Degradation of domains with sequential field application
DDegradation of domains with sequential field application
N. Caballero
Department of Quantum Matter Physics, University of Geneva,24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland (Dated: September 2,2020)Recent experiments show striking unexpected features when sequences of alternating magneticsquare field pulses are applied to ferromagnetic samples: domains show area reduction and domainwalls change their geometrical properties. In this work, we use a very simple scalar-field model, inwhich no physical quantities need to be specified a priori , and which only considers two preferentialvalues for an order parameter, short-range exchange, disorder, and temperature. By proposing anumerical protocol that mimics the experimental one used to observe domains with polar magneto-optic Kerr effect microscopy, we show that domains described by the model are also subject toarea reduction under sequential field application. We study two domain geometries common inferromagnetic films: a bubble and a stripe domain. In both cases, the area reduction as a functionof the number of alternating field cycles follows a linear combination of an exponentially decreasingfunction and a linearly decreasing function, as reported for the experiments. We characterize thedomain walls geometry by computing its roughness exponents. The obtained roughness exponentsare indistinguishable from the ones observed experimentally, not only under alternating fields, butalso when the standard protocol to measure velocities is numerically emulated.
I. INTRODUCTION
Ferromagnetic domains are extensively used as mem-ory units to store information [1], from the initial emer-gence of magnetic-bubble-based devices [2], to more re-cent breakthroughs in racetrack memories based on do-main wall and magnetic skyrmions [3]. Controlling suchdomains and understanding their behaviour is thus ofgreat applied as well as fundamental interest. A synergis-tic interplay between basic research and technological de-velopment has led to advanced knowledge of how to effi-ciently create and manipulate magnetic domains [4–6], inparticular with fixed and alternating magnetic fields [7].However, several key aspects of domain behaviour remainless well understood, and from a basic physics view point,these objects still present striking features which appeardifficult to elucidate.In particular, the reaction of ferromagnetic domainsto sequential application of different magnetic fields hasnot been extensively explored [8]. Recent experiments inultra-thin ferromagnetic films with perpendicular mag-netic anisotropy revealed an unexpected domain dynamicresponse under the application of alternating magneticfields: Polar magneto-optical Kerr effect microscopy im-ages showed that initially circular domains evolved to-wards distorted domains with irregular shape and con-comitant domain area reduction [8].Motivated by these experimental results, in this workwe use a scalar-field model [9–11] to numerically emu-late the experimental protocol. The advantage of us-ing this model is that no material specific parametersneed to be a priori defined, and it allows us to studyin detail how domains and domain walls behave underfield application. It was already shown that the numeri-cal emulation of the typically used experimental protocolto measure domain walls velocities by polar magneto-optic Kerr effect microscopy produces velocity-field re-
FIG. 1.
Protocol to study domain walls statics and dynam-ics numerically : We emulate typical protocols used in polarmagneto-optic Kerr effect microscopy. First, we imitate thenucleation process by creating a domain of radius R i . We ap-ply a field to let the domain grow until it reaches an effectiveradius R . After letting the system evolve at zero field fora time t (relaxation process), we apply N DC cycles followedby N AC cycles. The applied field is shown on the top figure,while in the central figure we show the effective radius of adomain as a function of time. The green dots indicate thetimes at which the images of the bottom where taken. Eachimage on the bottom is of size L × L . In the shown case, T = 0 . (cid:15) = 1, t = 10 , τ = 10 , L = 4096 (see text forunits description). sponses with characteristics of experimentally measuredcurves [11]. These curves are typically characterized bythree dynamical regimes as a function of the field: a flow a r X i v : . [ c ond - m a t . d i s - nn ] S e p regime where the velocity grows linearly, a depinningregime with signatures of the zero-temperature behav-ior, and a creep regime where the velocity grows expo-nentially [12]. In the low-field creep regime, which onlyemerges in presence of disorder, the ultra-slow motionof domain walls occurs trough thermal activation. Acti-vated events that involve collective reorganizations, trig-ger the domain wall and as a result the interface showsa non-zero average velocity. This velocity is highly non-linear and displays a stretched exponential behavior asa function of the applied field H : ∼ e − H − µ , where µ isa universal exponent [13]. In the case of ferromagneticultrathin films, µ (cid:39) / ζ that char-acterizes the domain wall geometry, through the relation µ = ( D + 2 ζ − / (2 ζ ), where D is the interface dimen-sion [18, 19].Theoretical and numerical studies of effects in thecreep regime are difficult to tackle due to the glassy na-ture of the problem [20]. The universality of these phe-nomena allows statistical-physics minimal models to suc-cessfully capture the main features of interfaces in thisregime trough “elastic line” type models [13]. Thesekinds of model are only able to describe interface prop-erties and fail to characterize important bulk features.In particular, some theoretical works based on this ap-proach, predict that the application of alternating pe-riodic magnetic fields would produce a periodic domainwall oscillation around the initial condition. Althoughthe response is expected to be nonlinear and hysteretic,the magnetic domain is expected to remain unchangedafter the application of an integer number of alternatingmagnetic field cycles [21–24].In a different description level, Ginzburg–Landau mod-els have been proven useful to understand many fea-tures of domain walls, especially for non-disorderedsystems [25–30]. A connection between a minimalGinzburg–Landau model and the Edwards–Wilkinsonequation, belonging to the category of elastic line models,was recently established for disordered systems [31]. Thiscould explain why several features of domain walls, as forexample the creep regime, are successfully described bythe more complex Ginzburg–Landau type models too.More complicated approaches, as the ones based on theLandau–Lifshitz–Gilbert equation [2] are also useful todescribe domain wall dynamics, especially if one is in-terested in microscopic details and internal structure ofdomain walls. However, it seems that to some extentit is not really necessary to consider the full micromag-netic description, since even for this model the quenchedEdwards–Wilkinson universality class arises [32], at leastfor the cases analyzed until now.The great advantage of Ginzburg-Landau type modelsis that bulk properties may be studied in combinationwith interface characteristics in a very simple way. Typ-ical protocols used in experiments to observe ferromag-netic domains in ultrathin films may be emulated, and effects in the different dynamical regimes may be stud-ied [11].In this work, we propose a numerical treatment basedon a Ginzburg–Landau type model to study domains un-der alternating fields that imitates the protocol used inexperiments. Under the application of a sequence ofalternating fields, we observe a domain area reductionwhich follows a linear combination of a decreasing ex-ponential and a linearly decreasing function. This iscompatible with what was reported for domain area lossin samples of Pt/Co/Pt and Pt/[Co/Ni]/Al [8]. More-over, we obtain roughness exponents characterizing thedomain walls geometry which are indistinguishable fromthe ones reported in the same experiments.The importance of our result resides in the fact thatthe area loss may be explained in very simple terms, andis material independent. It applies to all ferromagneticsamples with an easy axis of anisotropy, where dipolarinteractions are not important.The article is organized as follows. We first describethe model used to study the effect of alternating fields indomains (Sec. II). Details of the experimental and numer-ical protocol are presented in Appendix A. In Sec. III wenumerically study the evolution of bubble and stripe do-mains under DC and AC fields (Sec. III B). In Sec. IV weanalyze the domain walls geometries for both bubble andstripe domains. In Sec. V we discuss important implica-tions of our results. We finally present our concludingremarks and perspectives in Sec. VI. II. EMULATING REAL EXPERIMENTS TOSTUDY DOMAIN WALL DYNAMICSNUMERICALLY
To study the effect of alternating magnetic fields ondomain dynamics, we use a modified Ginzburg–Landaumodel with disorder [9–11]. In this model, a non-conserved order parameter ϕ ( (cid:126)r, t ), describing the localstate of a two-dimensional system ( (cid:126)r ∈ R ), is governedby the Langevin equation η ∂ϕ∂t = γ ∇ ϕ + ( Dαϕ + H )(1 − ϕ ) + ξ. (1) ξ = ξ ( (cid:126)r, t ) is a Gaussian white noise with zero meanand two-point correlator (cid:104) ξ ( (cid:126)r , t ) ξ ( (cid:126)r , t ) (cid:105) = 2 ηT δ ( (cid:126)r − (cid:126)r ) δ ( t − t ) . (2)Structural quenched disorder, compatible with the so-called random bond disorder [19, 31], is introduced byconsidering a perturbation of the double-well potentialas D = D ( ε, (cid:126)r ) = (1 + (cid:15)ζ ( (cid:126)r )). In the following, time isgiven in units of ηα , space is in units of (cid:112) γα , field is givenin units of αη , and temperature is in units of √ γηα .Numerically solving equation (1), with a protocol em-ulating the experimental ones typically used in polarmagneto-optic Kerr effect microscopy, allows us to obtainvelocity-field domain walls responses. We find that underthe application of square constant field pulses, the wallsof a bubble domain acquire a velocity with particular fea-tures as a function of the field. The domain wall velocityhas the same characteristics as experimentally observedvelocity-field curves in ultra-thin ferromagnetic systemswith strong perpendicular magnetic anisotropy [11].In this work, we solve numerically equation (1) [11],and explore domain dynamics in the creep regime. Wefollow a numerical protocol which is equivalent to an ex-perimental one recently used in polar magneto-optic Kerreffect microscopy experiments to study the effect of alter-nating magnetic fields on driven domains [8], as detailedin Appendix A 1. The numerical protocol is summarizedin Fig 1 and described fully in Appendix A 2. III. DOMAIN DYNAMICS
For the proposed field-pulse protocol summarized inFig. 1, we measure the domain area A , defined as thenumber of simulation cells for which the order parameterhas a positive value. The evolution of the effective radius R eff = (cid:112) A/π , of a bubble domain is shown in Fig. 1.Images taken after all relaxation processes, imitating theway in which images are typically obtained experimen-tally, are also shown. The advantage of performing thiskind of simulations is that we now have evolution detailsthat are not easily accessible experimentally. In partic-ular, we can precisely track the domain area evolutioneven in presence of an external field. By tracking theevolution of R eff as a function of time, many interestingfeatures of the domain area evolution arise. First, as wasalready observed before [11], immediately after removingthe field, the domain area is slightly increased, highlight-ing the importance of taking into account a relaxationtime t in the simulations. Second and more important,the velocity at which the domain area is reduced whenapplying a negative field is larger than the velocity atwhich the domain grows when a positive field is applied.This point raises intriguing questions about field drivendomain dynamics which we address in the following sec-tions. A. DC dynamics
The observation of larger velocities when a circular do-main contracts, compared to the ones observed for do-main expansions, raises one important question: Whatis the role of the circular shape of the domain in the dy-namics? To answer this question we first focus on themore traditional DC approach, by comparing how a cir-cular and a stripe domains grows.As described in Appendix A 2, our numericalnucleation-like process consists of starting with an ini-tial configuration where the order parameter takes the value -1, except for a region Ω where it takes the value 1.To analyze the evolution of a circular domain, we startwith Ω defined as a region of radius R i = 100 centered inthe middle of the system. As a first test, we let the sys-tem evolve at zero field, and no significant changes in thedomain area are observed, i.e. , even for this “small” cir-cular domain the force due to the curvature is not enoughto contract it in presence of disorder.Another possibility is to consider a stripe domain. Inthis case Ω is given by a rectangular region of size L × L i .We repeat the same protocol for this case, by choosing L i and L so that the initial and final domains in thenucleation-like numerical step have the same area com-pared to the circular case. We then apply N DC = 10cycles to both systems and we compute the domain areain both cases as a function of time, as shown in Fig. 2.To emulate the way in which velocities are obtained inthe experiments, we select one area point per DC cycle(the last one of each cycle, after the relaxation time). Wefit these points with a linear function, and the slope ofthis fit gives the domain wall velocity. For both domaingeometries, the computed velocities during DC cycles arethe same within numerical errors, and give v DC = 0 . B. AC dynamics
As expected, no surprising features arise on the analy-sis of domains under the application of the DC numericalprotocol. However, striking features are present underthe application of AC dynamics. In this section, we an-alyze in detail the evolution of domain area under thenumerical protocol (discussed in A 2) selected to emulatethe experimental one.Starting with a circular and a stripe domain whichwere subjected to the previously descibed DC cycles, weanalyze their area under the application of multiple ACcycles [33]. Both systems are subject to the same condi-tions ( i.e. both simulations are performed with exactlythe same parameters, and the only difference is given bythe initial condition for the order parameter).As shown in Fig. 3, both systems show area loss duringthis process. In particular, the bubble domain completelycollapses after 143 AC cycles, while the stripe domainalso shows area loss, although to a less drastic degree.In the experiments shown in Ref. [8], the area loss isdescribed by a linear combination of an exponential decayand a linearly decreasing function. In the case of oursimulations, the curves are also very well fitted by such acombination for the bubble (over a wide range) and thestripe domains, as indicated with dashed lines in Fig. 3.A detailed inspection of the area evolution during thefour first cycles reveals that in both cases the expansion
FIG. 2.
Domain expansion under the application of square field pulses (DC cycles) : Under the application of DC cycles adomain grows, and its velocity and domain wall geometry (characterized by the roughness exponent) are independent of theglobal domain geometry. We compute the area A of domains as a function of time, from which we extract the effective radius( R eff = (cid:112) A/π ) in the case of the bubble, or the effective width in the case of the stripe ( L eff = A/L ) (gray line in the centralfigures, shown in detail in the insets). After each relaxation time in one DC cycle (diamonds in the central figures), imitatingthe experiments, we extract the domain wall position when no external field is applied (left figures). A fitting of this evolvingdomain wall position (shown in dashed gray line in the central figures) gives a velocity for the domain walls which is independentfrom the domain geometry. We compute the roughness function B ( r ) of each of the obtained relaxed interfaces (right figures,averaged over both domain walls in the case of the stripe). These curves display a power-law behavior B ( r ) ∼ r ζ in the range r = [10 , ζ we fit each curve in this range (the curves obtained with the fitting procedureare shown in solid gray lines). From each fit we obtain a roughness exponent shown in the inset of the figures on the right. Theaverage of these values (dashed gray line in the inset) is ζ DC = 0 .
77 for the bubble domains and 0 .
79 for the stripe domains. velocity of the domains is slightly larger than the velocityacquired by a contracting domain. Surprisingly, and tothe best of our knowledge, this is something which is notusually tested in experiments. However, the observationspresented in [8] are compatible with this claim.
IV. DOMAIN WALL GEOMETRY
The roughness function, defined as B ( r ) = (cid:104) [ u ( z + r )] − u ( z )] (cid:105) (3)measures the quadratic correlations of u ( z ), the positionof the interface of a domain for a given coordinate z . Thisobservable is widely used to study diverse systems withdomain walls [34–37].Usually, the roughness function presents power-law be-haviors with signatures of the underlying physics of thesystem. In particular, at short length-scales, B ( r ) is ex-pected to show a thermal regime. In this regime B ( r )follows a power law ∼ r ζ th , with ζ th = 1 /
2. At larger scales, disorder plays a key role inducing a different powerlaw-scaling r ζ [38, 39]. The roughness exponent ζ isused to characterize domain wall geometries, which areusually determined by the competition between the fewkey ingredients that govern the system.When no fields are applied, domain walls described bythe model given by Eq. (1) display, at low-temperatures,the features of an interface described by the Edwards–Wilkinson [40] (EW) universality class [31]. However,the geometry of domains walls when non-zero fields areconsidered, to the best of our knowledge, has not yetbeen studied for this model. Moreover, in the same ex-periments with ferromagnetic samples where the imple-mentation of the DC-AC dynamic protocols was estab-lished, the roughness exponents of domain walls werereported [8]. In this work, we compute the roughnessexponents observed for the simulated domain walls andwe compare our results with the experiments. To do so,we define the domain wall as the contour in which theorder parameter ϕ is equal to zero. As an example, adetected domain wall is shown in Fig. 4 for the bubble FIG. 3.
Area loss under the application of alternating fields :An AC cycle consists of the application of two square fieldpulses of the same intensity and duration, but with oppositesign, followed by a relaxation time. A bubble and a stripedomain under the same simulation conditions show area lossduring this process. In the top figures, the domain boundariesafter N AC cycles are shown. The domain area loss, measuredat the end of each cycle with respect to the initial value, is alsoshown as a function of the number of AC cycles. The dashedlines are fittings of the curves with a linear combination ofan exponential decay and a linearly decreasing function. Thebubble domain collapses after 143 cycles. The stripe domainalso shows a marked area loss, but to a less drastic degree thanthe bubble domain. Details of the domain area evolution forthe bubble and the stripe domains are shown for the first 4cycles in the inset. domain case. To study the domain wall geometry, and aswas proposed to analyze the experimental domain wallsin Ref. [8], we define a circle of radius R eff = (cid:112) A/π ,where A is the domain area, centered in the centroid ofthe domain. We define the function u ( z ) as the differencebetween the domain wall boundary and this circle. Anobtained interface u ( z ) is shown in Fig. 4, after the nu-merical nucleation-like process for the same system stud-ied in the previous sections. The roughness function, de-fined in Eq. (3), was computed for this interface and fittedin a region where it shows a power-law behavior. In thiswork we focus on the large-scale power-law scaling of theroughness, since we are interested in a comparison with FIG. 4.
Domain wall geometry analysis : The boundary ofthe domain is established (cyan curve, top right image) andits difference, u ( z ), with respect to a perfect circle (dashedpink line) which has the area of the domain, and is centeredin the centroid of the domain, is computed. z is a standardlinear coordinate defined in terms of polar coordinates. Theroughness B ( r ), defined as the average of the quadratic corre-lations of u ( z ), is fitted in a restricted range (shown in orangein the bottom right figure) with a power-law ∼ r ζ , whichgives the roughness exponent ζ . The images correspond totime 14 . × , after the nucleation-like numerical process(top left figure). the experimental observations. How to choose the fittingregion for a given roughness function B ( r ) is not a trivialtask (the reader might refer to [37] for a detailed discus-sion of this issue in the analysis of experimental interfacesin ferromagnetic systems). A crossover regime betweenthe thermal low-scale regime and the large-scale regimemay be present (as discussed, for example, in Ref. [39]).In this case, we chose the fitting region in order to maxi-mize the number of roughness functions that can be fittedin the same region, as will be shown later. In Fig. 4, thefitting region is r = [10 , ζ = 0 .
68, which is remarkablyclose to the value 2 /
3, corresponding to the ‘random-bond’ regime for the EW universality class.In Fig. 2, we show the roughness functions obtained af-ter each DC cycle for the bubble and stripe domains. Inthe case of the stripe domain, the function u ( z ) is definedas all the points where the order parameter ϕ is equal tozero. After each DC cycle, B ( r ) is independently com-puted for the left and right domain walls, and an averageover these two functions is taken to compute the rough-ness function after each cycle. By fitting these functions FIG. 5.
Geometry of interfaces of a bubble domain during AC cycles . We separate the interface evolution under AC cycles inthree cases: cycles 1-35 (first row), cycles 36-85 (second row), cycles 86-142 (third row). The first column shows a snapshotof the system configuration along with the detected interface and a circle of equivalent area to the domain centered on thedomain centroid (pink dashed line). The second column shows the domain area evolution, and highlighted with crosses are thepoints analyzed on the row. The third column shows u ( z ), defined as the fluctuations of the domain wall with respect to aperfect circle with the same area. The fourth column shows the roughness function obtained for each u ( z ), represented withthe same color. A fit of these curves in the region indicated by dotted lines is shown in gray for each curve. The fitting functionis a power-law ∼ r ζ . The obtained ζ values are shown in the insets. The dashed gray line in the insets corresponds to ζ AC ,the average value of ζ over the 142 analyzed AC cycles. The dashed gray lines on the main plots of the fourth column areproportional to r ζ AC . In the central row, the interface is highly pinned (see the protuberance indicated by the orange arrow).This protuberance induces an increase of B ( r ) at short distances (also indicated by an orange arrow), and a shift of the fittingregion. This motivated the division of the analysis in three parts. After 143 N AC total cycles, the system is in the saturatedstate and no domain is observed. The roughness exponent, defined as the mean value of the roughness exponent obtained foreach individual B ( r ), is ζ AC = 0 . ± . in the interval [10 , u ( z ) which become shorterat the end of each AC cycle, the roughness functions dis-play similar power-laws in the region [10 , B ( r ) is observed after eachAC cycle. The subsequent roughness functions (corre- System ζ DC ζ AC Simulations Bubble 0 . ± .
04 0 . ± . . ± .
09 0 . ± . . ± .
04 0 . ± . /Al 0 . ± . sponding to cycles 36 to 85) show a high increase at veryshort distances exactly when the interface seems to behighly pinned. This strong pinning center seems to shiftthe region where B ( r ) can be reasonably fitted with apower-law to larger values of r , [30 , B ( r )again recovers the behavior observed during the first 35cycles. The power-law behavior is obtained in the re-gion [10 , B ( r )decreases after each AC cycle. During the whole process,the roughness exponents obtained do not deviate signif-icantly from their mean value, as shown in the inset inFig. 5.In the stripe domain case, the roughness functions ob-tained after each cycle are similar between each other,and only a global increase on B ( r ) with the number ofAC cycles is observed. In this case we fit each function inthe range [10 , V. DISCUSSION
Based on our analysis, we find that a simple numer-ical model captures quite effectively the experimentallyobserved behavior of ferromagnetic domains under theeffect of alternating magnetic fields. In the experimentsand in our simulations domains under repeated AC cy-cling exhibit pronounced area reduction. In the experi-ments, many AC cycles can be applied to a ferromagneticsample before the domain collapses. It is natural that inour work this effect happens at a shorter number of cy-cles, since the systems we are simulating are smaller thanthe experimental ones. We can estimate the orders ofmagnitude of our simulated systems by using some micro-magnetic concepts. In a micromagnetic model, where themagnetization is ruled by the stochastic Landau-Lifshitz-Gilbert equation, the domain wall width when dipolar in-teractions are negligible is given by ˜∆ = (cid:113) ˜ A n ˜ K [2], where˜ A n is the material anisotropy constant and ˜ K its stiff-ness. For the model considered in this work (Eq. (1)),the equilibrium domain wall width is ∆ = (cid:113) γα [31].By choosing our model parameters in order to recoverthe domain wall width predicted by the micromagnetic model, we can write the model parameters in terms of ex-perimental quantities, and thus compare our numericalresults with a specific material.In particular, for Pt/Co/Pt where ˜ A n = 14 pJ/m and ˜ K = 364 kJ/m [41], the analogy shows that oursimulated system has a lateral size of (cid:39) µm , whichis smaller than the domains studied experimentally inRef. [8], where the initial domain area after nucleationis ∼ µm (giving an effective radius of ∼ µm ).This difference could explain why our circular simulateddomain collapses completely after a hundred N AC cycles,while in the experiments thousands of cycles can be stud-ied before the domain collapses. However, the domainarea reduction in the experiments and in our simulationsfollows the same functional behavior as a function of thenumber of applied AC cycles. A linear combination of anexponentially decreasing and a linearly decreasing func-tions fits the area reduction very well but a justificationof the microscopic origin of behavior is still lacking. Inthis direction, our work provides valuable information todevelop a theory describing these phenomena.The roughness exponents characterizing the geometryof domain walls are in excellent agreement with the onesreported in the experiments. However, we note that themean values of the exponents we obtain are systemat-ically slightly higher than those observed in the experi-ments, even if the discrepancy lies within the error bars ofboth data-sets. This discrepancy could potentially leadto the question if this is a sign of new physics which needto be understood, or if the model requires additional im-provement. This question should be addressed in furtherexperimental and theoretical investigations. It would beinteresting to check if the model could be improved: Onepossible approach is to consider a similar model, but withother disorder types (as discussed for example in [11]).The relation between exponents describing dynamic andstatic features of domain walls could give insight into howthe geometry and dynamics of domains are related. VI. CONCLUSIONS AND PERSPECTIVES
Motivated by recent experimental observations of pro-nounced domain area reduction under the application ofalternating magnetic fields in ferromagnetic thin films [8],hitherto not predicted theoretically, we propose a numer-ical protocol to study the same effect in a very simpletwo-dimensional scalar-field model. The model only con-siders the main and basic ingredients of ferromagneticsystems [11], and has the advantage of being material-independent (no physical units need to be specified apriori). Its simplicity allows us to do extremely long sim-ulations to study domain area loss under several combi-nations of magnetic pulses.Our approach allows us to observe the same effect re-ported experimentally: under the application of alter-nating magnetic fields of equal duration and intensitybut opposite direction (AC cycles), a domain area lossoccurs. We study the effect of AC cycles over differentdomain geometries: a bubble and a stripe domains. Sev-eral repetitions of AC cycles lead to a collapse of thebubble domain. The stripe domain does not collapse inthe same interval, but a significant area loss is nonethe-less observed. As had been proposed for the analysis ofthe experimental results, the domain area evolution forboth domain geometries is very well fitted by a linearcombination of an exponentially and linear decreasingfunctions. Moreover, we find that extrapolating this fitpredicts also an eventual collapse of the stripe domainbut at longer time scales. The slower decrease of thestripe domain area could indicate that this domains arebetter preserved compared to the bubble domains, de-pending on the fields treatments that could be appliedto a ferromagnetic sample. The difference in the areareduction for the two domain types could be importantto determine the domain wall surface tension, a quantitythat is difficult to asses experimentally, but which hasseveral important technological applications [42].We also analyze the evolution of domain wall geometryunder the application of sequential field pulses with thesame direction (DC dynamics) and under AC dynamics.We characterize the bubble and stripe domain walls withthe roughness exponent ζ . Both geometries give similarresults: a lower exponent value in the DC scenario, anda higher one under the application of AC fields. Remark-ably, the exponent values in both cases are indistinguish-able, within error bars, from the ones reported experi-mentally for Pt/Co/Pt and Pt/[Co/Ni]/Al. This resulthas important consequences. The relation between do-main walls dynamics and geometry clearly needs furtherexploration. In this sense, our work can mark a courseof how this problem can be approached theoretically.From the experimental point of view, our work con-tributes to the systematization of experiments. With thesame experimental set-up usually used to study domainwalls dynamics with polar magneto-optic Kerr effect mi-croscopy, more information can be extracted from thesame sample: besides of the usual velocity determinationfrom domains expansion, our work shows that it is alsoimportant to determine the contraction velocity.In addition, the result presented in this work couldhave important technological implications. Although wedid not discuss any particular device implementation, ac-cording to our simulations the effect of area loss underthe application of alternating magnetic fields should beobserved in any disordered ferromagnetic material withstrong easy axis of magnetization. Of course, this claimshould be verified experimentally and numerically forspecific devices.It would be of extreme interest to understand whichinteractions could compensate the observed effect in or-der to prevent domain area loss. In this direction, theeffects of long-range dipolar interactions (which can beeasily included in the model used in this work, as for ex-ample proposed in Ref. [9]) could be explored. In thiscase, the area loss effect, if present, may be modified in the presence of more than one domain. The effects ofcurrents will also be interesting to assess. In this case,more deep modifications need to be done to the modelused in this work.Our work triggers new directions to explore and at thesame time, gives insight in how to theoretically under-stand the observed phenomena. VII. ACKNOWLEDGEMENTS
I gratefully acknowledge J.-P. Eckmann, T. Giamarchi,and P.Paruch for their critical reading of the manuscriptand their insightful comments. I also acknowledge valu-able discussions with S. Bustingorry, E. E. Ferrero andA. B. Kolton about code implementation. I am gratefulto G. Pasquini for pointing me to her paper [8], whichwas the initial seed that stimulated me to carry out thepresent project. I am also grateful to Vincent Jeudy andJavier Curiale for giving me insight in how PMOKE ex-periments are carried out. I acknowledge support fromthe Federal Commission for Scholarships for Foreign Stu-dents for the Swiss Government Excellence Scholarship(ESKAS No. 2018.0636). This work was supported inpart by the Swiss National Science Foundation under Di-vision II. All the simulations presented in this work wereperformed in the
Mafalda cluster, at the University ofGeneva.
Appendix A: Protocol to study domain walls staticsand dynamics under alternating magnetic fields1. Experimental protocol
To study domain wall response under alternating mag-netic fields by means of polar magneto-optic Kerr effectmicroscopy (PMOKE), the following sequence of stepswas recently proposed [8].DC experimental protocol: In the experiment, thesample is i) saturated by the application of a high mag-netic field perpendicular to the sample plane. ii) A mag-netic field in the opposite direction is applied to nucleatea domain. iii) After nucleation, a square pulse of mag-netic field H during a time τ results in the expansionof the domain(s). iv) After this step, an image of (usu-ally one) domain is optically captured (with no externalmagnetic field applied). The application of field pulsesof equal magnitude and of the same duration τ is re-peated N DC times. From the average displacement ofthe domain wall, the mean domain wall velocity for thatparticular field H is extracted.AC experimental protocol: After the DC protocol, N AC cycles are applied to study the domain response.An AC cycle consists of the application of a square pulseof field H during a time τ , followed by a second squarepulse of field − H of the same duration. After each suchcycle, an optical image is captured with no applied ex-ternal field. The sequence of applied magnetic fields isshown in Fig. 1.
2. Numerical emulation of the experimentalprotocol
To reveal the causes of domain area reduction observedin the experiments, we imitate the experimental protocolin our simulations. The method consists of a series of cy-cles of two types which are called “DC” and “AC”. TheDC experimental protocol is the standard one used tomeasure domain walls velocities [16, 17, 41, 43], while theAC one is applied afterwards to probe domain wall dy-namics and geometry under alternating magnetic fields.We start with a system where ϕ ( (cid:126)r, t = 0) takes thevalue −
1, except inside a circular region of radius R i ,where it is 1. At fixed temperature T , and disorder in-tensity ε , as shown in Fig. 1, we apply a square pulse offield H i (above the depinning field), favouring the phasecorresponding to the value 1 (in a real PMOKE exper-iment, this step would be equivalent to the nucleationprocess with a high field), until the domain reaches aneffective area πR . We then let the system evolve atzero field for a simulation time t , to ensure a station-ary value of the domain area. After time t , we apply a square pulse of field H during a time τ , which results inan expansion of the initial domain. This cycle is repeated N DC times. Afterwards, we apply alternating fields cy-cles N AC times. Each of these cycles consists of a squarefield pulse H during a time τ , followed by a square pulseof the same duration and intensity, but in the oppositedirection. Subsequently, the domain is relaxed at zerofield during a time t , as shown in Fig. 1.We consider a system of size L × L simulation cells,with L = 4096 and periodic boundary conditions in bothdirections. To obtain the system evolution as a functionof time, we integrate Eq. (1) by following a semi-implicitEuler method with integration time-step ∆ t = 0 . ε = 1, the velocity-field response of a domain withthese characteristics was shown to display a depinningfield H d (cid:39) . R i =100, and we let it grow until it reaches a radius R =500 under the application of a field H i = 0 .
06 at T =0 .
01. The velocity in this case is v ( H i = 0 . , ε = 1 , T =0 .
01) = 0 . T = 0 .
01, the response of domains underfield H = 0 .
05. The velocity of a domain under constantsquare field pulses at this field is v = 0 . ∼
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