J. Stat. Mech. (2021) 033211 ecnicas omica , ıa At´ 2 ıficas y T´ 1742-5468/21/033211+18$33.00 ıo Negro, Argentina and 5 ıa, CNEA-CONICET, Centro , 4 on Nacional de Energ´ ,NirvanaCaballero 4 ∗ , 1 cs, equilibrium and non-equilibrium ,JavierCuriale 3 ıo Negro, Argentina Along with experiments, numerical simulations are key to gaining omico Bariloche, Comisi´ eParis-Saclay,CNRS,LaboratoiredePhysiquedesSolides, stacks.iop.org/JSTAT/2021/033211 [email protected] omico Bariloche, Av. E Bustillo 9500, R8402AGP San Carlos de ıo Negro, Argentina At´ Bariloche, R´ 9500, R8402AGP San Carlos de Bariloche, R´ (CNEA), Consejo Nacional de Investigaciones Cient´ Ernest-Ansermet, CH-1211 Geneva, Switzerland 91405 Orsay, France (CONICET), Av. E BustilloR´ 9500, R8402AGP San Carlos de Bariloche, Instituto de Nanociencia y Nanotecnolog´ Instituto Balseiro, Universidad Nacional de Cuyo-CNEA, Av. E Bustillo Centro At´ Department of Quantum Matter , University of Geneva, 24Universit´ Quai 4 5 1 2 3 Vincent Jeudy Sebastian Bustingorry insight into theferromagnetic underlying systems. mechanisms However, a governing directtion comparison domain of between model wall numerical systems simula- motion andHere, experimental in we results thin still present representsdynamics a a of tuned great domain challenge. Ginzburg–Landau walls in model quasi to two-dimensional ferromagnetic quantitatively systems study with the Abstract. Received 24 September 2020 Accepted for publication 20Published January 15 2021 March 2021 Online at https://doi.org/10.1088/1742-5468/abe40a Pamela C Guruciaga E-mail:

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ferromagnetic materials Tuning Ginzburg–Landauto quantitatively study thin theory PAPER: Classical statistical mechani ©2021IOPPublishingLtdandSISSAMedialabsrl J. Stat. Mech. (2021) 033211 2 9 4 2 13 15 ], and 4 ]relatedto ]materials. 11 10 , 9 ...... prescription for thermal fluctuations, ]andferromagnetic[ 8 er of the material, and thermal activation. ature. They appear in systems as diverse as ], vortices in type-II superconductors [ 3 ineering of magnetic materials. ], ferrimagnetic [ 7 – ...... 5 ], earthquakes [ 2 ...... interfaces in random media, defects, dynamical processes, numerical , ...... 16 Tuning Ginzburg–Landau theory to quantitatively study thin ferromagnetic materials 1 ...... allowing us to perform material-specificuniversal simulations and features. at the We sameous time show recover experimental that velocity-field ouranisotropy data Pt/Co/Pt model in ultra-thin films the quantitativelywall in reproduces archetypal the perpendicular motion previ- three magnetic dynamicalanalysis (creep, regimes of of the depinning domain domainvide wall and detailed width nano-scale flow). parameter,statistical information showing In disordered that while model. our addition, retaining model the can we complex pro- Keywords: present behavior of a a statistical epniua antcaiorp.This model material and anisotropy. magnetic incorporates perpendicular exper- imental parameters and the micromagnetic simulations References 17 Acknowledgments 2.1. Ginzburg–Landau theory for2.2. magnetic Micromagnetic systems . theory ...... 52.3. . . . Linking . . the . . models...... 7 ...... 4 The latter, particularly, present very promising technological applications [ importance for the design and eng Having a theoretical approachand that allows accounts for for a the direct interplay of comparison all between these simulations ingredients and experiments is of great currents or magneticdepends fields. not The only thebetween moments, magnetic behavior pinning disord on of these the external velocity parameters in but these also systems, on however, the relevant interactions domain walls in ferroelectric [ the possibility of tuning domain wall motion with controllable parameters such as electric 5. Conclusions 4. Domain wall width 3. Domain wall dynamics https://doi.org/10.1088/1742-5468/abe40a Elastic driven interfaces arecontact ubiquitous lines in in n wetting [ 1. Introduction 2. From micromagnetism to the Ginzburg–Landau model 1. Introduction Contents J. Stat. Mech. (2021) 033211 3 ]. eq ζ 29 regime creep ,thevelocity d =0 transition. H ]. However, these T ψ T ∼ ) 4, which has been found T regime. The proportional- mulations are particularly / , d H ]. When considering disorder, ,weareabletosimulatethe =1 ( 1 flow v 30 µ ,andinthepresenceofquenched H regime, just above and ,wheretheuniversalcreepexponent µ β ) − the dimension of the interface and d H d H rials, with particular care of material and nics, a suitable model to study magnetic is scalar-field approach presents the same ]. Within this formalism, the domain wall ∝ ,thetwolattercharacterizingthevelocity namics for the order parameter can be con- − edynamics(e.g.precession)ofthemagnetic ) ψ 19 v al predictions are in good agreement for the depinning H – ( )with 12 and ∼ eq ζ β 0) − , µ ]. In the → ]. Although micromagnetic si (2 / T 26 36 , – – —called the field—below depinning which no domain wall 2) ]impliesacreepexponent d H 22 31 ( − 21 H , v Tuning Ginzburg–Landau theory to quantitatively study thin ferromagnetic materials d 9 , 0Afiieeprtr, thermallyactivated =0.Atfinitetemperature,a + 20 T eq 3[ ]. ζ / ]. This theory was originally proposed as a mean-field approach to con- 29 at higher fields, in what is known as the 29 =(2 =2 – H µ eq 27 ζ ∝ v Ageneraltheoryforthevelocity-fieldresponseinmagneticsystemswasderivedin In this work, we present a connection between micromagnetism and Ginzburg– Models such as the aforementioned elastic line and Ginzburg–Landau theory have From the standpoint of statistical mecha appears at low fields following the law ln( the system. Moreover, simplesidered, dissipative dy allowing to study time-dependent phenomena [ tinuous phase transitions, relating the order parameter to the underlying symmetries of https://doi.org/10.1088/1742-5468/abe40a the equilibrium roughness exponent.ical In value a one-dimensional interface, then, the theoret- experimentally as well [ When considering an applied externaldisorder, magnetic there field is a field ture is neglected, thusmoments ignoring that any compose possibl it. Whileis in simply a perfectly linear, ordered quenched system disorder the velocity-field is relation responsible for a much more complex behavior. the context of theis modeled elastic-line as model an [ elastic interface by considering solely its position; any internal struc- growth of magnetic domains and the concomitant domain wall dynamics, which can be incorporates thermalresults fluctuations in following a non-trivial the noise micromagnetic term. prescription, As shown which in figure Landau theory that allowsmeans us of to quantitatively a study tuned ferromagnetic scalar-field ultra-thin model. films by In particular, this material-dependent approach useful to describe the dynamicstaking of magnetic into textures account in the flowto contributions regimes obtain at of zero critical domain temperature, behavior wall is pinning not and straightforward. thermal fluctuations nection to experiments. Micromagneticinto the theory, physical on properties itsexperimental of turn, parameters magnetic provides [ mate important insight statistical models miss material-dependent characteristics, thus falling short in the con- motion can exist at is given by in the depinning regime [ non-linear response (creep, depinningin and experiments [ flow) for the velocity-field curve as observed the obtained domain wall dynamics using th the media [ domains is given by Ginzburg–Landau theory, with a proper inclusion of the disorder of presents universal power-lawFinally, behavior associated to the underlying proven useful to understandIn the universal particular, characteristics experiments ofvalues and domain wall theoretic of dynamics the [ critical exponents without quenched disorder. ity constant in this linear behavior is the mobility, and is the same than in the system J. Stat. Mech. (2021) 033211 4 (1) )is t , r 19 and ( . ξ for a two- =0 z ϵ =12 and ns, m δt )istakentorepresent t , r ( φ 45 T during . =0 z H 0 µ s. While the white strip corresponds to the ation. In the limit of strong perpendicu- µ ldynamicsinultra-thinmagneticmaterials 30Kbfr n fe pligan out- applying after and K=300 before T =+1,grayrepresentsitsgrowthafterthepulse. =10 z ], this generalized order parameter follows the m δt 37 )at 3 1. ). We use previous experimental velocity-field data in 3 − = ]totestourmaterial-dependentmodel.Notonlywefind ξ. z 22 + m ory for magnetic systems Tuning Ginzburg–Landau theory to quantitatively study thin ferromagnetic materials 007 T during GL . δφ in section H Spatial distribution of the out-of-plane magnetization δ =0 1 Γ z − H 0 = µ φ 3n,seesection =30nm, t ∂ (b) Black stands for Figure 1. dimensional system with quenched Gaussian Voronoi disorder ( initial relaxed domain with ℓ of-plane magnetic-field pulse of (a) 2. From micromagnetism to the Ginzburg–Landau model with perpendicular . wall width fluctuationsexperiments. In this in way,perform of our simulations numerical this domain wal tuned Ginzburg–Landau system, model which shows great are versatility to typically not exposed by PMOKE Pt/Co/Pt ultra-thin films [ good agreement in thesal three features regimes as of domain the wall creep motion, exponent. but In we addition, also we recover report univer- new results regarding domain https://doi.org/10.1088/1742-5468/abe40a evolution equation the out-of-plane componentlar of anisotropy the and magnetiz strong damping [ symmetries of theSince problem we are and interesteddicular considering in easy-axis studying anisotropy, the the the interactions non-conserved case scalar via of field a effective ferromagnetic quantities. thin film with perpen- As mentionedGinzburg–Landau in theory can the be used previous to section, study magnetic from systems by a relying statistical on the physics standpoint, 2.1. Ginzburg–Landau the directly compared withwill polar return magneto-optical to Kerr figure ffect e (PMOKE) experiments (we an additive, uncorrelated white noise that represents thermal fluctuations and satisfies Here, Γis a damping parameter that sets the time scale of the problem, and J. Stat. Mech. (2021) 033211 . 5 (3) βα ,the 2 is the β √ γ 3) / ,(5) 1) by means of a ( ), (2) ) ′ ± t th . = r − H d t φ ,itsphysicalinterpreta- ( + φ δ ]. Nonetheless, the lack of yields domain walls with ) eff ′ 1. In order to promote this, ! 39 r ], the free-energy Hamilto- φ H – model’ and consists of three h ( ! per se − 29 4 | 37 – − r × φ φ , ( | r ,(4) ]. Taking all these ingredients into 27 ˜ ξ d 29 M Tδ 28 is the vacuum permeability, – S # + 0 0 4 η % M 4 27 µ φ 2 φ Γ =2 + tunsuitableforquantitativecomparisons ⟩ ]andreferencestherein)itisgivenby )+ − way to model the evolution of the magneti- ) anisotropy. A ferromagnet is a system that 2 taining a great number of atomic magnetic ′ th 2 .Thelastterm,onitsturn,amountstothe t 1 φ 40 , ,whereweestablishtheconnectionbetween $ α ′ H perpendicular to the film. − ) r ( h " 2.3 + h ξ and a domain wall energy equal to (4 ) + eff t ! , H α r ( ( αφ ]. As discussed in [ α β/ ' ξ + anisotropy (favoring the values 2 ⟨ ], the equilibrium value of 30 r × +( & d ought of as composed by cells of a given volume which is 30 agnetization implies that is the saturation magnetization. While theffective e field is φ does not need to be bounded 2 M 2 | S 2 φ φ 0 2 ∇ 0 η Tuning Ginzburg–Landau theory to quantitatively study thin ferromagnetic materials M )intheevolutionequation[ β |∇ 2 and micromagnetism. γµ φ = 1+ ! =0 and − φ − β t ⟩ ) ∂ (1 t = = h , Γ) r )ineachcellbyfollowingthestochasticLandau–Lifshitz–Gilbert(SLLG) / ( GL M t t ξ 3canbesubtractedfromtheexternalfieldcontribution,resultinginaterm , Γ. As shown in [ ⟨ H (1 ∂ / r can be modeled by following the modified ‘ 3 is the temperature [ ( ξ/ is the adimensional damping constant, 0 GL hφ = T M η ˜ ξ H With some variations, this model has been widely used to generically model the Although the range of proportional to magnetization in quasi two-dimensional systems [ this scalar-field model https://doi.org/10.1088/1742-5468/abe40a moments. Then, micromagnetism provides a zation equation. In the Landau formulation (see [ presents a net magnetization (thatof is, external magnetic field. moment per Itlarge unit can compared volume) be to in the the th absence atomic scale, and con In this section, wenetic present thin the film micromagnetic with approach perpendicular to magnetic the dynamics of a ferromag- 2.2. Micromagnetic theory where gyromagnetic ratio and inclusion of an external magnetic field tion as a component of the m While the first termsecond incorporates one the represents rigidity easy-axis two-well of potential the system with with an elastic energy stiff ness barrier contributions: material and experimental parameterswith make real i systems. These results will be useful in section account, the Langevin equation for the modified Ginzburg–Landau scalar-field model is where with aterm nian awidthparametergivenby J. Stat. Mech. (2021) 033211 , ) 6 5 A (11) ]. 41 ,(9) . r % x )d the volume r m =1). ( 2 V | ∇ m y · m | ]. However, for a discrete z m ˆ e 31 ) − z ! y m S y m 2 )(7) (known as the anisotropy M m ′ z t ∇ S x − H − 0 0 m t µ $ ( /η δ 0 x % K/M − a component of the magnetization 2 z η ij r m δ z ( m ,and d − =2 κλ 2 y z * − , f )] k 2 y r ) Dδ 1 , , z ( S $ x )+ ,andZeemancouplingtoanout-of-plane ) m m z of the system by the relation z formalism introduced above is adapted to ndent from each other, as well as the three =2 ,(10) · = K ∇ ectivefield,equation( ff )tocalculatethee (with the norm constraint m ⟩ H z 6 H ) x j N A/M ′ z ˆ e , [ t + ˆ +( e i m ( + z z 2 λj =2 ! % ) − m ), and the norm constraint. Within this formalism, 2 z y f the temperature of the system and 0 a ) km K t m + m 10 ( T /η y − y ∇ − κi +( ˆ e .Thefreeenergyofthesystemisgivenby f r y z 1 m z ⟨ d ( $ +( m m 2 x z H | 2 2 0 f f ) ) + x µ ∇ r ,(6) x ( is a random vector field that represents thermal noise. The + + ,(8) a m ˆ e Tuning Ginzburg–Landau theory to quantitatively study thin ferromagnetic materials 2 0 z H x m M = µ ∇ δ ˆ e δ otropy with strength T S ( z z m |∇ B ) f − k z m = t =0 and 0 ! + = η ∂ S ⟩ y 2 0 γVM ) 0 am A ,Cartesiancoordinates t ˆ eff e η η λ ( y 0 /M = = = f H κi 0 f γµ M 1+ D ⟨ H µ N + ,analogoustoequation( and 6 x y = component of the magnetization can be written ˆ e the Boltzmann constant, κ m x z f m B k = and In this work, the general micromagnetic Following the prescription of equation ( th x It is not straightforward to apply this methodology to the continuous theory of micromagnetism [ in a quasi two-dimensionaleffective system. easy-axis We anis consider exchangemagnetic interactions field with offfness sti intensity study the magnetization dynamics inWe ultra-thin films are with dominant interested uniaxial anisotropy. in studying the evolution of the H https://doi.org/10.1088/1742-5468/abe40a 6 derived from the free-energy Hamiltonian where formulation like ours, the thus defined random field has been shown to correctly reproduce equilibrium thermodynamics [ components, and they arewhite uncorrelated noise in are time. then The statistical properties of this vector fluctuating terms of dierentff cells are indepe where for simplicity we have defined of the cell. with m The full micromagnetic description is comprised of the other two coupled equations for field), and for cells for the J. Stat. Mech. (2021) 033211 z 7 7 . m (c)) (12) that ∇ 2 AK θ apriori √ by writing =4 ]). =0)isequal (a), we do so ε .Onthecon- 42 z 2 0 is significantly m m ;seefigure d using one single /η (see [ π 1 H z K ∝ m v =0, θ ( | ,themicromagneticformu- ) ) z z 1 eel ( ˆ ˆ e e z z .Toachievethis,weuncou- m m in each cell. Indeed, although we ( ( 2.1 z (a), the proposed parametrization 2 (b)), N´ m ). As shown in figure 2 ∇ y ∇× that only one evolution equation—in |∇ × , θ x to ( z xy eparametrizationimpliesdeciding m ]. For a driving field lower than the so-called m dbydomainwallwidthparameterequalto +sin ≡ ). For most experiments (as those reported in 42 | of a domain wall in the system: any finite 2 0 z z z η 2; see figure m m m )andadomainwallenergygivenby 0 π/ ∇ |∇ ∆ is decomposed in one component in the direction of (1 + θ / π 2, 3 0 η m π/ cos ∝ ' = 2 z v θ m Tuning Ginzburg–Landau theory to quantitatively study thin ferromagnetic materials ), the orientation of the magnetization within the domain wall − ,butitwillallowustolinkthemicromagnetictheorywith with W —is needed to describe the of magnetic properties the system. The as a function of 1 z .Inthecasewhereadomainwallispresent,theanglebetweenits and one component orthogonal to it. Note that fixing the value of y 2.2 W + ,thuslocallyfixingthedirectionofthein-planemagnetization.In )fromtheonesfortheothercomponentsof xy m z ˆ e θ y magnetization inside the domain wall precesses during the motion. The H )isusedtoparameterizethemagnetizationintermsofanangle ,soabovethecreepanddepinningregimesonlythelinearasymptotic = m 10 m m W W 12 H