b–d)The location of the domain boundary as the time evolves. Itcan be seen that the red domain grows until it fills the wholecrystal.

3.4 Breaking the rotational symmetry

The magnetic free energy together with the coupling tothe density gives rise to a rotationally-symmetric xy modelwhich has peaks on the atomic sites. If, in the free energyof equation (3), we set ↵ = 0 and ✏ = 0 the magnetiza-tion vector would be independent of the density and theconfiguration of Figure 4a is obtained. On the other hand,if we set the parameters ↵ = 1, i.e., to a non-zero value,then the vectors will be localized on the atomic sites andthe configuration of the Figure 4b is obtained.

By setting an initial condition for magnetization thatbreaks the rotational symmetry of the xy model, we canmake the domains move by application of an externalmagnetic field. We set the free energy parameters as men-tioned in the first section except that we set (↵, ✏) = (1, 0).We initiated the magnetization as m = 0.1x̂ in a cir-cle centered at the middle of the box and m = �0.1x̂in the rest of the box. Application of an external mag-netic field of H = 0.003x̂ moved the boundary betweenthe domains. The configuration of the magnetization isshown in Figure 5. The colors show the boundary betweenthe spin left and spin right. It is possible to analyze thescaling behaviour for the growth of the domains. From theKardar-Parizi-Zhang equation it can be shown that in thepresence of a magnetic field, H, the radius of a droplet,R, follows the equation [30,31]

dR

dt= H � �

R, (14)

where � is the surface tension. This equation is derivedfrom the KPZ equation for the time scales before anynucleation would take place, and fluctuations and bub-bles (multiple nucleation sites) are neglected [30]. Onecan solve this di↵erential equation to obtain Ht = R +L ln(1 � R/L) + c, where L = �/H, and c is a constant.

Fig. 6. Scaling behaviour of the shrinking and the growingmagnetic domain. (a) A droplet of m = 0.1x̂ is placed inthe box of m = �0.1x̂, and H = 0, so the droplet shrinkswith time. (b) Droplet of m = 0.1x̂ is placed in the box ofm = �0.1x̂ and an external magnetic field of H = 0.0022x̂ isapplied. As a result the droplet grows with time.

The value of c can be obtained from the size of the dropletat time t = 0: c = �R(0)� L ln(1�R(0)/L).

It follows that, for the case of L ! 0, R = Ht+ c. Forthe case of L ! 1 or H ! 0, R2 = R(0)2 � 2�t. Wecalculated the radius of the droplet as a function of timefor two cases. The first case is when the external magneticfieldH = 0. In this case the droplet shrinks with time untilit disappears. Figure 6a shows the fit of the equation forR in the case of L ! 1, i.e., R2 = 2�t+ c in a log-scaleplot. Figure 6b shows the fit of the data from a growingdroplet in the presence of an external magnetic field ofH = 0.0022 to the equation obtained in the presence of anexternal magnetic field, i.e., Ht = R+ L ln(1�R/L) + c,in a log-scale plot as well.

4 Conclusion

We combined the two-mode phase-field-crystal approachand the Landau-Ginzburg free energy for ferromagneticphase transition to study the influence of the magneticfield on the di↵erent phases produced by the two-modephase-field-crystal free energy, i.e., uniform, square and

Eur. Phys. J. B (2018) 91: 55 Page 7 of 7

hexagonal phases. It is shown that magnetic field caninduce a phase transition from a uniform to square phase.By expanding the density field in the Fourier space, weanalytically calculated the magneto-elastic strain energyof the system. Furthermore, by adding powers of the term1/k(m · rn)k, we examined the existence of magneticanisotropy in the system, by studying whether the energyof the system depends on the direction of the magneticmoments with respect to the crystalline direction and wefound that the energy changes periodically with angle ofthe magnetization relative to the x axis. We also stud-ied how magnetic field can e↵ect the domain growth in acrystalline system and showed that the growth rate is inagreement with the Kardar-Parizi-Zhang equation.

This work was supported by the Natural Sciences and Engi-neering Research Council of Canada and by le Fonds derecherche du Québec – Nature et technologies. We also acknowl-edge Compute Canada, particularly Sharcnet, for computingresources.

Author contribution statement

The authors all contributed on the development andanalysis of the model.

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