We now have a basic understanding of the exchange bias effect in an AF/F bilayer, and how to describe it with the Meiklejohn-Bean model. However, this interpretation of exchange bias can-not interpret most of the observed effects in such systems. The Meiklejohn-Bean model does can-not explain deviations in exchange bias magnitude, temperature dependence, F or AF-layer thickness dependence, enhanced coercivity in the F-layer or the polycrystalline structure of the AF-layer, all of which are observed experimentally in different exchange bias systems [17]. In the following sections, exchange bias observations from literature will be discussed and explained by alterna-tive exchange bias theories, if possible. Many of these observation are also observed in the results presented in chapter 5.
2.4.1 Exchange bias magnitude
For a simple Co/CoO system, as studied by Meikljohn and Bean themselves, the theoretical ex-change bias magnitude can be calculated via the Meiklejohn-bean model with equation (2.15).
A 10 nm thick Co layer with a saturation magnetization of 1.4 · 106A/m is assumed. JEBcan be calculated directly from the direct exchange interaction at the F/AF interface via: JEB= N JAF,F/A with N the number of atoms per lattice unit area A. Theoretical values for Co/CoO are: N = 4, A =3.2 nm and JAF,F=2.98 · 10−22J. This results in a value of µ0HEB=270 mT.
While this may look as a reasonable value, the result of this simple calculation of HEBis much larger compared to the values found in almost all experimentally studied systems [18]. The first explanation for the discrepancy is the simplification of the coupling between the F and AF-layer.
In the Meikljohn-Bean model it is assumed that all the AF-spins are fully compensated with the F-spins (all spins are directly coupled).
A second explanation can be found considering the interface between the F and AF-layer. A perfectly flat surface without any defects is assumed, in which the F-layer is only considered as one rotating spin (macrospin model). In reality, the AF or F-layer can consist of different grains and surface effects can play a considerable role.
Over the last 50 years, many researchers have tried to improve on the simple calculation of Meiklejohn-bean to get a more realistic value of the exchange bias. Most models take a more realistic approach to the interfacial coupling between the F and AF-layer [17].
For example, Malozemoff et al. [19], introduced random defects in the interface between the F and AF-layer. In figure 2.7 these defects are schematically shown. They form a rough interface, effectively decreasing the energy of the exchange coupling. This results in a more realistic value of HEB.
Another approach is taken by Mauri et al. [20]. They introduced a gradual rotation of the AF spins parallel to the interface. Such a domain wall can also lower the interfacial energy between the F and AF-layer. This assumption results in more reasonable values of HEB. However, this model assumes AF-layers thick enough to form a domain wall, while exchange bias is also observed in very thin layers of only a few atoms thick [21].
More of these interface theories are proposed during the last 30 years, but most of them are only applicable to certain F/AF systems in which they predict the exchange bias correctly. A small
Antiferromagnetic layer
Ferromagnetic layer
Hext
Figure 2.7: Schematic representation of random interface defects according to the Malozmoff model. The dashed line represents the rough interface.
deviation in system conditions can cause unpredictable results. Until now, no comprehensive theory of exchange bias exists that can predict the exchange bias magnitude in all system equally as good.
2.4.2 Enhanced coercivity
The previously introduced interpretations of the F/AF interface can explain the value of HEB rea-sonably well. However, the common experimental observation of an increase in coercive field HC of the F-layer when an AF-layer is placed on top of it is not explained by these theories. A literature example of such an increase is shown in figure 2.8, in which Pt/Co multilayers are studied, coupled to a CoO AF-layer at 10 K. In the experimental part of this thesis, such a coer-civity increase is also observed. A particularly successful model called the Spin-Glass (SG) model proposed by Radu et al. tries to explain this coercivity enhancement [17].
Figure 2.8: Observed coercivity enhancement and exchange bias with the insertion of a CoO AF-layer. Figure adapted from [22].
Just as many alternative theories, this theory tries to describe the interface between the F and
AF-AF-layer
Figure 2.9: Schematic representation of the F/AF interface according to the Spin-Glass model. A low anisotropy region at the interface is present, caused by impurities and surface roughness. This low anisotropy region can rotate together with the F-layer under an applied field.
layer in a more realistic manner. More specifically, the SG-model states that at the interface, the AF has a lower anisotropy than in bulk of the AF. This results in AF spins that rotate freely when external field is applied which is strong enough to rotate the F spins. This is schematically shown in figure 2.9. Two applied field directions are sketched in which a part of the AF-layer with a low anisotropy rotates coherently with the F-layer. Arguments for such a low anisotropy region can be derived from the imperfect nature of the AF/F interface. For example, chemical intermixing or other defects are likely present at such an interface. Moreover, structural inhomogeneities at the interface can lead to a frustrated spin-glass like state. Such a frustrated SG-state can be the result of a spin structure in which there is no perfect AF ordering such as seen in figure 2.2, but where there is an energy competition between spin configurations or crystalline anisotropies.
Due to this instability, a part of the spins will have a random orientation, resulting in a lower anisotropy. The transition from the F to AF-layer is therefore not immediate, but gradual in the form of a low anisotropy region, in which the AF spins are in a frustrated spin-glass like state.
This SG-like behavior can be modeled by adding an effective anisotropy energy constant to the Meiklejohn-bean model in equation (2.10), which describes the rotatable part of the AF-layer and the resulting imperfect interface. In appendix B, more details about the extended SG-model are given.
In figure 2.10, the SG-model is compared to the standard Meiklejohn-Bean model. The following parameters are used for the F-layer: Ms =1.4 MA/m, KF=4 · 104 J/m3, tF =1.25 nm, which correspond to a typical Co layer. The exchange constant is estimated to be JEB=1 · 10−4J/m2. The SG-model adds an extra energy term that describes the gradual rotation of a part of the AF-layer. A clear coercivity enhancement and exchange bias reduction are seen for the SG-model, which resemble the experimental observations (figure 2.8).
2.4.3 Thickness dependence
Another experimental observation, is the exchange bias dependence on the thicknesses of the F and AF-layer. Because exchange bias is an interface effect, the magnitude of the exchange bias is expected to decrease if the thickness of the F-layer increases. An 1/tF dependence is found in the Meiklejohn-bean model, see equation (2.15). Surprisingly, considering the failure
MB-model SG-model
Figure 2.10: Calculation of hysteresis loops of a F/AF bilayer with the SG-model compared to the MB-model. For the SG-model, corresponding to a less ideal inter-face, a coercivity enhancement and a exchange bias reduction is found. The field is swept along the easy-axis of the F-layer.
of the prediction of HEB, this dependence is actually observed experimentally in a vast amount of systems, see for example figure 2.11[20]. Deviation from this dependence is found in very thin (a few nm) or very thick (> 1µm) F-layers. Noguès et al. suggests that the derivation for thin F-layers is due to the F-layer becoming discontinuous [18] .
While the thickness tAF of the AF-layer is not included in simple exchange bias models, in most experiments a dependence on this thickness is found. See for example figure 2.11b in which a temperature and thickness dependent exchange bias is found for an FeMn AF-layer.
Some theories, such as the previously introduced SG-model, can model the observed behavior of figure 2.11ab, as it includes both the F and AF thickness in its model. The result of the SG-model is shown in figure 2.12. For details about the parameters or the SG-model, see appendix B.
As expected, the exchange bias follows a 1/tF dependence. This can intuitively explained by it being a surface effect. Increasing the volume of the F-layer reduces this relative strength of this surface. The tAF dependence is more complicated. For small thicknesses, the AF-layer rotates with the F-layer due to its low total anisotropy. As a result, the exchange bias disappears, and an increase in coercivity below a critical thickness is observed. The behavior is comparable to the experimental results in figure 2.11. For thicker AF-layers, the exchange bias can decrease. This can be explained by the AF domain wall model from Mauri et al. as discussed in section 2.4.1.
A change in anisotropy due to the increase in F-layer thickness is not taken into account in this model, but will of course play a role in experimental results.
2.4.4 Temperature dependence & field-cooling
Temperature can have a strong effect on the exchange bias. An important parameter of exchange bias systems is the blocking temperature TB. At this temperature, the exchange bias completely
a. b.
Figure 2.11: Literature examples of observed F and AF-layer thickness dependence on coercivity and exchange bias. a: F-layer thickness dependence. Note the non-linear scale. Figure taken from [20]. b: AF-layer thickness dependence. The ex-change bias emerges after a critical thickness. Figure taken from [23].
a. b.
Figure 2.12: Layer thickness dependent exchange bias, simulated using the SG-model.
vanishes. One might expect that this is equal to the previously introduced Neèl temperature TN, which describes the temperature at which the AF-layer becomes paramagnetic (disordered).
However, studies show that for thin (< 10 nm) AF-layers, the blocking temperature can be con-siderably lower than the Neèl temperature [24]. Causes for this behavior have been sought in finite size effects, AF grains or super-paramagnetic behavior [25]. An example of low blocking temperature for thin AF-layers is shown in figure 6.11 for a CoO antiferromagnet.
The discrepancy between TN and TB appears to be very useful when setting the exchange bias in a desired direction. This can be done by a so-called field-cooling procedure, in which the temperature is heated to above TB and subsequently cooled under a strong applied magnetic field. This sets the direction of the exchange bias parallel to that of the applied field. As TBis usually lower than TN, lower field-cooling temperatures are needed. This process is extensively discussed in section 4.2.
The models that are considered in this thesis do not include any temperature dependence. In light of the previously introduced SG-model, the discrepancy between the blocking and Neèl temperature can be interpreted as follows: The SG model is based on frustration at the interface, resulting in a low anisotropy region. The conversion factor f that describes this, is probably temperature dependent, as thermal energy can influence the frustration. For thin AF-layers, this can result in a large low anisotropy region and the disappearance of exchange bias. AF grains can also play a role in the temperature dependence of AF-layers. This will be further explored in the next section.
Figure 2.13: Blocking temperature TB as a function of the AF thickness in a Fe3O4/CoO bilayer. The curve is a guide to the eye. Figure taken from [25].
2.4.5 Polycrystalline antiferromagnetic thin films
Most of the experimentally observed effects can be explained by theory quite well. However, they still consider only a macrospin model of the magnetization reversal. Some AF materials such as IrMn (which is used extensively in this thesis) tend to grow in a polycrystalline manner (especially with the fabrication method used in this thesis). A macrospin model can only account for the average of such differences and does not account for the microscopic behavior of the AF-layer and the observed magnetization reversal. Such a polycrystalline structure can result in vastly different results [26]. Crystalline grains can have different sizes and crystal orientations.
Therefore, the AF spins direction can differ among grains as they are bound to the crystalline axis at which the crystalline anisotropy energy is lowest. Likewise, this can also have an effect on the local exchange bias, blocking temperature or on an underlying ferromagnetic layer. More details on the effects of AF grains will follow with the interpretation of results in chapter 5 and 6.