The goal of this paper is to investigate the short time-scale, thermally-induced probability of magnetization reversal for an biaxial nanomagnet that is characterized with a biaxial magnetic anisotropy. For the first time, we clearly show that for a given energy barrier of the nanomagnet, the magnetization reversal probability of an biaxial nanomagnet exhibits a non-monotonic dependence on its saturation magnetization. Specifically, there are two reasons for this non-monotonic behavior in rectangular thin-film nanomagnets that have a large perpendicular magnetic anisotropy. First, a large perpendicular anisotropy lowers the precessional period of the magnetization making it more likely to precess across the x^=0 plane if the magnetization energy exceeds the energy barrier. Second, the thermal-field torque at a particular energy increases as the magnitude of the perpendicular anisotropy increases during the magnetization precession. This non-monotonic behavior is most noticeable when analyzing the magnetization reversals on time-scales up to several tens of ns. In light of the several proposals of spintronic devices that require data retention on time-scales up to 10’s of ns, understanding the probability of magnetization reversal on the short time-scales is important. As such, the results presented in this paper will be helpful in quantifying the reliability and noise sensitivity of spintronic devices in which thermal noise is inevitably present.

The interaction of the magnetic order parameter with the underlying thermal fluctuations of the magnetic body has been analyzed extensively in prior literature.1–4 Most prominently, W. Brown, in his seminal works in 1963 to 1979, developed the “Brownian motion” model of thermal noise. Using the Fokker-Planck analysis, Brown showed that the probability of the thermal reversal in fine ferromagnetic bodies (i) is a single exponential function with respect to time and (ii) it varies monotonically with the energy barrier of the nanomagnet.5,6 Brown’s analysis is specifically applicable in cases when the magnetization energy is “independent of the radial angle”5 implying the absence of demagnetization field caused by the shape anisotropy of the magnetic body. The lack of the demagnetization field decouples the dimensions of the Landau-Lifshitz-Gilbert (LLG) equation and allows analytical solutions of the thermally-induced magnetization reversal probability.7 However, thin-film biaxial nanomagnets contain both a negative uniaxial anisotropy along its free-axis, and a positive shape anisotropy oriented perpendicular to the free-axis. As such, the magnetization energy depends on both the azimuthal and polar angles of the magnetization vector, which does not allow for the dimensional decoupling of the LLG equation.

While building upon prior works, including those of Brown, in this work, we specifically analyze the magnetization reversal probability of thin-film nanomagnets that are characterized with a biaxial magnetic anisotropy. The analysis is conducted for a variety of material parameters of the magnetic body such as the magnetic saturation and the uniaxial energy density. For the first time, we show that for a given energy barrier, the magnetization reversal probability varies non-monotonically with the magnitude of the perpendicular shape anisotropy. Specifically, we consider two cases of the instability in magnetization resulting from the thermal noise that eventually lead to magnetization reversal within a given time period. These two cases of thermal reversals are defined as follows.

  • Case-I: reversal occurs when the magnetization vector crosses the x^ = 0 plane

  • Case-II: reversal occurs when the magnetization energy exceeds the energy barrier of the nanomagnet

In Case-I, an increase in the demagnetization field increases the frequency of precessional energy orbits, which makes it more likely for the magnetization to cross x^=0 plane once the magnetization energy exceeds a particular energy threshold. However, we find that the reversal probability in Case-II also varies non-monotonically with the demagnetization energy. This is because a larger perpendicular anisotropy shapes the energy orbit, minimizing its z^ components, and allows the thermal field to induce a greater torque on the magnetization. Finally we test the limits of this non-monotonic behavior and show that while this behavior is especially present in small-time scale (sub-100 ns) measurements, it is diminished at very large time-scales.

We consider a single-domain thin-film nanomagnet of size 60 nm × 45 nm × 2 nm whose magnetization evolves under the influence of thermal noise. It is assumed that the easy-axis is along the x^ axis while the hard- and out-of-plane axes are oriented along the y^ and z^ directions, respectively.

We use the phenomenological LLG equation to describe the evolution of the magnetization of a monodomain nanomagnet subject to an effective magnetization field, Heff. Mathematically, the LLG equation is given as8–10 

dm^dt=γμ0(m^×Heff)+α(m^×dm^dt),
(1)

where m^ is the magnetization vector normalized to the saturation magnetization (Ms), γ is the gyromagnetic ratio, μ0 is the free space permeability, α is the Gilbert damping coefficient, and q is the elementary charge. For all simulations, α=0.01. Heff is the sum of the fields acting on the nanomagnet and is given as

Heff=HK+HD+HT,
(2)

where HT is the thermal field. HK is the uniaxial anisotropy field and is given from the Stoner-Wahlforth model as HK=(2Kuμ0Msmx)x^, where Ku is the uniaxial anisotropy energy density.11,HD is the field due to the shape anisotropy (demagnetization) and is given as HD=MsNxmx,Nymy,Nzmz, where Nx, Ny, and Nz are the demagnetization factors determined by the shape of the nanomagnet.12 The magnetization energy of the nanomagnet can be written as:2,13,14

E(m^)=Kumx2+12μ0Ms2(Nxmx2+Nymy2+Nzmz2).
(3)

We consider that the demagnetization field of the thin-film nanomagnet is oriented entirely in the z^ direction. This approximation is justified when the thickness of the nanomagnet is significantly smaller than the planar dimensions. Therefore, the demagnetization coefficients are given as Nx=Ny=0 and Nz=1, and the energy model reduces to a biaxial anistropy model. The energy barrier for biaxial magnetic anisotropy is given as Eb=Ku. The thermal noise manifests itself as fluctuations in the internal anisotropy field and is included with the effective field of the magnet.5 The thermal field can be modeled as 3-dimensional Wiener process (W):6,13,15

HT=2αkBTμ02γMSV(WXtx^+WYty^+WZtz^),
(4)

where V is the volume of the nanomagnet. We solve the stochastic LLG equation numerically using the Heun method.4,16 Due to the computational intensity of numerical integration, transient simulations were coded in CUDA and run in parallel.17 Each probability data point is calculated using 1000 simulations.

To discuss the magnetization relaxation, we focus on the precessional trajectories of magnetization dynamics as shown in Fig. 1. The precessional trajectories are the solutions to the LLG equation assuming α=0 and that the magnetization lies on a constant energy orbit18 The blue trajectories correspond to the high energy region, while the red trajectories denote the two energy basins in the energy landscape. The separatrix (plotted in black) shows the orbit associated with the energy barrier. In context of the magnetization reversal, we discuss the precessional trajectories in Section III of the paper.

FIG. 1.

Sample reversal trajectories for nanomagnet under the influence of thermal noise. In all cases Ku=102Jm3. Background light-colored lines denote precessional trajectories for the corresponding Ms/Ku values. Green line denotes sample magnetization trajectory under thermal noise.

FIG. 1.

Sample reversal trajectories for nanomagnet under the influence of thermal noise. In all cases Ku=102Jm3. Background light-colored lines denote precessional trajectories for the corresponding Ms/Ku values. Green line denotes sample magnetization trajectory under thermal noise.

Close modal

The magnetization reversal due to thermal noise in the thin-film nanomagnet is analyzed for two cases and the results are shown in Figs. 2 and 3. In Case-I, thermally-induced magnetization reversal occurs when the magnetization crosses the x^ = 0 plane within a specific time period for which the probability of reversal is being examined. As evident in Fig. 2, for a fixed energy barrier, an increase in Ms leads to a non-monotonic trend in the magnetization reversal probability. It is known that a larger Ms increases the thermal stability of the nanomagnet because the magnitude of the thermal field is decreased; hence, an increase in Ms results in an overall decrease in reversal probability.6 However, once Ms exceeds a certain threshold, there is an increase in the reversal probability of the magnetization. This increases is partly explained by observing the sample magnetization reversals as plotted in Fig. 1. In the case of lower Ms, the magnetization trajectory may cross the x^=0 plane due to the random walk of the thermal noise. However, in the large-Ms cases, the thermal noise simply forces the magnetization beyond the energy barrier, where the magnetization will precess across the x^=0. As shown in the inset of Fig. 2, for large values of Ms, the precessional time period decreases and becomes comparable or even smaller than the observation time. Thereby, it is more likely for the magnetization at a high-energy trajectory to cross the x^=0 plane during the observation time which leads to a higher probability of reversal. In Case-II, the thermal reversal probability is defined such that the magnetization will cross the energy barrier atleast once within a given time period. In this case, the probability of magnetization reversal for various Ms/Ku parameters is plotted in Fig. 3. Case-I reversals are a subset of Case-II reversals. Similar to Case-I dynamics, the non-monotonicity of reversal probability on Ms for a fixed Ku is also present in the magnetization dynamics of Case-II. This shows that a perpendicular anisotropy not only alters the precessional period, but also changes the distribution of the magnetization energy and both these aspects contribute to the positive reversal probability trend affecting large-Ms nanomagnets.

FIG. 2.

Probability that the magnetization will cross the x^=0 plane atleast once within 1 ns. While Eb is solely proportional to Ku, it is shown that a larger Ms also increases the probability of Case 1 reversal. Inset: Precessional period associated with nanomagnet with corresponding Ms/Ku parameters at the orbit whose associated energy =0.01Eb.

FIG. 2.

Probability that the magnetization will cross the x^=0 plane atleast once within 1 ns. While Eb is solely proportional to Ku, it is shown that a larger Ms also increases the probability of Case 1 reversal. Inset: Precessional period associated with nanomagnet with corresponding Ms/Ku parameters at the orbit whose associated energy =0.01Eb.

Close modal
FIG. 3.

Probability that the magnetization energy for a nanomagnet will exceed Eb atleast once within a given time. Three simulation times are considered according to title of plots. While Eb is solely proportional to Ku, a larger Ms also increases the probability of Case 2 reversal. The subplots together also demonstrate that the effect of this non-monotonic behavior is diminished as the observed time is increased.

FIG. 3.

Probability that the magnetization energy for a nanomagnet will exceed Eb atleast once within a given time. Three simulation times are considered according to title of plots. While Eb is solely proportional to Ku, a larger Ms also increases the probability of Case 2 reversal. The subplots together also demonstrate that the effect of this non-monotonic behavior is diminished as the observed time is increased.

Close modal

The non-monotonicity of magnetization reversal probability on Ms for a fixed Ku is present only when the dynamics at short time-scales (sub-100 ns) are considered. As shown in Fig. 3, the probability curve tends toward the classical energy-barrier dependent model as the observation time is increased. This suggests that Brown’s model is still valid when considering the long-term stability of nanomagnets as in the case of spintronic memory devices. Yet, in many proposals of spintronic logic devices, data retention on short time-scales is relevant. Hence, the non-monotonic behavior of magnetization reversal must be considered to accurately analyze the noise sensitivity and reliability of these logic devices.

In Case-II dynamics, the non-monotonic behavior of magnetization reversal results primarily from the altered thermal-field torque for a precessional orbit at a specific energy E(m^). To illustrate this point, we can assume that the three-dimensional Wiener process in (4) is instead a constant vector dW=[1,1,1]. We then apply the LLG equation (assuming no damping) along each point in the precessional orbit associated with a particular energy to measure the change in magnetization energy due to the thermal field.

As shown in Fig. 4, the thermal-field torque is enhanced when the nanomagnet has a larger perpendicular anisotropy. This is because the perpendicular anisotropy shapes the precessional orbits affecting the z^-components of their trajectories as shown in Fig. 1. Spatial distance between the magnetization and free-axis, alters the thermal-field torque imposed on the magnetization. This enhanced torque can cause a greater change in the magnetization energy, thereby increasing the probability for E(m^) to exceed the energy barrier, Eb. We also capture the position of the magnetization vector when it first exceeds the energy barrier to further corroborate the results. As shown in the inset plot of Fig. 4, the point at which the magnetization first crosses the energy barrier is spread evenly along the separatrix for smaller perpendicular magnetic anisotropy (low-Ms). However, when the perpendicular anisotropy is increased, the crossover point of the magnetization becomes clustered closer to the free-axis.

FIG. 4.

Change in magnetization energy due to thermal field assuming Wiener process in thermal field, dW=[1,1,1]. Change in energy calculated for each point along precessional orbit whose associated energy =0.01Eb. Precessional periods for each Ms value normalized along x axis. Assuming Ku=102Am a dt=1012s. Inset: Scatter plot of magnetization position when magnetization energy first exceeds energy barrier for various values of Ms.

FIG. 4.

Change in magnetization energy due to thermal field assuming Wiener process in thermal field, dW=[1,1,1]. Change in energy calculated for each point along precessional orbit whose associated energy =0.01Eb. Precessional periods for each Ms value normalized along x axis. Assuming Ku=102Am a dt=1012s. Inset: Scatter plot of magnetization position when magnetization energy first exceeds energy barrier for various values of Ms.

Close modal

In this paper, we have examined the magnetization reversal probability of biaxial nanomagnets due to the inherent thermal noise in magnetic bodies. We observe that the reversal probability of an biaxial nanomagnet for a fixed energy barrier is a non-monotonic function of the perpendicular magnetic anisotropy. We show that a large perpendicular anisotropy increases the frequency of the precessional orbits of the magnetization, which increases the likelihood for the magnetization to cross the x^ = 0 plane leading to reversal. On the other hand, a large anisotropy also shapes the precessional orbits by bringing the sepratrix closer to the free-axis. As the thermal-field torque is enhanced closer to the free axis, the probability that the magnetization energy will exceed the energy barrier increases. We also show that such non-monotonicity is observed at short time-scales (sub-100 ns), while at longer time-scales the model presented in this paper converges to the well-known random-walk model proposed by W. Brown.

The authors acknowledge the support of Semiconductor Research Corporation (SRC) Nanoelectronics Research Initiative for funding this research. Thanks to Prof. Andrew Kent and his group at New York University and Dr. Daniele Pinna at The National Center for Scientific Research (CNRS) for insightful discussions. We would also like to thank Dr. Giorgio Bertotti at the Istituto Nazionale di Ricerca Metrologica (INRIM) for his very insightful comments and advice during the investigation of this and other research topics.

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