Another kind of anisotropy can arise when the previously described F and AF-layer are placed adjacent to each other. This was first discovered by Meiklejohn and Bean in the 1950’s when they conducted experiments on Co particles embedded in anti-ferromagnetic CoO [15]. When ana-lyzing the behavior of Co in a magnetic field, they discovered an extra unidirectional anisotropy in the system. The ferromagnetic Co appeared to have an extra preferential direction along one specific direction, not just along one axis as discussed in section 2.2.1 (uniaxial anisotropy).
The explanation of this observation can be found at the interface between the F and AF-layer. If the AF interface layer has uncompensated spins, they can couple to the spins at the F interface layer via the exchange interaction, shown schematically in figure 2.3. The strength of this cou-pling is determined by the exchange strength JEBand can be written in the form of the exchange anisotropy energy Eexchange anisotropy:
Eexchange anisotropy=−JEBcos(ψ − β), (2.8) with β the angle of the F-layer magnetization, and φ the angle of the uncompensated AF spins.
This adds an extra unidirectional anisotropy to the ferromagnet. When applying an external field to the system, the added anisotropy term results in an increase in the switching field in a certain direction. This effect is called the exchange bias effect.
The exchange bias effect is one of the most widely studied effects in magnetism over the last 50 years. Subsequently, exchange bias has found its way to many modern applications in magnetic devices [16]. Still, no comprehensive theory exists to fully describe this phenomenon. In the
Antiferromagnetic
Figure 2.3: Exchange coupling between a F and AF-layer. Uncompensated spins at the AF interface couple to the F spins via the exchange interactions. This coupling induces an extra unidirectional anisotropy in the F-layer.
following sections, an attempt will be done to give a general overview of the exchange bias effect in magnetic multilayers. First, a simple model for magnetization reversal by magnetic fields in a F/AF bilayer system will be introduced to give a phenomenological understanding of the ex-change bias effect. This model is called the Meiklejohn-Bean model and is one of the first models which tried to grasp the effect in a simple calculation. It describes the magnetization behavior in a bilayer system by energy considerations involving the previously introduced Zeeman energy, anisotropy energy, and exchange energy. This model, however, is an extremely simplified model and cannot explain many observed effects in real systems. To get a more realistic picture of ex-change bias, more properties and observation of exex-change bias have to be explored, as discussed in section 2.4.
2.3.1 Magnetization reversal in the Meiklejohn-Bean model
The Meiklejoh-Bean model describes the magnetization reversal process in a F/AF bilayer. It takes into account different energy contributions to determine the equilibrium angle β of the magnetization ~Mof the F-layer. This model is a macrospin model, in which the F-layer is described by one average spin. Hence, it assumes coherent rotation of the magnetization of the F-layer. In real systems, more complicated reversal processes might occur involving domain nucleation and domain wall motion. An introduction to domain nucleation and domain wall motion is given in appendix A. The simplistic nature of this model asks for some more assumptions regarding the F/AF system. Two important assumptions are: (i) a perfect interface between the F and AF-layer and (ii) the spins in the AF-layer remain unchanged during magnetization reversal. This implies that KAF=∞and the AF interface is fully uncompensated.
A schematic overview of an ideal two-dimensional exchange coupled AF/F system is given in figure 2.4. Panel 2.4a shows the definition of different angles relative to the x-axis. The angles φ and ψ determine the easy-axis of the ferromagnetic and anti-ferromagnetic material, respectively.
Different angles of φ and ψ result in different configurations of the exchange bias. In figure 2.4b, the case of an OOP F-layer coupled to an OOP AF-layer is shown. Other configurations can also exist and will be described in section 2.5.
K
AFK
FM
FH
extψ=90o =90o
a. b.
y
x θ
β ψ ,
Figure 2.4: a: Representation of the different angles in the Meiklejohnbean modelin two dimensions. b: Example of a AF/F configuration with corresponding angles φ and ψ.
The total energy per unit area of the bilayer system consists of the Zeeman energy (the magneti-zation aligns to an external field Hext, see equation (2.1)), the unidirectional exchange anisotropy energy at the interface between the ferromagnetic and antiferromagnetic layer, and the anisotropy energy of the ferromagnet:
Etotal= EZeeman+ Eanisotropy,F+ Eexchange anisotropy. (2.9) This energy can be written as a function of the angle β, the magnetization angle of the ferromag-net. For an infinite sheet of material with a thickness tF, the total energy per unit area can now be written as:
Etotal(β) = −µ0HextMstFcos(θ − β) + KFtFsin2(φ − β) − JEBcos(ψ − β), (2.10) in which θ is the external magnetic field angle.
To find the equilibrium position of the magnetization vector, the energy is minimized with respect to β:
∂ Etotal(β)
∂ β =0, (2.11)
and the stability condition:
∂2Etotal(β)
∂ β2 >0. (2.12)
If both conditions are satisfied, the magnetization angle β is in a stable regime. Equations (2.11) and (2.12) can be numerically solved to find the magnetization behavior under various applied magnetic fields.
Two important limiting cases are an external magnetic field applied along the easy-axis or hard-axis of the material. First, the case is considered in which (i) the magnetic field is applied along the easy-axis (θ = π/2, y-direction), (ii) no coupling between the F-layer and AF-layer (JEB=0) exists and (iii) the easy-axis of the F-layer is oriented out-of plane (φ = π/2). For each magnetic field value, one or more local energy minima are found. A plot of the energy function Etotal(β) is shown in figure 2.5. By increasing the external field, the magnetization can switch from one energy minimum corresponding to β = +π/2 to another energy minimum corresponding to β = −π/2. In other words, the magnetization in the y-direction switches.
β
E
total(β )
Hext(units μ0KF/Ms)=Figure 2.5: The energy function plotted as a function of the magnetization angle β.
The sphere indicates a stable configuration for the magnetization. By applying an external field of −2KF/µMsthe magnetization switches from β = π/2 to β = −π/2.
The curves have been shifted vertically for clarity. Figure adapted from ??.
The resulting normalized magnetization in the y-direction My/Ms can be related to β via My = sin β and can be plotted as a function of the applied magnetic field Hext, y, see figure 2.6a. The resulting graph is called a hystresis curve along the easy-axis of the material. Due to the uniaxial anisotropy of the F-layer, the history of the applied field determines the magnetization at zero applied field. This is called the remanent magnetization MR. It is generally given as a percentage of the total saturation magnetization Ms. In the case of the easy-axis hysteresis curve, this value is 100%. The field at which the magnetization switches between the states is called the coercive field HC. For an ideal easy-axis hysteresis curve HC can be calculated with equation (2.11) with φ = π/2, θ = π/2 and JEB=0:
µ0HC=±2KF
Ms . (2.13)
Note that only a slight deviation in applied field direction or, for example, thermal fluctuations can significantly reduce the coercive field in real systems. The anisotropy constant KFis therefore hard to determine from a easy-axis hysteresis curve. For applied fields smaller than the coercive field, two local minima exist and the initial condition determines in which state the magnetization is.
In figure 2.6b, the same system is analyzed except the field sweep direction is along the x-axis instead of y-axis (θ = 0). This results in a so-called hard-axis loop of the magnetization. The uniaxial anisotropy is now perpendicular to the field direction resulting in no hysteresis and no coercive field. The remanence is also equal to zero. For strong applied fields the magnetization is completely saturated. The field at which this happens is called the anisotropy field HK. Solving equation (2.11) for φ = π/2, θ = 0 and JEB=0 results in the following equation for HK:
µ0HK=±2KF
Ms . (2.14)
With this equation it is generally possible to estimate the effective anisotropy KFfrom a hard-axis loop.
Figure 2.6:Hysteresis curves for an F-layer with OOP anisotropy. The magnetization and field sweep direction is indicated for each figure. a: Due to the OOP anisotropy in the F-layer, magnetic hysteresis is present. The field sweep direction corresponds to the easy-axis of the material. The coercivity HCand remanence MRare indicated in the figure. b: A hard-axis hysteresis loop of an OOP F-layer is found when the field is swept in the x-direction. The magnetization is saturated at large applied fields, corresponding to the saturation magnetization Ms.c: Resulting easy-axis hysteresis curve when an F-layer is combined with an AF-layer. Due to the induced unidirec-tional anisotropy, the curve is shifted to the left compared with figure a. This shift is called the exchange bias HEB.
Important results from the Meiklejohn-bean model are found when the coupling between the F and AF-layer is taken into account (JEB>0). The field is swept in the y-direction corresponding to the easy-axis, just as in figure 2.6a. As shown in figure 2.6c, the hysteresis curve shifts to the left. The magnitude of this shift is called the exchange bias field HEB. This field is a direct result of the unidirectional anisotropy induced by the exchange coupling between the F and AF-layer.
Solving equation (2.11) for φ = π/2, θ = π/2 and JEB= JEBresults in:
µ0HEB=− JEB
µ0MstF. (2.15)
This gives the possibility to calculate the exchange constant strength JEBfrom an exchange biased easy-axis loop.