In this section, the theory behind the SAF is explained. As mentioned in the first chapter, the pinned multilayers consist of an exchange biased FM layer and an SAF.
The SAF is used in order to create a pinned multilayer that has no net magnetic moment which makes it less susceptible to external fields. Figure 2.14 shows the location of the SAF within a (simplified) GMR/TMR stack.
First, a general introduction on coupling mechanisms in magnetic multilayers is given. Next, the so-called Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction is introduced. This interaction occurs in a system with magnetic impurities embedded in a non-magnetic metalic material, and it will be shown that these magnetic impu-rities can be coupled via the RKKY-interaction. In section 2.4.3 a simplistic model is introduced to apply the RKKY-theory to magnetic multilayers to show that it is possible to couple two FM layers through a non-magnetic metallic spacer layer.
Figure 2.15: Illustration of the dipole-dipole interaction between two magnetic mo-ments seperated by a distance r.
Next, Bruno’s model will be briefly introduced, which takes the band structure of the spacer layer into account to quantify the RKKY coupling strength. Section 2.4.5 explains the influence of roughness at the FM/NM interfaces and the thermal stability is discussed. Finally in section 2.4.6 the magnetic behavior of an SAF versus an applied field is discussed.
2.4.1 Interlayer exchange coupling mechanisms
Two FM layers can be coupled through a non-magnetic metal spacer. Several cou-pling mechanisms are known to be present in FM multilayers, such as dipole-dipole interactions, direct exchange coupling through pinholes, orange peel coupling and genuine indirect exchange coupling.
The dipole-dipole interaction is a long-range interaction for which the interaction energy can be written as
The dipole-dipole interaction falls of as 1/r3 with increasing distance r between the magnetic moments. Bloemen calculated[16] that the dipole coupling strength of two in-plane Co monolayers at a distance d = 2Å is in the order of 10−3mJ/m2, which is far less than the commonly observed coupling strengths (≈ 0.1mJ/m2at d = 10Å).
The dipole-dipole interaction is not sufficient to explain the coupling strengths which were experimentally observed in the magnetic multilayers and is therefore neglected in this thesis.
A second coupling mechanism which is present is the direct exchange interaction, which couples magnetic layers through pinholes in the spacer layer. When a spacer
Figure 2.16: Illustration of pinhole coupling of two FM layers. When the spacer layer is thin and inhomogeneous, direct channels of FM material between the FM layers can exist and result in a direct FM coupling of the two magnetic layers.
Figure 2.17: Illustration of orange peel coupling. This coupling results from fringe fields and correlated roughness at the FM/spacer layer interface.
layer seperating two FM layers is not homogeneous, pinholes can occur through which the FM layers are directly coupled as is shown in figure 2.16. Pinhole coupling depends strongly on the quality of the layers and will decrease rapidly for increasing spacer layer thickness. Therefore, pinhole coupling has usually to be considered only for thin spacer layers[17].
A third coupling is orange peel coupling. Orange peel coupling is caused by fringe fields that are the result of correlated roughness at the FM/spacer layer interface, as illustrated in figure 2.17. Orange peel coupling results in an FM coupling of the magnetic layers.
In order to create an SAF, the FM layers need to be antiferromagnetically cou-pled. This is possible by considering the indirect exchange interaction. The indirect exchange interaction can couple the FM layers either parallel or antiparallel, which depends on the thickness of the spacer layer. The indirect exchange interaction bears
An5ferromagne5c)coupling) Ferromagne5c)coupling) Distance)r)between)the)Co)atoms)
)
J)
m1)
m2) r)
Co)atom)1)
Co)atom)2)
Figure 2.18: Illustration of the interaction strength J between two Co atoms em-bedded in a Cu crystal versus the distance between the Co atoms. Depending on the distance between the Co atoms, the interaction strength is either positive, governing an FM coupling, or negative, governing an AF coupling.
much resemblance with the RKKY-interactions between magnetic impurities em-bedded in a conducting material. In the next section, the RKKY-interaction will be introduced.
2.4.2 The RKKY interaction between magnetic impurities
A succesful theory behind the oscillatory exchange interaction was developed by Ruderman, Kittel, Kasuya, and Yosida in the 1960’s, after which this interaction was called the RKKY-interaction[18][19][20]. In their papers they discuss the coupling between magnetic impurities embedded in a non-magnetic metallic material. To understand the coupling mechanism, consider a Cu crystal, in which two Cu atoms are replaced by Co atoms. The coupling between the Co atoms as a function of the distance between the Co atoms is shown in figure 2.18.
Conducting electrons in the Cu experience a potential step when approaching the Co atom. Due to the wave properties of electrons, the scattering of electrons at the potential results in a phase shift. Since the Co atom has a non-zero magnetic moment, they will interact differently with a spin-up electron then with a spin-down
not cancelling out anymore, and after a summation over all the wave vectors up to the Fermi surface this results in an oscillatory spin density. A second Co-atom, if placed not too far from the first Co atom, will experience either a net spin-up density or a net spin-down density depending on the distance between the two Co atoms and therefore is (indirectly) coupled to the first Co atom. Yoshida showed that the interaction strength J can be described by[20]:
J ∼ cos(2kFr)
r3 (2.20)
with kF the Fermi wave vector of a free electron gas and r the distance between the two Co atoms. When J > 0 the spins of the Co atoms are coupled ferromagnetically, and when J < 0 the spins are coupled antiferromagnetically.
In order to translate the RKKY-interaction between magnetic impurities to a system of magnetic layers seperated by a spacer layer, a simple 1D model will be used[21].
2.4.3 The interlayer exchange coupling: a simple model
The 1-dimensional system is depicted in figure 2.19. It contains two FM monolayers F1 and F2, seperated by a non-magnetic metallic layer.
When a conducting electron in the spacer approaches the first magnetic layer it will experience a potential step due to a difference in band structure between the spacer and the magnetic material. The electron wave will partly be reflected (er) and partly be transmitted (et). The wave equations of the electron in the spacer layer (−12d < x < 12d) can be calculated by solving the time-independent Schrodinger equation and results in
ψ(x) = eikxx+ Re−ikxx (2.21) in which k is the wavevector of the incoming and reflected electron and R is the reflection coefficient. The probability of finding an electron at a certain position x is given by
| ψ(x) |2= 1 + R2+ Re−2ikxx+ Re2ikxx
= 1 + R2+ 2Rcos(2kxx) (2.22)
∝ 2Rcos(2kxx) (2.23)
when only considering the position dependent term. From this equation it can be seen that the electron density within the spacer varies due to the last cosine term:
the density oscillates as a function of x with a period of π/k.
Figure 2.19: A schematic overview of the FM/NM/FM trilayer system. The con-ducting electrons in the non-magnetic layer will partly be reflected and partly be trans-mitted due to the potential difference at the interface.
Due to the non-zero magnetic moment of the FM layer at the interface, the incoming electrons with spin up will experience a different potential than electrons with a spin down. The different potentials lead to different reflective coefficients (R↑ and R↓)). The different reflective coefficients lead to two spin dependent charge density waves
| ψ↑(x) |2∝ 2R↑cos(2kxx)
| ψ↓(x) |2∝ 2R↓cos(2kxx) (2.24) Substracting these two standing spin density waves leads to the net spin density wave at a position x given by
| ψ↑(x) |2 − | ψ↓(x) |2∝ 2(R↑− R↓)cos(2kxx) (2.25) This is the spin charge density for only one electron, and in order to calculate the total spin density due to all electrons present in the spacer we have to integrate this over all possible kx’s up to the Fermi wave vector kF, resulting in
kF
ˆ
0
2(R↑− R↓)cos(2kxx)dkx = (R↑− R↓)sin(2kFx)
x (2.26)
The spin density has an oscillatory behavior with respect to the distance from the point of reflection, and has a period of π/kF. As the distance to the second magnetic layer increases, the sign of the spin density changes from positive to negative and back. If the total spin density is positive at a certain position, a net spin up is found and when the total spin density is negative a net spin down is found. The second FM layer placed at a certain distance from the first FM layer will experience a net
d) 2d) 3d) 4d) 5d) 6d)
JRKKY(x))
x)
Figure 2.20: Example of aliasing. The spacer layer thickness is quantified as a multiple of the interatomic distance d. This results in a larger period for the measured oscillation.
spin up or net spin down density, and the magnetization of the second layer will align itself with the local spin polarization, being either parallel or anti-parallel, depending on the distance between the two magnetic layers and thus leading to an oscillatory coupling.
2.4.4 Bruno’s model
The model described in the previous section is a simplified phenomenological model, however it shows that two FM layers can be coupled via conduction electrons in the spacer layer. Bruno developed a more realistic model based on a bulk band structure of the spacer material and a spin-dependent reflection amplitude at the FM/NM interfaces. With this approach, he showed that the interlayer exchange coupling for large spacer layer thickness x is then given by[5]
JRKKY(x) = −J0
x2sin(2kFx) (2.27)
For a complete derivation of this formula the reader is refered to [5]. The period resulting from this theory (λ2F) is shorter than any experimentally observed period.
However, it has been shown[7][8] that in this theory, the spacer thickness x is assumed to be a continuous variable and that in reality the distance between the FM mono-layers is x = (N +1)d, with d the space between the atomic planes and N the number
of atomic planes in the spacer. This results in the fact that the interlayer distance is discrete, which leads to an effective period Λ resulting in a q vector given by
2π
Λ =| q − n2π
d | (2.28)
where n is such that Λ > 2d. This effect is also known as ’aliasing’ and is shown in figure 2.20.
Note that in this model, the magnetic layers are single monoatomic layers, and experimentally these layers are thicker. However, in his paper Bruno shows that the exchange coupling strength is in a first approximation independent of the thickness of the magnetic layers.
2.4.5 Influence of roughness on the RKKY-coupling
In the past section we assumed that the layers are perfect and the location of the interface (and thus the point of reflection/transmittion) is well defined. In reality imperfections are present, such as roughness and interdiffusion. Roughness at the interface can lead to a varying thickness of the spacer layer. As the exchange coupling depends on the spacer layer thickness, this means that the net exchange coupling is an average over all the different couplings present. Another important effect of roughness at the interface is that the phase of the reflected electrons at the interface varies. The net spin density is the result of the summation over all the spin density waves of the individual electrons, and when the phases of these electrons vary widely the summation and thus the oscillating spin density will damp out quickly, resulting in a quick loss of the indirect exchange interaction.
Roughness at the interface can also lead to additional coupling effects. As ex-plained in section 2.4.1, correlated roughness can lead to orange peel coupling. An-other coupling that can occur due to roughness is the biquadratic coupling which is the result of frustration between spins. The energy per unit area due to the biquadratic coupling is given by
Ebiq = Jbiqcos2(θ) (2.29)
where Jbiq is the biquadratic coupling strength and θ is the angle between the mag-netizations of the two FM layers. The minimum energy is found when the magneti-zations have an angle of 90° between eachother.
2.4.6 The thermal stability of the RKKY-coupling effect
As explained in Chapter 1, in order to build a stable SAF-based spin valve sys-tem applicable in the automotive industry, the RKKY-coupling between the two FM layers must be stable at high temperatures. When the temperature is increased there are three main effects that can affect the interlayer coupling which should be
temperature dependence.
2.4.6.1 The intrinsic temperature dependence
As shown in the previous section, the oscillatory behavior of the exchange coupling strongly depends on the Fermi wavevector. The theory is based on a fermi distribu-tion for the occupied states, and for non-zero temperatures the distribudistribu-tion of states is equal to
ni = 1
e(Ei/kbT )+ 1
In which Ei is the energy of the specific state, kb is the boltzmann factor and T is the temperature. Finite temperatures smooth out the step function for the occupied density of states, which smooths out the oscillations of the interlayer exchange cou-pling JRKKY. The amplitude of the exchange coupling is thus expected to decrease for increasing temperature, while the period remains unchanged.
2.4.6.2 The temperature dependent disordering of the magnetic mo-ments
The interaction of the magnetization with the conducting electrons at the interface is the mechanism that creates the RKKY-coupling. Besides the intrinsic temperature dependence of the free electrons, the temperature dependence of the magnetization has to be considered as well. At finite temperatures, the magnetization will decrease due to thermal agitation. The amount of reduction is related to the temperature: for increasing temperature, the magnetization will decrease, untill the Curie temperature Tcis reached. At this temperature, the magnetic ordering disappears and the system is in a paramagnetic state. Especially around Tc the loss of magnetization increases rapidly and it has been shown that[23] for thicker FM layers the RKKY-coupling decays linearly and slowly with T, and for the monolayer limit the RKKY-coupling decays with T ln(T ).
2.4.6.3 Extrinsic temperature dependence
When an SAF is annealed at high temperatures, interdiffusion of the FM layers and the spacer layer can occur. The interdiffusion of the layers leads to a less well defined interface between the spacer layer and FM layers, thus leading to more roughness. As explained in section 2.4.5, this can result in the RKKY-coupling damping out more rapidly and additional coupling interactions such as pinhole coupling for thin spacer layers, and additional biquadratic coupling due to frustrations between spins. An additional coupling which can lead to biquadratic coupling is the so-called ’loose spin coupling’[24]. This coupling originates from loose spins that are present in the spacer
layer, for instance when a FM spin is diffused into the spacer layer material due to annealing. The loose spins interact with the FM layers via the RKKY interaction, and the loose spins contribute to an effective exchange coupling between the FM films. The presence of loose spins in the spacer layer can thus effect the RKKY-coupling between the FM layers, resulting biquadratic RKKY-coupling of the FM layers.