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Quantifying the Thermal Stability in Perpendicularly Magnetized Ferromagnetic Nanodisks with Forward Flux Sampling

L. Desplat and J.-V. Kim
Phys. Rev. Applied 14, 064064 – Published 22 December 2020
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  • INTRODUCTION
  • GEOMETRY AND METHOD
  • VARIATION OF THERMAL STABILITY WITH DMI
  • DISCUSSION AND CONCLUDING REMARKS
  • ACKNOWLEDGMENTS
    • References

    Abstract

    The thermal stability in nanostructured magnetic systems is an important issue for applications in information storage. From a theoretical and simulation perspective, an accurate prediction of thermally activated transitions is a challenging problem because desired retention times are of the order of 10 years, while the characteristic timescale for precessional magnetization dynamics is of the order of nanoseconds. Here, we present a theoretical study of the thermal stability of magnetic elements in the form of perpendicularly magnetized ferromagnetic disks using the forward flux sampling method, which is useful for simulating rare events. We demonstrate how rates of thermally activated switching between the two uniformly magnetized “up” and “down” states, which occurs through domain-wall nucleation and propagation, vary with the interfacial Dzyaloshinskii-Moriya interaction, which affects the energy barrier separating these states. Moreover, we find that the average lifetimes differ by several orders of magnitude from estimates based on the commonly assumed value of 1 GHz for the attempt frequency.

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    • Received 28 July 2020
    • Revised 23 October 2020
    • Accepted 2 December 2020

    DOI:https://doi.org/10.1103/PhysRevApplied.14.064064

    © 2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Magnetization dynamicsMicromagnetismSpintronics
    1. Physical Systems
    FerromagnetsNanodisksUltrathin films
    1. Techniques
    Landau-Lifschitz-Gilbert equationMicromagnetic modeling
    Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

    Authors & Affiliations

    L. Desplat1,2,* and J.-V. Kim1,†

    • 1Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, Palaiseau 91120, France
    • 2Institut de Physique et Chimie des Matériaux de Strasbourg, CNRS, Université de Strasbourg, Strasbourg 67034, France

    • *louise.desplat@ipcms.unistra.fr
    • †joo-von.kim@c2n.upsaclay.fr

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    Vol. 14, Iss. 6 — December 2020

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    • Figure 1
      Figure 1

      Schematic of the forward flux sampling method. (a) Minimum energy path for magnetization reversal between the up and down states through domain-wall nucleation and propagation. Progression along the reaction coordinate is parameterized by the order parameter ζ. (b) Sequence of interfaces {λ} separating basins A and B. We record N0 crossings at the interface λ0 when the system starts in A and successfully reaches λ0. (c) Simulated trajectories between subsequent interfaces. The micromagnetic configuration is stored at λi+1 for each successful crossing from λi to λi+1, which serves as a starting point for Langevin dynamics simulations toward the next interface. Failed crossings involve returns to basin A.

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    • Figure 2
      Figure 2

      Averaged magnetization configurations at different interfaces λ for different runs involving three values of the DMI, D. Here λ7 corresponds to the interface at which ζ=0. Each configuration is rotated about the disk center such that wall propagation proceeds from left to right during reversal before the averaging procedure is performed. Note that, although we represent the edges of the disk as smooth, in reality, the boundary exhibits a staircase effect due to the discretization.

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    • Figure 3
      Figure 3

      In the underdamped limit with DDIs: (a) Conditional probabilities P(λi∣λi−1) as a function of the interface λi for three different values of the DMI D. (b) Cumulative conditional probabilities P(λi∣λ0), determined from the cumulative products of P(λi∣λi−1) in (a), as a function of the order parameter ζ for three values of D.

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    • Figure 4
      Figure 4

      In the overdamped limit with effective anisotropy, cumulative probability P(λi∣λ0) as a function of the order parameter ζ for different values of D: (a) 0.5, (b) 1.0, (c) 1.5, and (d) 2.0mJ/m2. The insets show the conditional probability P(λi∣λi−1) as a function of the interface λi. The colored lines represent single FFS runs, while the solid black lines represent averages over the different runs.

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    • Figure 5
      Figure 5

      Example of interface optimization in FFS. (a) The new values of the order parameter {ζiopt} are obtained by inverting the interpolation function in Eq. (11), which is plotted with a dotted line. The {ζiopt} are designed to approach a constant flux of partial trajectories through each interface. (b) Partial fluxes P(λi∣λi−1) as a function of the order parameter ζ resulting from the interface optimization procedure. With the exception of the first and last interfaces, the values are found within the shaded area. (c) Evolutions of the order parameter ζ (left axis, triangles) and the energy normalized by the thermal energy at 300 K, β300E (right axis, circles), as a function of the reaction coordinate χ along the minimum energy path. The interfaces are optimized again after the first FFS run with the interfaces shown in (a). The number of interfaces per interval of ζ is represented by the colored background, where a darker color corresponds to more interfaces. Denser interfaces are needed before the saddle point, around ζ=0.58.

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    • Figure 6
      Figure 6

      In the underdamped limit with effective anisotropy, cumulative probability P(λi∣λ0) as a function of the order parameter ζ for different values of D. The inset shows the same probabilities normalized by PB=P(λB∣λ0). The normalized probabilities for different values of D are superimposed because the flux of barrier recrossings remains a constant fraction of the rate constant, independently of D.

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    • Figure 7
      Figure 7

      Average lifetime τ as a function of the DMI D at T=300 K computed with FFS and direct Langevin (DL) dynamics simulations. (a) Lifetimes for α=0.01 with full dipole-dipole interactions. (b) Lifetimes for α=0.01 and α=0.5 with effective perpendicular anisotropy. The average lifetime computed from full Langevin dynamics simulations is given for all three cases at D=2mJ/m2. The insets give a comparison of the FFS data with the constant f0=1 GHz approximation and the energy barriers respectively computed in Refs. [5] and [43].

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    • Figure 8
      Figure 8

      Direct Langevin dynamics simulations of the dwell time τ for D=2mJ/m2. The results correspond to (a),(b) α=0.01 with full dipolar interactions, (c),(d) α=0.01 with effective perpendicular anisotropy, and (e),(f) α=0.5 with effective perpendicular anisotropy. For each case, 100 simulations are performed with a successful switching event. (a),(c),(e) Probability distribution of the dwell time. The solid line represents a fit to an exponential function. (b),(d),(f) The mz component of the magnetization as a function of time, where the curves are shifted along the time axis such that the first crossing of −mz,0 occurs at t=0. The colored curves represent the individual Langevin dynamics simulations, while the solid black line represents an average of these curves.

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    • Figure 9
      Figure 9

      Effects of (a) damping and (b) dipole-dipole interactions on the cumulative conditional probabilities normalized by PB, P(λi∣λ0)/PB. The transparent colored lines show the cumulative probabilities for all values of D as a function of the order parameter ζ, where the color of the line represents the value of D with the same colorscale as that of Fig. 6. In both graphs, the solid colored lines correspond to the underdamped case with effective anisotropy, while the dashed colored lines correspond to the other case. The black lines show the average cumulative conditional probabilities as a function of the average order parameter for each case.

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