Fluctuations in a model ferromagnetic film driven by a slowly oscillating field with a constant bias
aa r X i v : . [ c ond - m a t . m t r l - s c i ] S e p Fluctuations in a model ferromagnetic film driven by a slowlyoscillating field with a constant bias
Gloria M. Buend´ıa and Per Arne Rikvold Department of Physics, Universidad Sim´on Bol´ıvar, Caracas 1080, Venezuela Department of Physics, Florida State University, Tallahassee, FL 32306-4350,USA
Abstract
We present a numerical and theoretical study that supports and explains recent experimen-tal results on anomalous magnetization fluctuations of a uniaxial ferromagnetic film in its low-temperature phase, which is forced by an oscillating field above the critical period of the associateddynamic phase transition (DPT) [P. Riego, P. Vavassori, A. Berger, Phys. Rev. Lett. , 117202(2017)]. For this purpose, we perform kinetic Monte Carlo simulations of a two-dimensional Isingmodel with nearest-neighbor ferromagnetic interactions in the presence of a sinusoidally oscillatingfield, to which is added a constant bias field. We study a large range of system sizes and supercriti-cal periods and analyze the data using a droplet-theoretical description of magnetization switching.We find that the period-averaged magnetization, which plays the role of the order parameter for theDPT, presents large fluctuations that give rise to well-defined peaks in its scaled variance and itssusceptibility with respect to the bias field. The peaks are symmetric with respect to zero bias andlocated at values of the bias field that increase toward the field amplitude as an inverse logarithmof the field oscillation period. Our results indicate that this effect is independent of the systemsize for large systems, ruling out critical behavior associated with a phase transition. Rather, it isa stochastic-resonance phenomenon that has no counterpart in the corresponding thermodynamicphase transition, providing a reminder that the equivalence of the DPT to an equilibrium phasetransition is limited to the critical region near the critical period and zero bias. . INTRODUCTION The hysteretic response when a uniaxial spin system with long-range order ( i.e. , belowits critical temperature) is subject to a symmetrically oscillating field of amplitude H andperiod P , depends crucially on P . If P is much longer than the response time of the system(which depends on the temperature and H ), a symmetric hysteresis loop centered on zeroresults. If P is much shorter than the response time, asymmetric hysteresis loops centeredaround the values of the system’s static order parameter are observed. Numerical studiesin the 1990’s showed that the transition between these two regimes is not smooth. Rather,there is a critical period P c , where the period-averaged order parameter h Q i (see formaldefinition in Sec. II) vanishes in a singular fashion. This phenomenon was first observedby Tom´e and de Oliveira [1] in a kinetic mean-field study of an Ising model, followed bykinetic Monte Carlo (MC) simulations by Rao, Krishnamurthy, and Pandit [2] and Lo andPelcovitz [3]. Early work in the field was reviewed by Chakrabarti and Acharyya in Ref. [4].Kinetic MC combined with finite-size scaling analysis [5–10], as well as further mean-fieldstudies of Ising and Ginzburg-Landau models [11–14], confirmed not only that this is a true,dynamic phase transition (DPT), but also that it is in the same universality class as thecorresponding equilibrium Ising model. The DPT has been confirmed experimentally in[Co/Pt] magnetic multilayers [15] and uniaxial Co films [16].With all the attention that has been given to the DPT and its universality class, one mightlose sight of the fact that the equivalence between the critical properties of the equilibriumIsing model and the DPT of the same model in an oscillating field does not necessarilyamount to equivalence outside the critical region. A warning was provided very recently byRiego, Vavassori, and Berger [17]. These authors fabricated Co films with (1010) crystallo-graphic surface structure with a single, in-plane magnetic easy axis, which they subjectedto a sinusoidally oscillating, in-plane magnetic field plus a constant bias field H b . Such aconstant bias field has previously been shown by MC simulations and finite-size scaling tobe (at least a significant component of) the field conjugate to h Q i in the critical region near P c [8], and this has later been confirmed for mean-field models [12–14] and in experiments[16]. It therefore seemed surprising that, in the experiments reported in Ref. [17], both thefluctuations in the order parameter and its derivative with respect to H b , for P ≫ P c , be-haved quite differently from the dependence of the equilibrium susceptibility on the applied2tatic field at temperatures above critical. Instead of the wide, smooth, unimodal maximumof the supercritical equilibrium susceptibility of the Ising model, two distinct peaks wereobserved at nonzero values of H b , symmetrical about zero [17]. In their article the authorsalso presented kinetic mean-field results that corroborate the presence of these peaks, whichthey dubbed “sidebands.”The purpose of the present paper is to investigate the long-period parameter regime withkinetic MC simulations of a two-dimensional Ising model with nearest-neighbor ferromag-netic interactions. To match the experimental conditions of Ref. [17] as closely as possible, wechoose the oscillating field to have a sinusoidal waveform. We are not aware that systematicsimulations in this regime have been performed previously. Our study reveals “sidebands”analogous to the experimental results. We thus conclusively confirm that the experimen-tally observed phenomenon is not caused by residual magnetostatic long-range interactions.Using simulations for a range of field periods and system sizes together with knowledge ofthe kinetics of magnetization switching by homogeneous nucleation and growth of antiphasedroplets [18], we demonstrate that the “sidebands” result from noncritical fluctuations dur-ing the half-cycles when the sign of the oscillating field is opposite to that of the bias field.This is essentially a stochastic resonance phenomenon [19–21].The rest of this paper is organized as follows. In Sec. II we describe the model and detailsof the simulation method, and we define the appropriate observables to be measured. Ournumerical results are presented in Sec. III. In Sec. III A we present numerical observation ofsidebands for a single, supercritical value of the field period. In Sec. III B we present shorttime series of the system magnetization for several values of bias and period, which enableus to propose a simple approximation for h Q i in the limits of weak bias and long period.In Sec. III C we present numerical results for h Q i vs H b for a wide range of supercriticalperiods, as well as the sideband positions H peak b as functions of period and system size. Thelatter are analyzed using results from the droplet theory of magnetization reversal. Ourconclusions are given in Sec. IV. A short summary of pertinent results from the droplettheory of magnetization reversal is given in Appendix A, and the case of extremely longperiods is discussed in Appendix B. A brief discussion of the mathematically simpler caseof a square-wave oscillating field is presented in Appendix C.3 I. MODEL AND MONTE CARLO SIMULATION
We consider a kinetic S = 1 / H = − J X h ij i s i s j − [ H ( t ) + H b ] X i s i , (1)where J > s i = ±
1, the first sum runs over all nearest-neighbor pairs, and the second oneover all sites. H b is a constant “bias field,” and H ( t ) is a symmetrically oscillating externalfield of period P . Here we choose H ( t ) = H cos (cid:18) πP t (cid:19) . (2)The system is simulated on a square lattice of N = L × L sites with periodic boundaryconditions. We perform Glauber single-spin-flip dynamics in a heat bath at temperature T .A spin at a randomly chosen site i is allowed to flip from s i to − s i with probability W ( s i → − s i ) = 11 + exp( β ∆ E i ) , (3)where ∆ E i is the change in the system energy associated with flipping the spin i , and β = 1 /k B T where k B is Boltzmann’s constant. The time unit is one MC step per site(MCSS), during which, on average, each site is visited once. Hereafter, H , H b , and T areall given in units of the interaction constant J (i.e., J = k B = 1), and P is given in units ofMCSS.The Glauber dynamic can be derived as the weak-coupling limit of the quantum-mechanical Hamiltonian of a collection of quasi-free Fermi fields in thermal equilibriumwith a heat bath [22]. However, the DPT with H b = 0 has been shown to be universal withrespect to dynamics that obey detailed balance in equilibrium, including Metropolis [23]and “soft Glauber” [9], as well as different forms of H ( t ) including square-wave [7, 9] andsawtooth [15].We calculate the time dependent, normalized magnetization per site, m ( t ) = 1 L X i s i ( t ) , (4)and by integrating it over each cycle of the magnetic field, we obtain the average magneti-zation during the k th cycle of the field, Q k = 1 P Z kP ( k − P m ( t ) dt . (5)4he dynamic order parameter of the model is the period-averaged magnetization, h Q i , de-fined as the average of Q k over many cycles. Its fluctuations are measured by the scaledvariance, χ QL = L ( h Q i − h Q i ) , (6)and its dependence on the bias field is measured by the susceptibility with respect to H b , χ bL = d h Q i /dH b . (7)In order to take advantage of temperature and field dependent parameters measured withhigh precision in previous MC simulations [6], our calculations are performed with H = 0 . T = 0 . T c , where T c = 2 / ln(1 + √ ≈ .
269 is the critical temperature of the standard,square-lattice Ising model in zero field. In the absence of a bias field, at this temperature,and for sufficiently large L , switching between the equilibrium values of m , following fieldreversal from − H to + H , occurs via a nearly deterministic and L -independent multi-droplet mechanism [18]. In Ref. [6], the characteristic switching timescale (the time fromthe field reversal until the system magnetization reaches zero) under Glauber dynamicswith the same parameters as we use here was measured by MC simulations as τ ≈ .
6. Inthe same work, the critical period in a sinusoidal field of amplitude H with zero bias wasmeasured as P c ≈ h Q i vanishes for P ≥ P c and H b = 0. Near critical-ity, the constant bias field H b is the field conjugate to h Q i , and the period P mimics thetemperature in the equilibrium phase transition. Simulations were performed for periodsbetween P = 258 and 28,000 and system sizes between L = 32 and 1024. Except for thesmallest values of P , the measurements were obtained by averaging over 800 field cycles,after discarding 200 cycles. This means that at least 800 × P MCSS were performed foreach measurement.
III. NUMERICAL RESULTS AND ANALYSISA. Observation of “sidebands”
Results of simulations with P = 1000 ≈ . P c for several values of L are displayed inFig. 1. “Sidebands” are observed, consistent with the experiments reported in Ref. [17].5he dependence of the order parameter h Q i on the bias H b is shown in Fig. 1(a). For weak H b , h Q i increases almost linearly with H b , but the slope of the curve increases considerablyaround | H b | ≈ .
09, followed by saturation of h Q i for | H b | & .
15. This behavior is reflectedin the bimodal shape of the susceptibility χ bL , shown by the lower set of curves in Fig. 1(b).Between the two peaks lies a flat-bottomed valley corresponding to the linear regime inpart (a), and a rapid approach to zero for large | H b | mirrors the saturation of h Q i also seenin (a). The scaled variance χ QL also displays peaks, whose positions coincide with those of χ bL . However, the ratio χ QL /χ bL for fixed P depends quite strongly on H b with maximumvalues near the peaks. This variable ratio precludes a straightforward interpretation interms of an effective, nonequilibrium fluctuation-dissipation relation with P playing the roleof “temperature.” For these values of L and P , finite-size effects are seen to be negligible,ruling out critical behavior associated with a phase transition. The relationships betweensystem size, field period, and finite-size effects will be discussed in further detail below. B. Magnetization time series
To gain a more detailed understanding of the relationships between bias, period, systemsize, and the order-parameter fluctuations, we present in Fig. 2 short time series of thenormalized magnetization, m ( t ). The total applied field, H ( t ) + H b , is shown as an orangecurve. In this figure we set H b >
0, so that the up-spin phase is favored and the down-spinphase is disfavored.Figure 2(a) shows data for P = 1000 and H b = +0 .
10, just on the strong-bias side ofthe fluctuation peak for this period length. For the smaller system sizes ( L = 32 and 64),the switching from the favored (up-spin) to the disfavored (down-spin) magnetization isstochastic and abrupt (mediated by a single or a few droplets of the down-spin phase [18])and occurs only in narrow time windows near the negative extrema of the total appliedfield. For the larger systems, the switching becomes more deterministic and gradual (multi-droplet [18]). However, the growing down-spin phase does not have time to completely fillthe system before the field again becomes positive. For the largest system studied, L = 1024,the extreme negative magnetizations during a period are close to − . P = 1000 and H b = +0 . L = 32 remains stochastic. However, the larger6ystems appear more deterministic, and their extreme negative magnetizations during aperiod are close to − . P = 1000 and H b = +0 .
08, just on the weak-bias side of thefluctuation peak. The switching for L = 32 remains stochastic. The larger systems behavemore deterministically, and the extreme negative magnetizations during a period approach − . L to a nearly deterministic multidropletmechanism for larger L , in agreement with known results for field-driven magnetizationswitching by homogeneous nucleation and growth of droplets of the stable phase [18].Figure 2(d) shows data for L = 128 with a weak bias, H b = +0 .
04, and two differentperiod lengths, P = 1000 and 14,000. In both cases, the switching is nearly deterministicand complete, so that the period-averaged magnetization h Q i depends mostly on the relativeamounts of time the system spends in the two phases. As P increases, the switching occursearlier in the half-period.The differences between the single-droplet and multidroplet switching modes are furtherillustrated in Fig. 3. In Fig. 3(a), time series for m ( t ) over five cycles with P = 1000at the corresponding peak position, H peak b = +0 . L = 32 and 1024.All the parameters are the same as in Fig. 2(b), except the seed for the random numbergenerator. When the total applied field, H ( t ) + H b , is negative, the down-spin phase, whichis disfavored by the positive bias, is the equilibrium phase. Nucleation and growth of thisphase may only occur during the time intervals of negative total applied field. Snapshotscaptured at m ( t ) = +0 . .
45, are shown in Fig. 3(b) for L = 32 and in Fig. 3(c) for L = 1024.For L = 32 we see a single down-spin droplet which, as seen from the time series inFig. 3(a), nucleated during the third period shown, near the time when the field had itslargest negative value. It barely reached the capture threshold of m = +0 . L = 1024 the picture is quite different. In the snapshot we see a large number7f growing clusters that have nucleated at different times during the negative-field timeinterval. Some of these have already coalesced by the time the snapshot was captured, whileothers are still growing independently. From the time series it is seen that this multi-dropletswitching mode leads to a nearly deterministic evolution of the total magnetization, with theunderlying stochasticity only evident in the slight variations of the minimum magnetizationvalues from period to period. This switching process is well described by the Kolmogorov-Johnson-Mehl-Avrami (KJMA) approximation [18, 24–28].Magnetization reversal from the favored to the disfavored direction is only possible whilethe total applied field, H ( t ) + H b , has the opposite sign of the bias, H b . This implies that − < H b /H ( t ) ≤
0. Switching from the favored phase to the disfavored one on averagetakes longer time than switching in the opposite direction. Thus, the time the system canspend in the disfavored phase during each period must be less than or equal to the time thatthe field has the disfavored direction, t Dmax = P (cid:20) − π sin − (cid:18) | H b | H (cid:19)(cid:21) . (8)In this limit of long period and weak bias, h Q i is simply determined by the sign of H b andthe difference between the fractions of the period that the total field has the same and theopposite sign as H b , respectively. This yields h Q i ≈ m π sin − (cid:18) H b H (cid:19) , (9)which is symmetric under simultaneous reversal of H b and h Q i . Here, m is the magnitudeof the magnetization in the favored phase. This approximation represents a lower bound onthe magnitudes of h Q i and χ b [29]. The former is included as a dashed curve in Fig. 4(a).However, the bounds depend on the waveform of the oscillating field, and as we show inAppendix C, they vanish in the case of a square-wave field.The corrections to this approximation are of O ( t FD ( H b , H ) /P ), where t FD ( H b , H ) isthe average time it takes the magnetization to switch to the disfavored direction, after thetotal applied field has changed sign. For | H b | ≪ H , the correction vanishes as 1 /P , asseen in Fig. 4(a). However, for larger | H b | , t FD ( H b , H ) ∼ P , and the “correction” becomesthe dominant part of h Q i , determining the sideband peak positions, H peak b . The details arediscussed below in Sec. III C. 8 . Dependence on H b , P , and L Results for L = 128 and a range of periods between P c = 258 and P = 28 ,
000 areshown in Fig. 4. In the critical region, H b is the field conjugate to h Q i [8, 12–14, 16]. At P = P c , h Q i therefore vanishes in a singular fashion as H b approaches zero. On the scaleof Fig. 4(a), this singularity appears as a jump in h Q i at H b = 0 for P = P c , resulting invery narrow central peaks in both χ bL and χ QL . We also found broad central peaks in bothquantities for P = 400, which are due to finite-size broadening of the critical region for thisrelatively modest system size. For clarity, these central peaks are not included in Fig. 4(b).Beyond P = 500, h Q i becomes linear for small H b , with a slope that approaches that of theasymptotic approximation in Eq. (9) as P increases. Simultaneously, the peaks in χ bL and χ QL increase in height, and their positions H peak b move in the directions of ± H , as seen inFig. 4(b). [For clarity, some of the values of P included in Fig. 4(a) are excluded from Fig.4(b).]The magnitudes of the peak positions, | H peak b | , are plotted vs P for different values of L in Fig. 5(a). We note two main features. First, | H peak b | increases quite rapidly with P for relatively short periods, and much more slowly for longer periods. This behavior isconsistent with the experimental data shown in Fig. 2 of Ref. [17]. Second, finite-size effectsare essentially negligible for P . P = 1000. For longerperiods, | H peak b | increases with L for smaller sizes, and then becomes size independent forlarger L .In order to explain this behavior quantitatively, we first recall from the time series shownin Fig. 2 that for bias near | H peak b | , the time it takes m ( t ) to change significantly towardthe disfavored sign is on the order of a finite fraction of P . For stronger bias, the totalfield driving the magnetization toward the disfavored sign is too weak and consequently thetime required for switching is much longer than P , so that reliable magnetization reversaldoes not occur. For weaker bias, the field in the disfavored direction is relatively strong,and complete and reliable magnetization reversal takes place on a timescale significantlyshorter than P . In other words, the peak positions correspond to bias values that producemagnetization reversal on a timescale of P . Equations for magnetization switching rates bythe stochastic single-particle mechanism that dominates for small systems [Eq. (A1)] andthe nearly deterministic multidroplet mechanism that dominates for large systems [Eq. (A2)]9re found in Appendix A. The nucleation rate for droplets of the disfavored phase variesvery strongly with the oscillating field, having appreciable values only in a narrow windownear the maximum field in the disfavored direction, | H | = H − | H peak b | . Using this value of | H | and ignoring less important prefactors, we can use these equations to write the followingrequirement for | H peak b | : L − a exp b Ξ H − | H peak b | ! ∼ P , (10)with a = 2 and b = 1 for single-droplet switching, and a = 0 and b = 3 for multidropletswitching. The meaning of the constant Ξ ≈ .
506 is explained in Appendix A. In eithercase, this equation is equivalent to a statement that | H peak b | should approach H asymptoti-cally as 1 / log P for long periods. (A caveat to this statement for the case of extremely longperiods is discussed in Appendix B.) Plotting 1 / ( H − | H peak b | ) vs log P therefore shouldproduce straight lines for large values of P . The ratio between the slopes of the lines rep-resenting multidroplet switching for large L and those representing single-droplet switchingfor small L should be 3/1. Such a plot is presented in Fig. 5(b). The slope ratio between thecurves representing L = 256 and L = 32 in the long- P regime is approximately 2.867, con-sistent with the theoretical prediction. This conclusion is confirmed by the short time seriesof m ( t ) for P = 20 ,
000 for these two system sizes, shown in Fig. 6. In the switching regions,the smaller system displays the stochastic, square wave form characteristic of single-dropletswitching [20], while the larger system shows the continuous wave form characteristic ofmultidroplet switching [6].To further support our conclusions, we calculated the transition times and the orderparameter in the multidroplet regime for the mathematically simpler case, in which thesinusoidally oscillating field has been replaced by a square-wave field. The details of thecalculations are given in Appendix C. In Fig. 7 we show that there is very good agreementbetween the theoretically calculated h Q i and the simulations, particularly when | H b | . | H peak b | . IV. SUMMARY AND CONCLUSION
Riego et al. [17] recently presented experimental data on Co films with a single, in-planemagnetic easy axis, which were subjected to a slowly oscillating magnetic field with an added10onstant bias. In this paper we have presented kinetic MC simulations and theoretical analy-sis of a two-dimensional Ising ferromagnet with only nearest-neighbor interactions, designedto closely mimic the experimental setup. At zero bias, such systems exhibit a dynamicphase transition (DPT) at a critical period P c , where the period-averaged magnetization h Q i vanishes in a singular fashion. It has previously been shown that the DPT belongs tothe equilibrium Ising universality class, with P playing the role of temperature and the bias H b being the field conjugate to h Q i . Following Riego et al. [17], we studied the dynamicsof the system at values of P above P c , and in agreement with the experiments we foundthat h Q i exhibits a strong bias dependence and fluctuation peaks at nonzero values of H b ,symmetrically located around zero bias.Since the simulated system has only nearest-neighbor interactions, our results show thatthe experimental results are not due to any residual magnetostatic interactions. The simu-lational approach also enables studies of the effects of finite system size. We found that, atfixed P , finite-size effects saturate beyond a P -dependent size limit. Using the droplet theoryof magnetization switching, we conclude that this saturation occurs at the crossover betweentwo different dynamic regimes. For small systems, the magnetization switching from the fa-vored to the disfavored direction occurs by a stochastic single-droplet mechanism. For largesystems, the switching occurs by the size-independent and nearly deterministic KJMA mech-anism, which involves a large number of simultaneously nucleating and growing droplets.We therefore conclude that this “sideband” phenomenon for supercritical values of P is not a critical phenomenon, but rather a stochastic-resonance phenomenon. We believe theseinsights will be important for the design and analysis of devices that involve magnetizationreversal by time-varying fields, such as memory elements, switches, and actuators. ACKNOWLEDGMENTS
We thank A. Berger for providing data from Ref. [17] before publication, and for usefulcomments on an earlier version of this paper. G.M.B. is grateful for the hospitality ofthe Physics Department at Florida State University, where her stay was supported in partby the American Physical Society International Research Travel Award Program (IRTAP).P.A.R. acknowledges partial support by U.S. National Science Foundation Grant No. DMR-1104829. 11 ppendix A: Mechanisms of magnetization reversal
When a d -dimensional Ising ferromagnet below its critical temperature is subjected tothe reversal of an applied field of magnitude | H | , the homogeneous nucleation rate per unitsystem volume for droplets of the new equilibrium phase is given by [6, 18, 20, 28, 30–32] I ( H ) ≈ B ( T ) | H | K exp (cid:20) − Ξ ( T ) | H | d − (cid:21) , (A1)where B ( T ) is a non-universal function of T . For d = 2, K = 3, and Ξ (0 . T c ) ≈ . /T ) [6]. The argument of the exponential function is the negativeof the free energy of a critical droplet of the equilibrium phase, divided by T . The inverseof L d I ( H ) is the average time between random nucleation events for a system of size L . Single-droplet reversal mechanism:
Under conditions of small system and/or mod-erately weak field, the time it takes for the first nucleated droplet to grow to fill the systemis much shorter than the average nucleation time. As a result, the magnetization reversal iscompleted by this single, first droplet.
Multidroplet reversal mechanism:
Under conditions of large system and/or moder-ately strong field, the average time between nucleation events is less than the time it wouldtake the first nucleated droplet to grow to fill the system. Therefore, many droplets nucleateand grow independently in different parts of the system until they coalesce and collectivelyfill the system. The result is a gradual and nearly deterministic growth of the new phasethrough a multidroplet process, well described by the KJMA approximation [18, 24–28].The characteristic reversal time is independent of the system size and given by h τ ( H ) i ∝ (cid:2) v d I ( H ) (cid:3) − / ( d +1) , (A2)where the propagation velocity of the droplet surface, v , is proportional to | H | in thisparameter range [33] as expected from the Lifshitz-Allen-Cahn approximation [34–36]. Appendix B: Extremely long periods
If the radius of the critical droplet reaches a size of about L/
2, it will not fit in the L × L system, and a new regime, called the coexistence regime , is entered [18]. In this regime, thedroplet is replaced by a slab of the equilibrium phase, and the nucleation time no longer12epends on | H | , but increases exponentially with L d − . The critical droplet radius in d dimensions is given by [18], R c ≈ (cid:18) ( d − T Ξ m Ω d (cid:19) /d | H | , (B1)where Ω d is the volume of the critical droplet, divided by R dc . Numerical values for theconstants with d = 2 at T = 0 . T c ≈ .
815 are found in Table I of Ref. [6]: Ξ ≈ .
506 andΩ ≈ . T is included in the numerator to cancel the factor 1 /T in Ξ .)Thus we have R c ≈ . | H | ≈ L . (B2)Replacing | H | by H − | H b | and setting L = 32, we thus find 1 / ( H − | H b | ) ≈ .
3. Finally,linearly extrapolating the large- P data for L = 32 in Fig. 5(b), we find that the single-droplet result from Eq. (10) should remain valid for periods up to approximately 10 ± .[The uncertainty in the exponent is the result of assuming a 10% uncertainty in the estimateof 1 / ( H − | H b | ).] Beyond this limit, H − | H b | should remain independent of P , at a value of O (1 /L ). For larger L , the single-droplet result should be valid up to even longer periods. Wedo not expect that these extremely long periods should be of great experimental relevancefor macroscopic systems. However, for nanoscopic systems the coexistence regime may beobservable with experimentally accessible periods. Appendix C: Square-wave oscillating field
Now, instead of a sinusoidally oscillating field, consider a square-wave field, such that H ( t ) = + H during one half-period, and − H during the other. Since the times that the totalfield is parallel and antiparallel to H b now each equal P/
2, the equivalent of the long-period,weak-bias approximation of Eq. (9) becomes h Q i ≈
0. Therefore, the value of h Q i for finite P and weak H b is determined by the difference between the average magnetization reversaltimes following a change of the total field from the favored to the disfavored direction, andthe opposite. Since the total field now has its full favored or disfavored strength during thewhole half-period, these average switching times will be shorter than the corresponding timesin the sinusoidally oscillating field case. With a square-wave field of amplitude H = 0 . . T c under Glauber dynamics, the critical period has been measured by MC simulations13s P c ≈
137 [7]. To calculate the transition times for a two-dimensional system in themultidroplet regime, we will again assume H b ≥ | H | = H − H b , we obtain the characteristic timescalefor transitions from the favored (parallel to the bias field) to the disfavored magnetizationdirection, after the total applied field has changed sign as t FD ( H b , H ) = τ (cid:18) − H b /H (cid:19) / exp (cid:18) Ξ H H b /H − H b /H (cid:19) ≥ τ , (C1)where τ is the magnetization reversal time for H b = 0. Analogously, the switching timefrom the disfavored to the favored magnetization direction is t DF ( H b , H ) = τ (cid:18)
11 + H b /H (cid:19) / exp (cid:18) − Ξ H H b /H H b /H (cid:19) ≤ τ . (C2)Both t FD and t DF reduce to τ ≈ . H b = 0.The order parameter h Q i is determined by P and the difference between t FD and t DF as h Q i ≈ m t FD − t DF P for t FD ≤ P m for t FD > P (C3)This approximation is shown together with simulation results in Fig. 7. The agreement isvery good for | H b | . | H peak b | . 14
1] T. Tom´e and M. J. de Oliveira, “Dynamic phase transition in the kinetic Ising model under atime-dependent oscillating field,” Phys. Rev. A , 4251 (1990).[2] M. Rao, H. R. Krishnamurthy, and R. Pandit, “Magnetic hysteresis in two model spin sys-tems,” Phys. Rev. B , 856 (1990).[3] W. S. Lo and R. A. Pelcovits, “Ising model in a time-dependent magnetic field,” Phys. Rev.A , 7471 (1990).[4] B. Chakrabarti and M. Acharyya, “Dynamic transitions and hysteresis,” Rev. Mod. Phys. ,847 (1999).[5] S. W. Sides, P. A. Rikvold, and M. A. Novotny, “Kinetic Ising model in an oscillating field:Finite-size scaling at the dynamic phase transition,” Phys. Rev. Lett. , 834 (1998).[6] S. W. Sides, P. A. Rikvold, and M. A. Novotny, “Kinetic Ising model in an oscillatingfield: Avrami theory for the hysteretic response and finite-size scaling for the dynamic phasetransition,” Phys. Rev. E , 2710 (1999).[7] G. Korniss, C. J. White, P. A. Rikvold, and M. A. Novotny, “Dynamic phase transition,universality, and finite-size scaling in the two-dimensional kinetic Ising model in an oscillatingfield,” Phys. Rev. E , 016120 (2000).[8] D. T. Robb, P. A. Rikvold, A. Berger, and M. A. Novotny, “Conjugate field and fluctuation-dissipation relation for the dynamic phase transition in the two-dimensional kinetic Isingmodel,” Phys. Rev. E , 021124 (2007).[9] G. M. Buend´ıa and P. A. Rikvold, “Dynamic phase transition in the two-dimensional kineticIsing model in an oscillating field: Universality with respect to the stochastic dynamics,” Phys.Rev. E , 051108 (2008).[10] H. Park and M. Pleimling, “Dynamic phase transition in the three-dimensional kinetic Isingmodel in an oscillating field,” Phys. Rev. E , 032145 (2013).[11] H. Fujisaka, H. Tutu, and P. A. Rikvold, “Dynamic phase transition in a time-dependentGinzburg-Landau model in an oscillating field,” Phys. Rev. E , 036109 (2001); erratum: , 059903 (2001).[12] R. A. Gallardo, O. Idigoras, P. Landeros, and A. Berger, “Analytical derivation of criticalexponents of the dynamic phase transition in the mean-field approximation,” Phys. Rev. E , 051101 (2012).[13] O. Idigoras, P. Vavassori, and A. Berger, “Mean field theory of dynamic phase transitions inferromagnets,” Physica B: Condensed Matter , 1377 (2012).[14] D. T. Robb and A. Ostrander, “Extended order parameter and conjugate field for the dynamicphase transition in a Ginzburg-Landau mean-field model in an oscillating field,” Phys. Rev.E , 022114 (2014).[15] D. T. Robb, Y. H. Xu, O. Hellwig, J. McCord, A. Berger, M. A. Novotny, and P. A. Rikvold,“Evidence for a dynamic phase transition in [Co/Pt] magnetic multilayers,” Phys. Rev. B , 134422 (2008).[16] A. Berger, O. Idigoras, and P. Vavassori, “Transient behavior of the dynamically orderedphase in uniaxial cobalt films,” Phys. Rev. Lett. , 190602 (2013).[17] P. Riego, P. Vavassori, and A. Berger, “Metamagnetic anomalies near dynamic phase transi-tions,” Phys. Rev. Lett. , 117202 (2017).[18] P. A. Rikvold, H. Tomita, S. Miyashita, and S. W. Sides, “Metastable lifetimes in a kineticIsing model: Dependence on field and system size,” Phys. Rev. E , 5080 (1994).[19] L. Gammaitoni, P. H¨anggi, and P. Jung, “Stochastic resonance,” Rev. Mod. Phys. , 223(1998).[20] S. W. Sides, P. A. Rikvold, and M. A. Novotny, “Stochastic hysteresis and resonance in akinetic Ising system,” Phys. Rev. E , 6512 (1998).[21] G. Korniss, P. A. Rikvold, and M. A. Novotny, “Absence of first-order transition and tri-critical point in the dynamic phase diagram of a spatially extended bistable system in anoscillating field,” Phys. Rev. E , 056127 (2002).[22] Ph. A. Martin, “On the stochastic dynamics of Ising models,” J. Stat. Phys. , 149 (1977).[23] E. Vatansever, “Dynamically order-disorder transition in triangular lattice driven by a timedependent magnetic field,” arXiv:1706.03351 (2017).[24] A. N. Kolmogorov, “A statistical theory for the recrystallization of metals,” Bull. Acad. Sci.USSR, Phys. Ser. , 355 (1937).[25] W. A. Johnson and R. F. Mehl, “Reaction kinetics in processes of nucleation and growth,”Trans. Am. Inst. Mining and Metallurgical Engineers , 416 (1939).[26] M. Avrami, “Kinetics of phase change,” J. Chem. Phys. , 1103 (1939); , 212 (1940); , 177(1941).
27] R. A. Ramos, P. A. Rikvold, and M. A. Novotny, “Test of the Kolmogorov-Johnson-Mehl-Avrami picture of metastable decay in a model with microscopic dynamics,” Phys. Rev. B ,9053 (1999).[28] K. Binder and P. Virnau, “Overview: Understanding nucleation phenomena from simulationsof lattice gas models,” J. Chem. Phys. , 211701 (2016).[29] P. Riego, P. Vavassori, and A. Berger, “Towards an understanding of dynamic phase tran-sitions,” Physica B: Condensed Matter, in press https://doi.org/10.1016/j.physb.2017.09.043(2017).[30] J. S. Langer, “Theory of the condensation point,” Ann. Phys. (N.Y.) , 108 (1967).[31] J. S. Langer, “Statistical theory of the decay of metastable states,” Ann. Phys. (N.Y.) , 258(1969).[32] N. J. G¨unther, D. A. Nicole, and D. J. Wallace, “Instantons and the Ising model below T c ,”J. Phys. A: Math. Gen. , 1755 (1980).[33] P. A. Rikvold and M. Kolesik, “Analytic approximations for the velocity of field-driven Isinginterfaces,” J. Stat. Phys. , 377 (2000).[34] J. D. Gunton and M. Droz, Introduction to the Theory of Metastable and Unstable States (Springer-Verlag, Berlin, 1983).[35] I. M. Lifshitz, “Kinetics of ordering during second-order phase transitions,” Sov. Phys. JETP , 939 (1962) [Zh. ´Eksp. Teor. Fiz. , 1354 (1962)].[36] S. M. Allen and J. W. Cahn, “A microscopic theory for antiphase boundary motion and itsapplication to antiphase domain coarsening,” Acta Metall. , 1085 (1979). b -1-0.8-0.6-0.4-0.200.20.40.60.81 < Q > L=64L=128L=256L=512 (a) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2H b Q64 X b64 X Q128 X b128 X Q256 X b256 X Q512 X b512 (b) FIG. 1. Results with P = 1000 ≈ . P c for system sizes L = 64, 128, 256, and 512. With thisperiod length and range of system sizes, finite-size effects are negligible, and the curves for differentsystem sizes practically coincide. (a) The order parameter h Q i vs H b . Error bars are smaller thanthe symbol size. (b) The scaled variance χ QL and susceptibility χ bL . See discussion of this figure inSec. III A. m ( t ) L=32L=64L=256L=1024 (a) m ( t ) L=32L=64L=256L=1024 (b) m ( t ) L=32L=64L=256L=1024 (c) m ( t ) P = 1000P = 14,000 (d)
FIG. 2. The time-dependent magnetization m ( t ) over a few cycles following a 200 P stabilizationrun, using systems with L between 32 and 1024. In all four parts, the bias is positive, and thetotal applied field, H ( t ) + H b , is shown as an orange cosine curve. A detailed discussion of thisfigure is given in Sec. III B. (a) P = 1000 and H b = +0 .
10, just on the strong-bias side of thefluctuation peak for this period length. (b) P = 1000 and H b = +0 . P = 1000 and H b = +0 .
08, just on the weak-bias side of the fluctuationpeak. (d) L = 128 and a weak bias H b = +0 .
04 with two different period lengths, P = 1000 and14,000. The switching is deterministic and complete, and as P increases, it occurs earlier in thehalf-period. This observation suggests the asymptotic weak-bias, long-period approximation for h Q ( H b /H ) i , given in Eq. (9) and included in Fig. 4(a). m ( t ) Total fieldL = 32L = 1024 (a) (b)
FIG. 3. A short time series and snapshots showing growing disfavored-phase clusters for P = 1000at the corresponding peak position, H peak b = +0 . m ( t ) over five cycles followinga 200 P stabilization run, showing data for L = 32 (green) and 1024 (maroon). The total appliedfield, H ( t ) + H b , is shown as an orange cosine curve. The snapshots were captured the firsttime past 200 P that m ( t ) fell below +0 . .
45. The times of capture are marked by black circles. Inthe following snapshots, regions of the up-spin phase are green, and down-spin are red. (b) L = 32.A single droplet of the down-spin phase has nucleated near the time when the total applied fieldhas its largest negative value. The highly stochastic nature of the single-droplet switching modeis also evident from the time series in part (a). (c) L = 1024. Many droplets of the down-spinphase have nucleated at different times and then grown almost independently. At the moment ofcapture, some clusters have coalesced while others are still growing independently. From the timeseries in part (a) it is seen that this multi-droplet switching mode leads to a nearly deterministicevolution of the total magnetization. This figure is further discussed in Sec. III B. b -1-0.8-0.6-0.4-0.200.20.40.60.81 < Q > Eq. (9)P = 258 = P c P = 400P = 500P = 800P = 1000P = 1400P = 2000P = 3000P = 5000P = 7000P = 14000P = 28000 (a) -0.3 -0.2 -0.1 0 0.1 0.2 0.3H b X Q b Q b Q b Q b Q b Q b P (b) FIG. 4. Results for L = 128 and a range of periods between P c = 258 and P = 28 , h Q i vs H b . Error bars are on the order of the line thickness. The dashedcurve is the weak-bias, long-period approximation of Eq. (9). (b) The scaled variance χ QL and thesusceptibility χ bL vs H b . The sideband peaks occur at values of H b that increase with P . For clarity,data for some values of P are omitted in (b), including a narrow critical peak for P = P c = 258 at H b = 0 and a broad central peak for P = 400. | H b | L = 32L = 64L = 128L = 256 (a) p ea k / ( H - | H b | ) L = 32L = 64L = 128L = 256 (b) p ea k FIG. 5. Peak positions | H peak b | as defined by the maxima of the scaled variance χ QL , shown vsperiod length P ≥ P c . (a) | H peak b | vs P , plotted on linear scales. (b) The peak positions plottedas 1 / ( H − | H peak b | ) vs log P , as suggested by Eq. (10). The blue and green dashed lines representthe slopes of the curves between P = 14 ,
000 and 28,000 for L = 256 and L = 32, respectively.The ratio of the slopes is approximately 2.867, close to the 3/1 ratio expected from droplet theory.This figure is analogous to Fig. 2 of Ref. [18]. m ( t ) L = 32, H b = +0.1868L = 256, H b = +0.21125 FIG. 6. Time series of m ( t ) over five periods with P = 20 , P stabilizationrun. Data are shown at their respective values of H peak b for L = 32 (green) and 256 (blue). Thecorresponding values of the total applied field, H ( t ) + H peak b are also shown in orange and magenta,respectively. The wave forms of m ( t ), characteristic of single-droplet and multidroplet switchingare seen for L = 32 and 256, respectively. -0.3 -0.2 -0.1 0 0.1 0.2 0.3H b -1-0.8-0.6-0.4-0.200.20.40.60.81 < Q > P = 550 = 4.0 P c P = 5500P = 55000
FIG. 7. Simulated results (solid) and approximate theoretical results from Eqs. (C1) – (C3)(dashed) for h Q i with a square-wave field of amplitude H = 0 .
3. System size L = 128 andthree different field periods P . In a square-wave field, P c ≈
137 [7].137 [7].