The presented theory about exchange bias is briefly summarized in this section. Basic magnetic concepts were introduced to describe a ferromagnetic and anti-ferromagnetic layer. The cou-pling between these layers was introduced, which results in exchange bias. The Meiklejohn-Bean model can describe the magnetization reversal of such a system and shows that the hysteresis curves is shifted if exchange bias is taken into account. However, this model appeared to be insufficient to explain numerous experimental results of exchange biased systems. No compre-hensive theory exists that can describe exchange bias for all systems, but assumptions on rough interfaces or rotation of the AF-layer can significantly improve theoretical predictions. Finally, a novel form of exchange bias was introduced in which the anisotropy of the F and AF-layer are oriented orthogonal. Two literature examples were shown that show such a configuration can exist.
Theory of current-induced magnetiza-tion reversal
The chapter introduces current-induced magnetization reversal, which is essential to achieve the goal of this thesis: to reverse the magnetization of a ferromagnetic layer by a current running trough an underlying non-magnetic layer, in combination with the effective field from an ex-change bias. This is based on the principle that a current can influence the magnetization in a ferromagnetic material.
There is still debate in the scientific community whether this current related effect on the mag-netization is due to the Rashba effect or the spin Hall effect. Therefore, in certain literature reports, one only speaks about Spin-Orbit Torques (SOT) which is a general term for both SHE and Rashba related effects [31]. As recent experiments in the group of FNA have shown that the SHE is the prevailing mechanism in Pt based samples, the SHE approach is followed [7] and the Rashba effect is not discussed.
In section 3.1, the spin Hall effect is explained. Via this effect, a spin-polarized current is gener-ated in a non-magnetic layer. This spin current can flow into an adjacent ferromagnetic layer and exert a torque on the magnetization. To describe the torque, the Landau-Lifshitz-Gilbert (LLG) equation is introduced in section 3.2. It describes the evolution of the magnetization under vari-ous torques. The LLG equation is used to simulate magnetization reversal via the spin Hall effect and an externally applied field. The last part of this chapter discusses the replacement of the external field by an exchange bias to induce field-free magnetization reversal: the final goal of this thesis.
3.1 The spin Hall effect
The spin Hall effect causes separation of electrons with opposite spin in a material with a high spin orbit coupling [8]. This results in a longitudinal charge current to generate transverse spin currents, shown schematically in figure 3.1. The direction of this spin current is perpendicular to the charge current direction and the spin polarization direction is perpendicular to both the charge current and spin current. This spin polarized current can be inserted into an adjacent
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layer if the interface between the layers is spin-transparent.
J J
y z
x
Figure 3.1: Sketch of the spin Hall effect. A longitudinal charge current J creates a transverse and perpendicular spin current. The polarization of the spin current is different for each direction. This magnitude of the spin currents depend on the spin Hall angle θSHEof the material.
The strength and sign of this effect is material dependent and is given by the dimensionless spin Hall angle θSHE, equal to the ratio of the resulting spin polarized current density Js divided by the total charge current density Jc.
The spin Hall effect is strongest in heavy metals like Pt or Ta. A typical value for the spin Hall angle in Pt is 0.07. This value is much debated in literature providing a range of different measured values for θSHE. The origin of these difference can be found in many different ways to measure θSHEin which spin-orbit or interface effects can influence the measurement. For an overview on the spin Hall angle magnitude debate, refer to [32].
Negative values of θSHEare also possible, for example Ta has a very high spin Hall angle of −0.15.
The spin Hall effect can even occur in antiferromagnetic materials like IrMn and PtMn or alloys such as CuBi [33][34].
The microscopic origin of the spin Hall effect comprises of three different contributions. Two extrinsic contributions are related to spin interactions with impurities inside a material: skew-scattering and side-jump. The skew-skew-scattering process arises from the fact that the momentum of an electron changes spin-dependently when scattering with an impurity, which leads to a spin-dependent force on the electron. The side-jump interaction as well as the third intrinsic contribution are more difficult to explain intuitively. It is related to the spin-dependent Berry curvature in k-space. A detailed explanation goes beyond the scope of this thesis. An attempt to explain these mechanisms in more details can be found in [7].
3.1.1 Spin Hall current in a thin Pt layer
In this section, a simple model for a spin current generated in a thin Pt layer is introduced.
This model provides a way to calculate the Pt thickness dependence for the total generated spin current, which depends on the initial charge current. The amount of spin current is a critical parameter in the forthcoming switching experiments. Hence, this model is used to predict the optimal Pt thickness.
Consider a Pt/Co bilayer, in which a longitudinal charge current flows. Due to the spin Hall effect, electrons with opposite spin drift in the corresponding transverse direction. If the Pt layer has a finite thickness, as depicted in figure 3.2a, the upward drifting spins (blue) drift to the Co layer at the top surface as this interface is spin-transparent, while the downward drifting spins (red) accumulate at the bottom surface if a non spin-transparent interface is assumed [35]. Due to this spin accumulation, a spin diffusion current (indicated by the dashed arrow) emerges in the upward direction. If the thickness of the Pt layer is much smaller than the spin diffusion length, the contribution of the drift and diffusion will cancel each other out, resulting in no spin current injection into the Co layer. However, the spin diffusion length λs for Pt is only ~1.4 nm.
Figure 3.2b depicts a Pt/Co bilayer in which the Pt layer is thicker than the spin diffusion length.
In this case, the spin diffusion from the bottom surface will not reach the top surface. A net spin current is injected into the Co layer. Increasing the Pt thickness well beyond the spin diffusion length (figure 3.2c) will not increase the injected current, as the spins from the bottom part of the Pt layer diffuse and don’t contribute to the total injected spin current. Moreover, if the spin current density relative to the total charge current through the Pt/Co layer is considered, more total current is running through the Pt for the same spin current density that is inserted into the Co. Therefore, efficiency of the stack decreases.
Spin Hall drift Spin diffusion
Figure 3.2: Schematic overview of the drift and diffusion of spin currents in a heavy metal layer. The spin Hall effect creates a spin current in the +~z direction for a charge current Ic running in the +~x direction. Spins that are drifting down accumulate at the interface and generate a diffusion current in the opposite direction. a: For thicknesses close to the spin diffusion length λs, the diffusion current will cancel the effective spin current injected in the the adjecent ferromagnetic layer. b: For thicknesses larger than the spin diffusion current, an effective spin current is injected into the F layer. For very thick layers (c) the charge current density is reduced. To keep the same injected spin current density, the total charge current through the heavy metal should increase.
The intuitive explanation of last paragraph follows from a simple drift-diffusion model. For a Pt layer, the diffusion term results in a reduction of (1 − sechtλPt
s)of the total spin current Js [36]. If a total current Itot is running through the Pt/Co bilayer of thickness d = tPt+ tCo and width w with total resistance RCo and RPt, the total charge current density through the Pt, Jc,Pt, is given
by:
Jc,P t= Itot RCo RCo+ RPt
1
tPtw. (3.1)
Via the drift-diffusion model, the spin current density through the Co layer can now be written as:
The resulting spin current density for a range of Pt thicknesses is shown in figure 3.3, with Itot equal to 10 mA, RCo=891 Ω, RPt=105 Ω, w = 1 µm, θSHE=0.07 and λs=1.4 nm. Bulk values for the resistance are taken, which can be different compared to the thin layer resistances. A fixed current is used as this limits Joule heating and the chance of device breakdown. The three regimes, intuitively explained in figure 3.2 can be seen in the graph. At tPt < λs the diffusion current is compensating the spin current, so almost not net spin current emerges. Increasing the Pt thickness will lead to a maximum at which the diffusion current does not reach the top surface.
Further increasing the thickness leads to a relative decrease in spin current, as the total current inserted into the stack remains constant. The total charge current density in the Pt becomes lower because of current shunting through the bulk of the Pt layer, where it does not contribute to Js. To conclude, the optimal Pt thickness for a spin Hall device is approximately 3 nm.
0 2 4 6 8 10
Figure 3.3: Calculated injected spin current density caused by the spin Hall effect as a function of the Pt thickness. The results correspond to the drift-diffusion model schematically drawn in figure 3.2. The optimal Pt thickness is around 3.5 nm. Below this thickness, diffusion of spins decrease the effective spin current. Above 3.5 nm the layer gets thicker but the total current is kept constant (to limit Joule heating and power consumption). Therefore, the charge current density through the Pt decreases resulting in a decreased injected spin current.