10.1 Cubic particles

Cubes are easy to handle with finite difference packages, because they have no curved boundaries. Thus, cubic particles are good candidates to compare the results of the finite difference program with those of the finite element package. In addition the results are compared with those of Nakatani et al. [56], whose material parameters have been used. They are chosen as $M_\mathrm{s}=0.4\times 10^{6}~\frac{\mathrm{A}}{\mathrm{m}}$ and $K_1=2\times 10^{6}~\frac{\mathrm{J}}{\mathrm{m^3}}$, $A=1\times 10^{-11}~\frac{\mathrm{J}}{\mathrm{m}}$. In all simulations the number of switching events was counted for at least 100 ns up to 1 $\mu$s and the results extrapolated to 1 $\mu$s.

Figure  shows the time dependence of the magnetization for a cubic particle of 32 nm edge length at 300 K. The magnetization fluctuates in the energy minimum around $\pm 1$. From time to time reversal processes occur when the magnetization crosses the energy barrier and switches to the other energy minimum. The probability per unit time, that $M_z$ jumps over the energy barrier $E$ in thermal equilibrium, is proportional to

$$\exp \left(-\frac{E}{k_B T} \right) \quad.$$

We consider a single energy barrier model and take only anisotropy into account. The reciprocal of the switching probability is the relaxation time $\tau$ which can thus be written in the form of the Arrhenius-Néel law [57]

$$\frac{1}{\tau}=f_0 \exp \left(-\frac{K_1 V}{k_B T} \right) \quad,$$ (10.1)

where $f_0$ is the characteristic dynamic frequency. The original estimation of Néel was $f_0 \approx 10^9 \mathrm{~s}^{-1}$, but recently it has become more customary to take $f_0 \approx 10^{10} \mathrm{~s}^{-1}$ up to $f_0 \approx 10^{12} \mathrm{~s}^{-1}$. As we will see, the characteristic dynamic frequency depends on the damping constant, which is a material parameter.

Figure: Magnetization reversal of a cubic particle

[$\alpha=1$] [$\alpha=0.1$]

The number of reversal processes should, of course, be independent of the time discretization. This has been verified and the results are shown in figure  for a $2 \times 2 \times 2$ discretization. As the time step is decreased from $10^{-12}$ s to $5 \times 10^{-14}$ s, the number of switching events increases and converges. Then, the space discretization dependence is investigated (fig. ) and we find for three different space discretizations ( $2 \times 2 \times 2$, $3 \times 3 \times 3$, $4 \times 4 \times 4$) consistent results (within fluctuations due to the stochastic nature of the underlying processes).

In addition, the results for a finite element model are plotted in figure . The cube has been discretized into 64 nodes and 135 tetrahedral elements and the results are in excellent agreement with those of the finite difference model.

If we fit the data of the smallest time step in the linear region in figure  with the Arrhenius-Néel law, we find a characteristic dynamic frequency of $f_0=3.52610\times 10^{11}$. The exponent is $-4.7\times 10^{25} \cdot V$ and it is in good agreement with the value

$$-\frac{K_1}{k_B T}= -4.8\times 10^{25}$$

which we would expect for a single (anisotropy) energy barrier. This approximation is sensible, if the magnetization reverses coherently. The magnetization vectors, which are represented by cones, at different times are plotted in figure  for a $64 \mathrm{~nm}^3$ particle and we see, that the particle switches coherently.

Figure: Snapshots of a switching event

[Initial magnetization ($t=0$ ps)] [Thermally perturbed magnetization distribution ($t=0.09$ ps)] [Switching starts ($t=0.1500$ ps)] [The reversal process progresses ($t=0.1556$ ps)] [Crossing the energy barrier ($t=0.160$ ps)] [Reversal process completed ($t=0.176$ ps)]

The characteristic dynamic frequency obtained above is quite high compared to the estimate of Néel. However, it is a question of the definition of a switching event. This fact is illustrated in figure . If the magnetization changes its sign and its absolute value exceeds $m_\mathrm{trh}$, then a switching event is counted. For $m_\mathrm{trh}=0.1$ we get a number of switching events which is one order of magnitude larger than that for $m_\mathrm{trh}=0.7$. This is due to the fact, that there are many switching events, in which the magnetization does not complete a full reversal, but it already switches back at an earlier stage. Such events can also be identified in figure , where we find ``spikes'' of incomplete switching events. Thus, the characteristic dynamic frequency depends on the definition of a switching event. The exponent of the Arrhenius-Néel law is not influenced, since the slope of the graphs in figure  remains the same.

Physically interesting is the dependence on the damping constant, because this is a material parameter, which can be obtained from ferromagnetic resonance experiments (cf. section ). As the damping constant is increased from $\alpha=0.01$, the number of switching events increases, too. At a temperature of 0 K the reversal time of a fine particle is proportional to $(1+\alpha^2)/\alpha$ [51]. Therefore, it is reasonable, that the characteristic dynamic frequency is proportional to $\alpha/(1+\alpha^2)$. The solid line in figure  is a fit of the $\alpha/(1+\alpha^2)$ law to the data obtained by computer simulations (circles).

Figure: Dependence of the number of switching events on the simulation parameters

[Time step dependence] [Discretization dependence] [Dependence on definition of switching]

Figure: Dependence of the number of switching events on damping constant and temperature

[Dependence on damping constant] [Dependence on damping constant for different particle sizes] [Temperature dependence]