Polycrystalline AF layers

The model described in the previous section assumes that the AF layer is a single domain. In reality, the AF layers are sputtered and have a polycrystalline structure.

In 1972, Fulcomer and Charap developed a model that accounts for this^[11]. The Ful-comer Charap model considers the AF layer as an assembly of small non-interacting particles, or grains. Each grain is exchange coupled to the FM moment of the FM thin film. Figure 2.9 shows an illustration of these AF grains on a FM layer. When assuming that the system is annealed and the easy axis of the FM layer and the AF grains are aligned, the energy per unit area of a single grain is given by

E_AF = t_gK_AFsin²(α) − cJ_ebcos(β − α) (2.15) where tg is the thickness of the grain, α is the angle of the magnetization of the grain and the easy axis of the FM thin film and c is the contact fraction. The first term is the magnetocrystalline energy term of the grain and the second term is the EB coupling term of the grain with the FM layer. Although two grains can have the same surface area, the AF surface moment can be different due to roughness.

Roughness present at the interface can lead to compensation of the EB coupling, for example when a different sublattice of the grain is in contact with the FM layer.

An example is depicted in figure 2.10. The coupling is significantly reduced, and the contact fraction accounts for this loss of EB coupling. For the grain depicted in figure 2.10, only ¹₅ of the grain is uncompensated, so c = ¹₅.

When considering AF grains, the energy per unit area of the exchange biased system can be written as^[12]

E = −µ₀HM_Ft_Fcos(θ − β) + K_Ft_Fsin²(β) + The first term in this equation is the Zeeman energy, the second term is the magne-tocrystalline energy of the FM layer, and the third term is a summation over all AF grains present in the AF layer.

In order to understand the magnetic behavior of the AF grains when the magnetic moment of the FM thin film is switched, the energy diagram of an AF grain shown in figure 2.11 has to be considered. The plot shows the energy of an AF grain EAF versus the angle α when the magnetic moment of the FM layer is switched (β = π). When the magnetization of the FM layer is switched, the orientation of the magnetization of the AF grains is in a local energy minimum (α = 0). In order to reach the global minimum (α = π) the energy barrier ∆E needs to be overcome. The barrier height

∆E is given by

Ferromagne5c)layer)

An5ferromagne5c)grains) M_F)

Easy)axis)

β)

M_AF) α)

Figure 2.9: Illustration of the AF grains in contact with an underlying FM layer for the Fulcomer Charap model. In this model, the AF anisotropy is considered finite, and when the magnetic moment of the FM layer is rotated, the EB coupling can result in the rotation of the spins of the AF grains over an angle α. Figure adapted from [11].

Figure 2.10: Schematic representation of coupling reduction due to roughness at the interface. Roughness at the interface can result in compensation of the EB coupling due to different sublattices of the AF grain being present at the interface. In this example, ³₅ of the grain is coupled in the positive x-direction, while ²₅ of the grain is coupled in the negative x-direction. The net contribution to the EB of this grain is ¹₅ in the positive x-direction. Figure adapted from [11].

0) π)

Figure 2.11: Energy diagram for the orientation of the magnetization of the AF grains when the magnetization of the FM layer is switched. In order to reach a global minimum, the grains have to overcome an energy barrier ∆E.

The process of overcoming the energy barrier is a thermally activated process which means that the switching of the magnetization of the AF grains takes a certain time. The grains approach a new equilibrium distribution with a relaxation time constant equal to

τ = ν₀e^−kbT∆E (2.18)

in which ν0 is the inverse of the switching rate of the magnetizations of the AF grains.

The relaxation time constant depends strongly on the temperature, and it is useful to introduce the concept of a blocking temperature TB. At low temperatures, the relaxation time is large and grains will not switch within the experimental measure-ment time. At high temperatures, the relaxation time is short and as soon as the magnetic moment of the FM film is switched, the grains will follow and the response seems instantly. This results in all grains switching within the experimental time.

The temperature at which τ is equal to the experimental time is called TB.

The switching of the grains during a measurement has great influence on the magnetic response of the system. It is important to distinguish three temperature regimes and the corresponding behavior of the grains in these regimes. At T  TB

the grains will not switch during the experiment, and the EB remains stable. This results in only a shift of the hysteresis loop. When the temperature is increased up to the blocking temperature grains are switching within the experimental time.

T_B)

Figure 2.12: Plot of the EB (white circles) and the coercive field (black circles) versus temperatures close to the blocking temperature of an AFM/FM bilayer. As the temperature increases, the EB decreases since more grains become unstable. Instead, these unstable grains contribute to the coercive field, which reaches a maximum at T = T_B. For T > TB grains are starting to go in a superparamagnetic state, leading to disorder and the contribution to the coercive field is lost until T ≥ TN and the grains no longer contribute to the EB or the coercive field. Figure adapted from [13].

These grains no longer contribute to the EB, but instead will switch together with the FM layer. This means that we do not only require a field to switch the FM layers, but the field needs to be large enough to switch the AF grains as well. This results in an increasing HC and leading to a maximum of HC when T = TB. When T > TN

the grains are no longer antiferromagnetically ordered and no longer contribute to the EB or the coercive field. At these temperatures, only the magnetic signal of the FM layer is measured. A typical plot of HC and Heb versus T is shown in figure 2.12.

2.3 The stability of the exchange bias effect against