Fluctuations in a model ferromagnetic film driven by a slowly oscillating field with a constant bias
Gloria M. Buendia Per Arne Rikvold
cond-mat.mtrl-sci cond-mat.stat-mech













apsrev







    #1⟨⟩#1√( #1) Tr
    
    ^1Department of Physics, Universidad Simón Bolívar, 
    Caracas 1080, Venezuela
    ^2 Department of Physics, Florida State University, 
    Tallahassee, FL 32306-4350,USA
    
    
    We present a numerical and theoretical study that supports and explains recent experimental 
    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 associated 
    dynamic phase transition (DPT) 
    [P. Riego, P. Vavassori, A. Berger, Phys. Rev. Lett.  118, 117202 (2017)]. 
    For this purpose, we perform kinetic Monte Carlo 
    simulations of a two-dimensional Ising model with nearest-neighbor ferromagnetic interactions 
    in the presence of a sinusoidally 
    oscillating field, to which is added a constant bias field. 
    We study a large range of system sizes and supercritical 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 the DPT, presents large 
    fluctuations that give rise to well-defined peaks in its scaled variance and its 
    susceptibility with respect to the bias field. The peaks are symmetric with 
    respect to zero bias and located at values of the bias field that 
    increase toward the field amplitude as an inverse logarithm of the 
    field oscillation period. 
    Our results indicate that this effect is independent of the system size 
    for large systems, ruling out critical behavior associated with a phase transition. 
    Rather, it is a stochastic-resonance phenomenon that has no counterpart in the 
    corresponding thermodynamic phase transition, providing a reminder that the equivalence 
    of the DPT to an equilibrium phase transition is limited to the critical region near the 
    critical period and zero bias.
    Fluctuations in a model ferromagnetic film driven by a slowly oscillating 
    field with a constant bias
        Gloria M. Buendía^1 and Per Arne Rikvold^2
        Received: date / Accepted: date
    =====================================================================================================
    
    
    
    
    § INTRODUCTION
    
    
    
    The hysteretic response when a uniaxial spin system with long-range order 
    ( i.e., below its critical temperature) 
    
    is subject to a symmetrically oscillating field of amplitude H_0 and period P, 
    depends crucially on P. If P is much 
    longer than the response time of the system (which depends on the 
    temperature and H_0), a symmetric hysteresis loop centered on zero results. 
    If P is much shorter than the response time, asymmetric hysteresis loops 
    centered around the values of the system's static order parameter are observed. 
    Numerical studies in 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 ⟨ Q ⟩ 
    (see formal definition in Sec. <ref>) 
    vanishes in a singular fashion. This phenomenon was first observed by 
    Tomé and de Oliveira <cit.> in a kinetic mean-field study of an Ising model, 
    followed by kinetic Monte Carlo (MC) simulations by Rao, Krishnamurthy, and Pandit <cit.> 
    and Lo and Pelcovitz <cit.>.  
    Early work in the field was reviewed 
    by Chakrabarti and Acharyya  
    in Ref. <cit.>.
    Kinetic MC combined with finite-size scaling analysis
     <cit.>, as well as further 
    mean-field studies of Ising and Ginzburg-Landau models 
    <cit.>, confirmed not only that this is 
    a true, dynamic phase transition (DPT), but also that it is in the same 
    universality class as the corresponding equilibrium Ising model. The DPT has 
    been confirmed experimentally in [Co/Pt]_3 magnetic multilayers 
    <cit.> and uniaxial Co films <cit.>.
    
    With all the attention that has been given to the DPT and its universality 
    class, one might lose sight of the fact that the equivalence between the 
    critical properties of the equilibrium Ising model and the DPT of the same 
    model in an oscillating field does not necessarily amount to equivalence  
    outside the critical region. A warning was provided very recently by Riego, 
    Vavassori, and Berger <cit.>. These authors fabricated Co films 
    with (1010) crystallographic surface structure with a single, 
    in-plane magnetic easy axis, which they subjected to a sinusoidally oscillating, 
    in-plane magnetic field plus a constant bias field H_b. Such a constant bias 
    field has previously been shown by MC simulations and finite-size scaling to be 
    (at least a significant component of) the field conjugate to ⟨ Q ⟩ in the 
    critical region near P_c<cit.>, and this has later been confirmed for 
    mean-field models <cit.> and in experiments <cit.>. 
    It therefore seemed 
    surprising that, in the experiments reported in Ref. <cit.>, 
    both the fluctuations in the order parameter and its derivative with respect to 
    H_b, for P ≫ P_c, behaved quite differently from the dependence of the 
    equilibrium susceptibility on the applied static field at temperatures above 
    critical. Instead of the wide, smooth, unimodal maximum of the supercritical equilibrium 
    susceptibility of the Ising model, 
    two distinct peaks were observed at nonzero values of H_b, symmetrical 
    about zero  <cit.>. In their article the
    authors also presented kinetic mean-field results that corroborate the presence of 
    these peaks, which they dubbed “sidebands."
    
    The purpose of the present paper is to investigate the long-period parameter 
    regime with kinetic MC simulations of a two-dimensional Ising model with 
    nearest-neighbor ferromagnetic interactions. 
    To match the experimental conditions of Ref. <cit.> as closely as possible, we 
    choose the oscillating field to have a sinusoidal waveform. 
    We are not aware that systematic simulations in this regime have been performed previously. 
    Our study reveals “sidebands" analogous to the experimental results. 
    We thus conclusively confirm that the experimentally 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 of the kinetics of magnetization switching by homogeneous 
    nucleation and growth of antiphase droplets <cit.>, 
    we demonstrate that the “sidebands"  result from noncritical fluctuations during 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 <cit.>. 
    
    The rest of this paper is organized as follows. 
    In Sec. <ref> we describe the model and details of the simulation method, and we define 
    the appropriate observables to be measured. 
    Our numerical results are presented in Sec. <ref>. 
    In Sec. <ref> 
    we present numerical observation of sidebands for a single, supercritical 
    value of the field period. 
    In Sec. <ref> we present short time series of 
    the system magnetization for several values of bias and period, 
    which enable us to propose a simple approximation for ⟨ Q ⟩ in the limits of 
    weak bias and long period. In Sec. <ref> we present numerical results for 
    ⟨ Q ⟩
    vs H_b for a wide range of supercritical periods, as well 
    as the sideband positions H_b^ peak as functions of period and system size. 
    The latter are analyzed using results from the droplet theory of magnetization reversal. 
    Our conclusions are given in Sec. <ref>. A short summary  
    of pertinent results from the droplet theory of magnetization reversal is given in  
    Appendix <ref>, and the case of extremely long periods is discussed in Appendix 
    <ref>. A brief discussion of the mathematically 
    simpler case of a square-wave oscillating field is presented in Appendix <ref>. 
    
    
    
    
    § MODEL AND MONTE CARLO SIMULATION
    
    We consider a kinetic S=1/2 Ising model with a time-dependent external 
    field and ferromagnetic nearest-neighbor interactions.  Its Hamiltonian is 
    
        ℋ=-J∑_⟨ ij ⟩ s_is_j - [ H(t)+H_b] ∑_i s_i ,
    
    where J>0, s_i=±1, the first sum runs over all nearest-neighbor pairs, 
    and the second one over all sites. H_b is a constant “bias field," and H(t) 
    is a symmetrically oscillating external field of period P. Here we choose
    
        H(t)=H_0cos( 2 π/P t )  .
    
    The system is simulated on a square lattice of N=L × L sites with 
    periodic boundary conditions.  
    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)=1/1+exp(βΔ E_i) ,
    
    where Δ E_i is the change in the system energy associated 
    with flipping the spin i, and β=1/k_ BT 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_0, H_b, and T are 
    all given in units of the interaction constant J (i.e., J=k_ B=1), 
    and P is given in units of MCSS. 
    
    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 equilibrium with a heat bath 
    <cit.>. However, the DPT with H_b=0 has been shown to be universal 
    with respect to dynamics that obey detailed balance in equilibrium, including 
    Metropolis <cit.> and “soft Glauber" <cit.>, as well as different 
    forms of H(t) including square-wave <cit.> and sawtooth 
    <cit.>.
    
    We calculate the time dependent, normalized magnetization per site,
    
        m(t)=1/L^2∑_is_i(t)  ,
    
    and by integrating it over each cycle of the 
    magnetic field, we obtain the average magnetization during the kth 
    cycle of the field,
    
        Q_k=1/P∫_(k-1)P^kP m(t)dt  .
    
    The dynamic order parameter of the model 
    is the period-averaged magnetization, ⟨ Q ⟩, 
    defined as the average of Q_k over many cycles. Its fluctuations are measured by the scaled variance,
    
        χ_L^Q=L^2(⟨ Q^2⟩ -⟨ Q ⟩^2)  ,
    
    and its dependence on the bias field is measured by the susceptibility with respect to H_b,
    
        χ_L^b=d⟨ Q ⟩/dH_b .
    
    
    In order to take advantage of temperature and field dependent parameters measured 
    with high precision in previous MC simulations <cit.>, 
    our calculations are performed with H_0=0.3 at T=0.8T_c, where 
    T_c=2/ln(1+√(2))≈ 2.269 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 field reversal from 
    -H_0 to +H_0, occurs via a 
    nearly deterministic and L-independent multi-droplet mechanism <cit.>. 
    In Ref. <cit.>, the characteristic switching timescale (the time from the field 
    reversal until the system magnetization reaches zero) under Glauber dynamics with the same 
    parameters as we use here was measured by 
    MC simulations as τ_0 ≈ 74.6. 
    In the same work, the critical period in a sinusoidal field of amplitude H_0 with zero bias 
    was measured as P_c ≈ 258. 
    
    The cycle-averaged magnetization 
    ⟨ Q ⟩ vanishes for P ≥ P_c and H_b=0. 
    Near criticality, the constant bias field H_b is the field conjugate to ⟨ Q ⟩, and 
    the period P mimics the temperature in the equilibrium phase transition. 
    Simulations were performed for periods between P = 258 and 28,000 and 
    system sizes between L = 32 and 1024.  
    Except for the smallest 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 for each measurement.
    
    
    
    
    § NUMERICAL RESULTS AND ANALYSIS
    
    
     §.§ Observation of “sidebands"
    
    Results of simulations with P = 1000 ≈ 3.9P_c 
    for several values of L are displayed in 
    Fig. <ref>. “Sidebands" are observed, consistent with the 
    experiments reported in Ref. <cit.>. The dependence of 
    the order parameter ⟨ Q ⟩ on the bias H_b is shown in Fig. <ref>(a). 
    For weak
    H_b, ⟨ Q ⟩ increases almost linearly with H_b, but the slope of the curve
    increases considerably around |H_b| ≈ 0.09, followed by saturation of 
    ⟨ Q ⟩ for |H_b| ≳ 0.15. 
    This behavior is reflected in the bimodal shape of the susceptibility χ_L^b, 
    shown by the lower set of curves 
    in Fig. <ref>(b). Between the two peaks lies a flat-bottomed valley corresponding 
    to the linear regime in part (a), and a rapid approach to zero for large |H_b| mirrors the 
    saturation of ⟨ Q ⟩ also seen in (a). 
    The scaled variance χ_L^Q also displays peaks, whose positions coincide with those of 
    χ_L^b. However, the ratio χ_L^Q / χ_L^b for fixed P 
    depends quite strongly on H_b with 
    maximum values near the peaks. This variable ratio precludes a straightforward interpretation 
    in terms of an effective, nonequilibrium fluctuation-dissipation relation with P playing the 
    role of “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 between system size, field period, and finite-size effects will be discussed 
    in further detail below. 
    
    
    
     §.§ Magnetization time series
    
    To gain a more detailed understanding of the relationships between bias, period, system size, 
    and the order-parameter fluctuations, we present in Fig. <ref> short time series of the normalized magnetization, m(t). 
    The total applied field, H(t) + H_b, is shown as an orange curve. 
    In this figure we set H_b > 0, so that the up-spin phase is favored and 
    the down-spin phase is disfavored. 
    
    Figure <ref>(a) shows data for 
    P=1000 and H_b=+0.10, just on the strong-bias side of the 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 is stochastic and abrupt 
    (mediated by a single or a few droplets of the down-spin phase <cit.>) 
    and occurs only in narrow time windows near the negative extrema of the total applied 
    field. For the larger systems, the switching becomes more deterministic and gradual 
    (multi-droplet <cit.>). 
    However, the growing down-spin phase does not have time 
    to completely fill the system before the field 
    again becomes positive. For the largest system studied, L=1024, the extreme 
    negative magnetizations during a period are close to -0.2.
    
    Figure <ref>(b) shows data for 
    P=1000 and H_b=+0.0915, at the maximum of the fluctuation peak. 
    The switching behavior for L=32 
    remains stochastic. However, the larger systems appear more deterministic, and their extreme 
    negative magnetizations during a period are close to -0.4. 
    
    Figure <ref>(c) shows data for 
    P=1000 and H_b=+0.08, just on the weak-bias side of the fluctuation peak. 
    The switching for L=32 remains stochastic. The larger systems behave more deterministically, 
    and the extreme negative magnetizations during a period approach -0.8. 
    
    These results illustrate how the switching behavior in the peak region crosses over from 
    a stochastic single-droplet mechanism for small L to a nearly deterministic multidroplet 
    mechanism for larger L, in agreement with known results for field-driven magnetization 
    switching by homogeneous nucleation and growth of droplets of the stable phase <cit.>. 
    
    Figure <ref>(d) shows data for 
    L=128 with a weak bias, H_b = +0.04, and two different period lengths, P=1000 
    and 14,000. In both cases, the switching is nearly deterministic and complete, so that the 
    period-averaged magnetization ⟨ Q ⟩ depends mostly on the 
    relative amounts of time the 
    system spends in the two phases. As P increases, the switching occurs earlier in the half-period. 
    
    The differences between the single-droplet and multidroplet switching 
    modes are further illustrated in Fig. <ref>.
    In Fig. <ref>(a),
    time series for m(t) over five cycles with P=1000 
    at the corresponding peak position, H_b^ peak = +0.0915 
    show data for L=32 and 1024. All the parameters are the same as in 
    Fig. <ref>(b), except the seed for the random number generator.
    When the total applied field, H(t) + H_b, is negative, 
    the down-spin phase, which is 
    disfavored by the positive bias, is the equilibrium phase. 
    Nucleation and growth of this phase may only occur during the time intervals of 
    negative total applied field. 
    Snapshots captured at m(t) = +0.1 during these growth periods, 
    corresponding to a down-spin fraction of 0.45, are shown in 
    Fig. <ref>(b) for L=32 and in Fig. <ref>(c) for 
    L=1024. 
    
    For L=32 we see a single down-spin droplet which, as seen from 
    the time series in Fig. <ref>(a), nucleated during the 
    third period shown, near the time when the 
    field had its largest negative value. It barely reached the capture threshold 
    of m=+0.1 before the field again became positive and caused it to 
    decay. The stochastic nature 
    of this single-droplet switching mode is also 
    clearly reflected by the time series. 
    During the five periods shown, the capture threshold was only reached twice. 
    And only once, during the fifth period, do we see full saturation of the 
    down-spin phase before the field again becomes positive.
    
    For L=1024 the picture is quite different. In the snapshot we see 
    a large number of growing clusters that have nucleated at different times 
    during the negative-field time interval. Some of these have already coalesced 
    by the time the snapshot was captured, 
    while others are still growing independently.    
    From the time series it is seen that this multi-droplet 
    switching mode leads to a nearly deterministic evolution of the total 
    magnetization, with 
    the underlying stochasticity only evident in the slight variations of the 
    minimum magnetization values from period to period.  
    This switching process is well described by the 
    Kolmogorov-Johnson-Mehl-Avrami (KJMA) 
    approximation <cit.>. 
    
    Magnetization reversal from the favored to the disfavored direction is only possible while 
    the total applied field, 
    H(t) + H_b, has the opposite sign of the bias, H_b. This implies that 
    -1 < H_b/H(t) ≤ 0. Switching from the favored phase to the disfavored one on average takes 
    longer time than switching in the opposite direction. 
    Thus, the time the system can spend in the disfavored phase 
    during each period must be less than or equal to the time that the field has the 
    disfavored direction, 
    
        t_ Dmax 
        = P/2[ 1 - 2/πsin^-1( |H_b|/H_0) ]  .
    
    In this limit of long period and weak bias, ⟨ Q ⟩ is simply determined by 
    the sign of H_b and the difference between the fractions of the period that the total field 
    has the same and the opposite sign as H_b, respectively. This yields  
    
        ⟨ Q ⟩≈2 m_0/πsin^-1( H_b/H_0)  ,
    
    which is symmetric under simultaneous reversal of H_b and ⟨ Q ⟩.
    Here, m_0 is the magnitude of the magnetization in the favored phase. 
    This approximation represents a lower bound on the magnitudes of ⟨ Q ⟩ 
    and χ^b<cit.>. The former 
    is included as a dashed curve in Fig. <ref>(a). 
    However, the bounds depend on the waveform of the oscillating field, and as we 
    show in Appendix <ref>, they vanish in the case of a square-wave field. 
    
    The corrections to this approximation are of O (t_ FD(H_b,H_0)/P ), where 
    t_ FD(H_b,H_0) is the average time it takes the magnetization to switch to the 
    disfavored direction,  after the total applied field has changed sign. For |H_b| ≪ H_0, the 
    correction vanishes as 1/P, as seen in Fig. <ref>(a). 
    However, for larger |H_b|, t_ FD(H_b,H_0) ∼ P, and 
    the “correction" becomes the dominant part of ⟨ Q ⟩, determining the sideband 
    peak positions, H_b^ peak. 
    The details are discussed below in Sec. <ref>.
    
    
    
    
     §.§ Dependence on H_b, P, and L
    
    
    Results for L=128 and a range of periods between P_c = 258 and P = 28,000  
    are shown in Fig. <ref>. In the critical region, H_b is the field conjugate to 
    ⟨ Q ⟩<cit.>.
    At P = P_c, ⟨ Q ⟩ therefore vanishes in a singular fashion 
    as H_b approaches zero. On the scale of Fig. <ref>(a), 
    this singularity appears as a jump in ⟨ Q ⟩ at 
    H_b = 0 for P=P_c, resulting in very narrow central peaks 
    in both χ_L^b and χ_L^Q. 
    We also found broad central peaks in both quantities for P = 400, which are due to 
    finite-size broadening of the critical region for this relatively modest system size. 
    For clarity, these central peaks are not included in Fig. <ref>(b). 
    Beyond P = 500, ⟨ Q ⟩ becomes linear for small H_b, with a slope that 
    approaches that of the asymptotic approximation in Eq. (<ref>) as P increases. 
    Simultaneously, the peaks in χ_L^b and χ_L^Q increase in height, and their positions 
    H_b^ peak move in the directions of ± H_0, as seen in Fig. <ref>(b). 
    [For clarity, some of the values of P included in 
    Fig. <ref>(a) are excluded from Fig. <ref>(b).]
    
    The magnitudes of the peak positions, |H_b^ peak|, are plotted vs P for different 
    values of L in Fig. <ref>(a). We note two main features. First, 
    |H_b^ peak| increases quite rapidly with P for relatively short periods, and 
    much more slowly for longer periods. 
    This behavior is consistent with the experimental data shown in Fig. 2 of Ref. <cit.>. 
    Second, finite-size effects are essentially 
    negligible for P ≲ 2000, as already shown in 
    Fig. <ref> for P=1000. For longer periods, |H_b^ peak| 
    increases with L for smaller sizes, and then becomes size independent for larger L. 
    
    In order to explain this behavior quantitatively, we first recall from the time series 
    shown in Fig. <ref> that for bias near |H_b^ peak|, the time it takes 
    m(t) to change significantly toward the disfavored sign is on the order of a finite fraction of P. 
    For stronger bias, the total field driving the magnetization toward the disfavored sign
    is too weak and consequently the time required for switching is much longer than P, 
    so that reliable 
    magnetization reversal does 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 significantly shorter than 
    P.  In other words, the peak positions correspond to bias values that produce 
    magnetization reversal on a timescale of P. Equations 
    for magnetization switching rates by the stochastic single-particle mechanism that 
    dominates for small systems 
    [Eq. (<ref>)] and the nearly deterministic multidroplet mechanism 
    that dominates for large systems [Eq. (<ref>)] 
    are found in Appendix <ref>.
    The nucleation rate for droplets of the disfavored phase 
    varies very strongly with the oscillating field, having appreciable values 
    only in a narrow window near the maximum field in the disfavored direction, 
    |H| = H_0 - |H_b^ peak|. 
    Using this value of |H| and ignoring less important prefactors, we can 
    use these equations to write the following requirement for |H_b^ peak|: 
    
        L^-aexp( 1/bΞ_0/H_0 - |H_b^ peak| ) ∼ P  ,
    
    with a=2 and b=1 for single-droplet switching, and a=0 and b=3 for 
    multidroplet switching. 
    The meaning of the constant Ξ_0 ≈ 0.506 is explained in Appendix <ref>. 
    In either case, this equation is equivalent to a statement that |H_b^ peak| 
    should approach H_0 asymptotically as 1/ log P for long periods. 
    (A caveat to this statement for the case of extremely long periods is discussed in 
    Appendix <ref>.) 
    Plotting 1/(H_0 - |H_b^ peak|) 
    vs log P therefore should produce straight lines for large values of P. 
    The ratio between the slopes of the lines representing multidroplet switching for large L 
    and those representing single-droplet switching for small L should be 3/1. 
    Such a plot is presented in Fig. <ref>(b). The slope ratio between the curves 
    representing L=256 and L=32 in the long-P regime is approximately 2.867, 
    consistent with the theoretical prediction. This conclusion is confirmed by the short 
    time series of m(t) for P=20,000 for these two system sizes, shown in 
    Fig. <ref>. In the switching regions, the smaller system displays the 
    stochastic, square wave form characteristic of single-droplet switching <cit.>, 
    while the larger system shows the continuous 
    wave form characteristic of multidroplet switching <cit.>. 
    
    To further support our conclusions, we calculated the transition times and the order parameter 
    in the multidroplet regime for the mathematically simpler case, 
    in which the sinusoidally oscillating field has 
    been replaced by a square-wave field. The details of the calculations are given in Appendix 
    <ref>. In Fig. <ref> we show that there is very good agreement between the theoretically 
    calculated ⟨ Q ⟩ and the simulations, particularly when 
    |H_b| ≲ |H_b ^ peak|.
    
    
    
    
    § SUMMARY AND CONCLUSION
    
    
    Riego  et al.<cit.> recently presented experimental data on Co films with a 
    single, in-plane magnetic easy axis, which were subjected to a slowly oscillating magnetic 
    field with an added constant bias. In this paper we have presented kinetic MC simulations and 
    theoretical analysis of a two-dimensional Ising ferromagnet with only 
    nearest-neighbor interactions, designed to closely mimic the experimental setup. 
    At zero bias, such systems exhibit a dynamic phase transition (DPT) at a critical period P_c, 
    where the period-averaged magnetization ⟨ Q ⟩ vanishes in a singular fashion. 
    It has previously been shown that the DPT belongs to the equilibrium Ising universality class, 
    with P playing the role of temperature and the bias H_b being the field conjugate to 
    ⟨ Q ⟩. 
    Following Riego  et al.<cit.>, 
    we studied the dynamics of the system at values of P aboveP_c, and in agreement with the experiments we found that ⟨ Q ⟩ 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 that the 
    experimental results are  not due to any residual magnetostatic interactions. 
    The simulational approach also enables studies of the effects of finite system size. 
    We found that, at fixed P, finite-size effects saturate beyond a P-dependent size limit. 
    Using the droplet theory of magnetization switching, we conclude that this saturation occurs 
    at the crossover between two different dynamic regimes. For small systems, the magnetization 
    switching from the favored to the disfavored direction occurs by a 
    stochastic single-droplet mechanism. For large systems, the switching occurs by the size-independent 
    and nearly deterministic KJMA 
    mechanism, 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 these insights will be important for the design and analysis of devices that involve 
    magnetization reversal by time-varying fields, such as memory elements, switches,  
    and actuators. 
    
    
    
    
    § ACKNOWLEDGMENTS
    
    We thank A. Berger for providing data from Ref. <cit.> before publication, 
    and for useful comments on an earlier version of this paper. 
    G.M.B. is grateful for the hospitality of the Physics Department at Florida State University, 
    where her stay was supported in part by 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. 
    
    
    
    § MECHANISMS OF MAGNETIZATION REVERSAL
    
    When a d-dimensional Ising ferromagnet below its critical temperature is subjected to 
    the reversal of an 
    applied field of magnitude |H|, the homogeneous nucleation rate per unit system volume 
    for droplets of the new equilibrium phase is given 
    by <cit.>
        I(H) ≈ B(T) |H|^K exp[ - Ξ_0 (T)/|H|^d-1]  ,
    
    where B(T) is a non-universal function of T. For d=2, K=3, and  
    Ξ_0(0.8 T_c) ≈ 0.506 (which includes a factor of 1/T)  <cit.>. 
    The argument of the exponential function is the negative of the 
    free energy of a critical droplet of the equilibrium phase, divided by T. 
    The inverse of 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 moderately weak field, 
    the time it takes for the first nucleated droplet to grow to fill the system 
    is much shorter than the average nucleation time. As a result, the magnetization reversal is 
    completed by this single, first droplet. 
    
     Multidroplet reversal mechanism:
    Under conditions of large system and/or moderately strong field, 
    the average time between nucleation events is less than the time it would take the first 
    nucleated droplet to grow to fill the system. Therefore, many droplets nucleate and grow 
    independently in different parts of the system until they coalesce and collectively fill the system. 
    The result is a gradual and nearly deterministic growth of the new phase through a multidroplet 
    process, well described by the KJMA 
    approximation <cit.>. 
    The characteristic reversal time is independent of the system size and given by 
    
        ⟨τ(H) ⟩∝[ v^d I(H) ] ^-1/(d+1) ,
     
    where the propagation velocity of  the droplet surface, v, 
    is proportional to |H| in this parameter range 
    <cit.> as expected from the Lifshitz-Allen-Cahn approximation 
    <cit.>. 
    
    
    
    § 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 <cit.>. In this 
    regime, the droplet is replaced by a slab of the equilibrium phase, and the nucleation time no longer 
    depends on |H|, but increases exponentially with L^d-1. 
    The critical droplet radius in d dimensions is given by  
    <cit.>,
    
        R_c ≈( (d-1) T Ξ_0/2 m_0 Ω_d)^1/d1/|H| ,
    
    where Ω_d is the volume of the critical droplet, divided by R_c^d. 
    Numerical  values for the constants with d=2 at 
    T=0.8T_c ≈ 1.815 are found in Table I of Ref. <cit.>: 
    Ξ_0 ≈ 0.506 and Ω_2 ≈ 3.152. 
    (The factor T is included in the numerator to cancel the factor 1/T in Ξ_0.)
    Thus we have 
    
        R_c ≈0.388/|H|≈L/2 .
    
    Replacing |H| by H_0 - |H_b| and setting L=32, we thus find 
    1/(H_0 - |H_b|) ≈ 41.3. Finally, linearly extrapolating the large-P data for L=32 in 
    Fig. <ref>(b), we find that the single-droplet result from Eq. (<ref>) 
    should remain valid for periods up to approximately 10^19 ± 2. 
    [The uncertainty in the exponent is the result of assuming a 10% uncertainty in the estimate of 
    1/(H_0 - |H_b|).]
    Beyond this limit, H_0 - |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. 
    We do not expect that these extremely long periods should be of great experimental relevance for 
    macroscopic systems. However, for nanoscopic systems the coexistence regime may be 
    observable with experimentally accessible periods.
    
    
    
    
    § SQUARE-WAVE OSCILLATING FIELD
    
    Now, instead of a sinusoidally oscillating field, consider a square-wave field, such that 
    H(t) = +H_0 during one half-period, and -H_0 during the other. 
    Since the times that the total field is parallel and antiparallel to H_b now each equal P/2, 
    the equivalent of the long-period, weak-bias approximation of Eq. (<ref>) becomes 
    ⟨ Q ⟩≈ 0. Therefore, the value of ⟨ Q ⟩ for finite P and 
    weak H_b is determined by the difference between the average magnetization reversal times 
    following a change of the total field from the favored to the disfavored direction, and the 
    opposite. Since the total field now has its full favored or disfavored strength during the whole 
    half-period, these average switching times will be shorter than the corresponding times in the 
    sinusoidally oscillating field case. With a square-wave field 
    of amplitude H_0 = 0.3 at 0.8 T_c under Glauber dynamics, the critical period has 
    been measured by MC simulations as P_c ≈ 137<cit.>. 
    To calculate the transition times for a two-dimensional system in the multidroplet regime, 
    we will again assume H_b ≥ 0 for concreteness. 
    
    From Eqs. (<ref>) and (<ref>) with |H| = H_0 - H_b, 
    we obtain the characteristic timescale for 
    transitions from the favored (parallel to the bias field) to the disfavored 
    magnetization direction, after the total applied field has changed sign as 
    
        t_ FD(H_b,H_0) = τ_0 ( 1/1 - H_b/H_0)^5/3exp(Ξ_0/3 H_0H_b/H_0/1 - H_b/H_0)  ≥τ_0  ,
    
    where τ_0 is the magnetization reversal time for H_b = 0.  
    Analogously, the switching time from the disfavored to the favored magnetization direction is 
    
        t_ DF(H_b,H_0) = τ_0 ( 1/1 + H_b/H_0)^5/3exp( - Ξ_0/3 H_0H_b/H_0/1 + H_b/H_0)  ≤τ_0  .
    
    Both t_ FD and t_ DF reduce to τ_0 ≈ 74.6<cit.> 
    for H_b = 0. 
    
    The order parameter ⟨ Q ⟩ is determined 
    by P and the difference between t_ FD and t_ DF as 
    
        ⟨ Q ⟩≈{[ 2 m_0 t_ FD - t_ DF/P                                   t_ FD≤P/2;                   m_0                                 t_ FD > P/2 ].
    
    This approximation is shown together with simulation results in Fig. <ref>. 
    The agreement is very good for |H_b| ≲ |H_b^ peak|. 
    
    
    
    
    
    
    
    
    
    
    
    
    34
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