4.7 Thermal switching mechanism in Fe particles

Thermal effects can also contribute to reduce the switching fields. The dynamic dependence of the coercivity is usually analyzed using the Sharrock law [Sharrock 90], which is derived from the Arrhenius-Néel law. The application of the Sharrock law requires the energy barrier dependence on the applied field. In this section we evaluate that dependence and study the thermal switching mechanism. We have calculated the energy barriers values using the Lagrange multiplier method discussed in Section 2.3.6, but in this case we have used the constraint $ \langle m_x \rangle=m_{x0}$, where $ \langle m_x \rangle$ is the average magnetization x component. This constraint has the advantage of having the form of an applied field in the X direction, allowing to obtain the energy barrier of an arbitrary applied field from the zero field energy barrier value [Lu 07].

Figure: Saddle point configuration for the $ 400\times 40\times 4 nm^3$ particle.
Figure: Saddle point configuration for the $ 400\times 175\times 4 nm^3$ particle.
Image saddle175

The saddle points as well as the energy barrier vary with the applied field. Therefore, in this section we will refer to zero field saddle points as, simply, saddle points. As in the field switching, the thermal switching mechanisms are also different for different particle widths. For the large aspect ratio particles the thermal switching proceeds through a $ 180^\circ$ domain wall, as a saddle-point configuration shown in Fig. 4.16. The nucleation of the domain wall starts from the structures that are created by the magnetostatics in the particle ends (see Fig. 4.6). In the small aspect ratio particles the saddle point configuration consists of two domains, which point in one of the local directions of the biaxial anisotropy, as shown in Fig. 4.17. The domain wall is not located at the center of the particle since the domain wall is stabilized in the center of the particle and such configuration is a shallow minimum of the energy. This minimum is not present if the particle presents any imperfection. As in the hysteresis loops, the change of behavior is for a width ca. $ 75 nm$ and is related to the length of the structures that minimize the magnetic charges. The zero field energy barrier as a function of the particle width is plotted in Fig. 4.18(b). The domain wall mechanism yields a linear dependence with the cross section. In our case there is a change in the slope due to the different effective anisotropy, being this slope larger for thinner particles due to their large shape anisotropy. Finally, for small aspect ratio the effective anisotropy is not dependent on the particle width obtaining a linear dependence of the energy barrier.

Figure 4.18: Energy barriers: (a) as function of the applied field for different particle sizes and (b) for zero applied field as a function of the particle width.
Figure: Energy barrier value normalized to the zero field value as a function of the normalized applied field for the $ 400\times 30\times 4 nm^3$ particle.

Figure: (a) Temperature dependence of the coercivity for the bcc-Fe ribbons with cross section $ 3.5 nm \times 27 nm$ (measured in the Instituto de Magnetismo Aplicado (IMA) by A. Martínez). (b) Comparison of experimental and simulational results for the temperature dependence of the coercivity normalized to the $ 5 K$ value.

In Fig. 4.18 (a) the energy barrier values are shown as a function of the applied field value and the particle width. Several authors [Skomski 06] have found the applied field dependence of the energy barrier value to be:

$\displaystyle E_B=E_0\left(1-\frac{H}{H_c}\right)^n$ (4.5)

where $ E_0$ is the zero field energy barrier value and $ n$-the scaling factor. As shown in Fig. 4.19, in the case of the domain wall thermal switching our simulations fit perfectly to a scaling factor $ n=2$. This factor also appears in other systems like in the case of a Stoner-Wohlfarth particle.

From the scaling of the energy barrier value and the Arrhenius-Néel law, M.P. Sharrock [Sharrock 90] obtained the following expression for the thermal dependence of the coercivity, due to thermal relaxation during the hysteresis process:

$\displaystyle H_c(T)=H_c\left[1-\left[\frac{k_BT}{E_0}\ln\left(\frac{f_0 t_0}{\ln 2}\right)\right]^{1/n}\right]$ (4.6)

where $ f_0$ is the attempt frequency and $ t_0$ the time scale of the measure. From Fig. 4.18(a), the thermal activation in the timescale of a magnetometer measure will be only appreciable in the case of large aspect ratio particles. From all this results we can calculate the thermal dependence of the coercivity applying the Sharrock law, considering a time of measure $ 0.1 s$ and $ f_0=10^{10} Hz$. The results for the particle with cross section $ 3.5 nm \times 27 nm$ are shown in Fig. 4.20. The dependence obtained from simulations is not in agreement with the experiment but indicates the existence of appreciable thermal activation in the samples. However, in order to obtain the real dependence of the coercivity the thermal dependence of the magnetization has to be taken into account in the simulation. These could be obtained from the experimental dependence of the magnetization as shown in Fig. 4.2. This simulation using temperature dependent parameters will be subject of future work.