MASTER
The temperature and field stability of exchange biased magnetic multilayers containing a synthetic antiferromagnet
Deen, L.D.P.
Award date:
2015
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Physics of Nanostructures group (FNA)
The temperature and field stability of exchange biased magnetic multilayers containing a synthetic antiferromagnet
L.D.P. Deen June 2015
Supervisors:
Aurélie Solignac and Jürgen Kohlhepp
sensors in cars, as the limits regarding the performance of regular magneto-resistance sensors that are currently used, are being reached. The conditions under which the magnetic sensors have to work properly in cars, differ a lot from the conditions in hard disk drives. The magnetic sensors consist of a pinned magnetic multilayer that needs to remain pinned, regardless of temperatures and external fields present. As the temperatures and fields are much higher in cars than in hard disks, this forms a challenge.
In this study, the stability of an exchange biased multilayer combined with a synthetic antiferromagnet is investigated. Exchange bias is an effect that pins a fer- romagnetic layer by linking it directly to an antiferromagnetic layer. This pinned ferromagnetic layer is used as part of a synthetic antiferromagnet. A synthetic an- tiferromagnet couples two ferromagnetic layers separated by a non-magnetic spacer layer antiferromagnetically via the RKKY-coupling effect. The RKKY-coupling co- efficient oscillates when the interlayer thickness is varied, and in order to couple the ferromagnetic layers antiferromagnetically, the spacer layer thickness needs to be tuned. When exchange bias is combined with a synthetic antiferromagnet it results in a pinned multilayer containing no net magnetic moment.
A growth study on Co/Ru/Co synthetic antiferromagnets is conducted. It is found that the sputter pressure used during the deposition of the Co layers has a large impact on the RKKY-coupling strength. Optimal growth conditions are found when depositing the Co layers at a pressure of 15 sccm and a power of 20 W.
The thermal stability of synthetic antiferromagnets is investigated. The RKKY- coupling strongly depends on the thickness of the spacer layer. Magnetic multilayers are grown containing a wedged spacer layer, which showed two antiferromagnetic coupling regimes. The first regime found at spacer layer thicknesses of between 0.5 and 1.5 nm disappeared at annealing temperatures of 320° C, whereas the second antiferromagnetic regime found between 2.1 - 3 nm remained stable at these tem- peratures. The instability of the first region is attributed to pinholes created at low spacer layer thicknesses.
The field stability of magnetic multilayers containing exchange bias and synthetic antiferromagnetic multilayers is investigated via relaxation measurements. The angle of the magnetization of the exchange biased ferromagnetic layer has a large influence on the field stability effect. Decay rates of 6.3% and 27.0% were measured for applied fields of 62.5 mT and 1T, respectively. Although spin flop could induce an instability of the exchange bias when a field along the field cooling direction is applied, this was not experimentally observed. This result is promising regarding the application of these sensors in automotive environments.
1 Introduction 1
1.1 Spintronics . . . 1
1.2 AMR-based sensors . . . 3
1.3 The next step: GMR/TMR based sensors . . . 4
1.4 This thesis: the stability of an SAF-EB system . . . 4
2 Theory 6 2.1 Stoner-Wohlfarth Model . . . 6
2.2 The exchange bias effect . . . 9
2.2.1 An idealistic model . . . 12
2.2.2 Crystalline AF layers . . . 13
2.2.3 Polycrystalline AF layers . . . 15
2.3 The stability of the exchange bias effect against temperature and field 19 2.3.1 The relaxation effect . . . 20
2.3.2 The training effect . . . 21
2.4 Interlayer exchange coupling . . . 21
2.4.1 Interlayer exchange coupling mechanisms . . . 22
2.4.2 The RKKY interaction between magnetic impurities . . . 24
2.4.3 The interlayer exchange coupling: a simple model . . . 25
2.4.4 Bruno’s model . . . 27
2.4.5 Influence of roughness on the RKKY-coupling . . . 28
2.4.6 The thermal stability of the RKKY-coupling effect . . . 28
2.4.6.1 The intrinsic temperature dependence . . . 29
2.4.6.2 The temperature dependent disordering of the mag- netic moments . . . 29
2.4.6.3 Extrinsic temperature dependence . . . 29
2.5 Theoretical magnetization curves of an SAF . . . 30
2.6 Combining the SAF and EB . . . 30
3 Experimental setup 34 3.1 The sputter system . . . 34
3.1.1 The deposition process . . . 34
3
tum Interference Device (VSM-SQUID) . . . 37
3.2.3 The SQUID-setup . . . 37
3.3 Annealing techniques . . . 38
3.3.1 The Argon oven . . . 38
3.3.2 The VSM-SQUID oven module . . . 39
3.4 Sample stack structure . . . 39
3.4.1 RKKY-coupled stacks . . . 40
3.4.2 EB-RKKY-coupled stacks . . . 40
4 Results 42 4.1 Magnetic properties of an SAF . . . 42
4.1.1 Typical hysteresis loops of an SAF . . . 42
4.1.2 Extracting the oscillatory RKKY-coupling . . . 45
4.2 Growth studies on the Co-layers . . . 48
4.2.1 Varying the pressure and power during the Co layer deposition 48 4.3 Investigating the thermal stability of the SAF . . . 51
4.3.1 Instability of the first transition region . . . 55
4.3.2 Interpretating the results of the SAF measurements . . . 55
4.4 Field stability measurements on EB-SAF coupled stacks . . . 57
4.4.1 Magnetic properties of EB-SAF stacks . . . 58
4.5 Investigating the stability of the exchange bias . . . 61
5 Conclusions 67
6 Outlook 69
GMR Giant magneto-resistance
TMR Tunnel Magneto-resistance
SAF Synthetic antiferromagnet
EB Exchange bias
MOKE Magneto-optical Kerr effect
VSM Vibrating sample magnetometer
SQUID Superconducting quantum interference device
AF Antiferromagnetic
FM Ferromagnetic
NM Nonmagnetic
Introduction
Ever since the production of the first commercial cars in the early 20th century, the production, sales and usage of cars have seen a consistent increase. Today, the car is the most popular used vehicle globally, and recent figures show that the amount of registered cars within the Netherlands has seen an 18% increase in the past 15 years. Soon there will be over 8 million registered cars, averaging at 420 cars per 1000 inhabitants[1]. Comparing this to other countries, we have an average amount of cars, whereas the USA tops the chart at 845 cars per 1000 inhabitants. The current amount of cars used worldwide has been subject of debate over the past years, as oil is becoming scarce and pollution is destroying the ecological system of our planet.
Therefore, the amount of pollution needs to be reduced in order to keep our planet healthy. However, it is not only the current situation which makes for concern: at the other end of the car-registration spectrum, China is found with an average of 37 registered cars per 1000 inhabitants[2]. China’s economy is rising fast, and at some point the amount of car usage will start to reach western european numbers, increasing the already present pollution problem to an immense height.
Solutions for these existing and upcoming problems need to be found: cars need to be smarter, less fuel consuming and less polluting. One approach to try start tackling this huge problem is to improve one of the tiniest parts of the car: the magneto- resistance (MR) sensors used in the electronic system of the car. The MR sensor technology is part of the field of spintronics. In this chapter, a brief introduction of the field of spintronics is given and the technology behind MR sensors will be explained. Afterwards the subject of the research described in this thesis will be discussed.
1.1 Spintronics
Spintronics covers the research field which adresses two important properties of electrons: its charge and its spin. An important discovery within this field is the so-called giant magneto-resistance (GMR) effect, which is illustrated in figure 1.1.
1
Figure 1.1: Simplified illustration of the GMR-effect for parallel and antiparallel configuration (top), and an example of the electrical resistance versus applied field for a CoPd/Ru/CoPd multilayer (bottom). Figure taken from [4].
This discovery marked the beginning of intensive spintronic research in the late 1990’s. However, long before the discovery of the GMR-effect, the ordinary and anisotropic magneto-resistance (OMR and AMR) effect were discovered. In 1856 William Thompson discovered that the resistance of a current which is sent through magnetic material varies with the direction of the magnetization within the magnetic material. He discovered that the resistance reaches it lowest value when the direction of the magnetization is parallel with the direction of the current. This effect originates from the spin scattering probability[3], which varies for different directions of the magnetization. This effect is material specific, and is still used as the main physical principle in sensor applications today. Over a century later the GMR effect was discovered by Albert Fert and Peter Grünberg, for which they received a nobel price in 2007.
θ)
Magne5za5on)
Figure 1.2: Simplified illustration of an AMR sensor. The resistance the current experiences depends on the angle θ between the direction of the current and the mag- netization.
The GMR effect is based on the fact that the electrical resistance an electron experiences when passing through two magnetic layers seperated by an non-magnetic spacer layer depends on the mutual orientation of the magnetization within these magnetic layers. Figure 1.1 shows the GMR-stack for two different orientations of the magnetization and a plot of the resistance versus the applied field. A current can be conceptually separated into two currents: a spin-up current and a spin-down current. When a current is sent through a ferromagnetic (FM) layer, electrons with a spin aligning with the magnetization of the FM layer experience less resistance than electrons with spins aligned antiparallel due to spin-dependent scattering[4]. When the current is sent through two FM layers, the resistance depends on the mutual orientation of the magnetizations of the magnetic layers. When the magnetizations are parallel, electrons with spins aligned with the magnetizations experience a low resistance while the spins that do not align with the magnetization experience a high resistance. If the magnetizations are aligned anti-parallel, all electrons experience a high resistance (either in the first or in the second layer). In the plot shown in figure 1.1 the dependence of the resistance on the mutual orientation of the two layers is clearly observed.
1.2 AMR-based sensors
In the automotive industry the AMR sensor is still the most used sensor today.
A simple representation of an AMR sensor is shown in figure 1.2. The resistance depends on the angle between the current and the magnetization in the magnetic material. The AMR-technology is known for over a century, and it is becoming more and more difficult to improve the AMR-sensor as the limit of its performance is being reached. A way to improve the sensors is to switch to the GMR and TMR- based sensors: the GMR and TMR-sensors are much more sensitive than the AMR
sensors. AMR-sensors are able to detect fields down to 10−7 Tesla, whereas GMR and TMR-sensors can detect fields down to 10−10 Tesla. Nevertheless, implementing GMR and TMR-sensors in the automotive industry requires these sensors to work at high temperatures and external fields. In this thesis the thermal and magnetic field stability of magnetic multilayers used in GMR and TMR sensors are investigated. In order to describe the subject in more detail, the principle of GMR and TMR-sensors will be explained first.
1.3 The next step: GMR/TMR based sensors
In a GMR/TMR-based sensor, a current passes through a stack composed of two FM layers seperated by a spacer layer. The resistance the current experiences depends on the mutual orientation of the magnetizations in the FM layers. By pinning one of the two FM layers and keeping the other FM layer rotating freely, (ultra)sensitive measurements can be done, since a small change of the magnetic orientation of the magnetization in the free layer results in a large change in resistance. In a GMR- sensor, the spacer layer is a metallic layer, allowing direct flow of current through the spacer layer. In a TMR-sensor, or tunnel magneto-resistance sensor, the spacer layer consists of insulating material, allowing only flow of current through a tunneling effect. One of the biggest and most important differences between an AMR-sensor and a GMR/TMR-sensor is the use of a pinning layer. One way to pin a FM layer is to make use of exchange bias (EB): the FM layer is magnetically coupled to an antiferromagnetic (AF) layer in order to pin it. The principle behind the EB will be explained in more detail in section 2.1. Sensors combining GMR and EB are shown in figure 1.3. The configuration as shown on the left is sensitive to external fields, as the stack has a net magnetic moment: when a magnetic field is applied, a torque is exerted on the pinned FM layer. A way to prevent this is to use a synthetic antiferromagnet (SAF) : by adding an additional FM layer and a spacer layer with the right thickness, the two FM layers are coupled antiferromagnetically, resulting in no net magnetic moment when considering the pinned part of the sensor. A sensor combining GMR, EB and an SAF is shown in the right part of figure 1.3. The principle behind the SAF will be explained in more detail in section 2.4.
1.4 This thesis: the stability of an SAF-EB system
In order for the sensor to work the pinned multilayer should remain pinned at all times, during mounting and its lifetime. Although GMR and TMR-sensors are al- ready produced and applied in hard disk technology, applying them in the automotive industry requires different properties of the GMR/TMR-sensors. The sensors have to work properly at high external fields and high temperatures. The stability of the SAF and EB is the main subject of investigation in this thesis.
Exchange)) bias)
Spacer)layer) Ferromagne5c)layer) An5ferromagne5c)layer) Free)layer)
SAF)
Exchange)) bias)
Ferromagne5c)layer) Spacer)layer) Ferromagne5c)layer) An5ferromagne5c)layer) Pinned)
mul5layer)
Figure 1.3: A regular GMR-based sensor (left) and a GMR-SAF sensor (right).
In the regular sensor, current passes through two FM layers (green and blue) and a spacer layer (yellow). The bottom F-layer is pinned by EB, whereas the top layer is free to rotate. In the GMR-SAF sensor, an additional FM layer and spacer layer are used and are configured such that this additional layer is antiferromagnetically coupled to the other FM layer. This results in no net magnetic moment for the pinned multilayer.
This thesis can be divided in two parts. First, the magnetic characteristics and thermal stability of the SAF are investigated. Later, the AF layer is added and the magnetic properties and field stability of stacks containing both EB and an SAF are investigated.
Chapter 2 will start by describing the theory of EB and synthetic antiferromagnets seperately, after which a system containing both EB and an SAF will be discussed.
Chapter 3 covers the experimental setups used to create and investigate both the SAF stacks and the EB-SAF multilayers. All multilayers are grown using a sputter system, and for the magnetic characterization a Magneto-optical Kerr effect (MOKE) setup and a vibrating sample measurement - superconducting quantum interference device (VSM-SQUID) are used. In order to set the exchang bias, an Argon oven as well as a build-in annealing module in the VSM-SQUID setup are used. Chapter 4 covers experimental results that are obtained during research. First, a growth study on different spacer materials is presented. Next, the thermal stability of an SAF-system is investigated. Finally, a study on the magnetic properties and field stability of an EB-SAF system is presented. Chapter 5 presents the conclusions of this study and Chapter 6 provides an outlook and recommendations for future research on EB-SAF systems.
Theory
This chapter provides the theoretical background necessary to understand the exper- imental results that will be described in Chapter 4. In the first section the Stoner- Wohlfarth model is introduced which can often be used to model the magnetic behavior of magnetic thin films in external fields. In section 2.2 the Meiklejohn-Bean model and the Fulcomer-Charap model describing the EB effect is introduced. The EB effect is used in the spin-valve systems to pin the magnetization of an FM layer in a pref- ered direction. The stability of this pinning effect is the one of the main subjects of this thesis. The stability of the EB effect is discussed in section 2.3. Next, the theory of interlayer exchange coupling is presented which is used to create the SAF based on Ruderman-Kittel-Kasuya-Yosida (RKKY)-interaction. The RKKY-interaction and a simplystic model are introduced to translate the RKKY-interaction to a system of FM layers seperated by a spacer layer. Bruno introduced a model to qualitively describe the RKKY-coupling in magnetic multilayers[5][6]. His model is briefly intro- duced in section 2.4. Finally, EB and the RKKY-coupling are combined to create an EB-SAF coupled stack. The magnetic properties of this stack are described in section 2.6.
2.1 Stoner-Wohlfarth Model
When describing the reversal of magnetization in magnetic thin films, the Stoner- Wohlfarth model is often used, which is a macrospin model based on the assumption of a uniform response of the magnetization when an external magnetic field is ap- plied. In the Stoner-Wohlfarth model, the magnetization of the thin film will rotate uniformly with the applied external field. Typical magnetic loops as described within the Stoner-Wohlfarth model are shown in figure 2.1 and 2.2.
6
H) +2KF/μ0MF) D2KF/μ0MF)
+MF)
DMF)
H) easy)axis)
M)
M)
Figure 2.1: Hysteresis loop for the Stoner-Wohlfarth model for a field applied along the hard axis. The magnetization slowly rotates towards the applied field direction, untill it completely aligns and saturation is reached.
M)
Hc) H) H)
Hard)axis) easy)axis)
M)
M)
+2KF/μ0MF) D2KF/μ0MF)
+MF)
DMF)
Figure 2.2: Hysteresis loop when the field is applied along the easy axis. A minimum applied field is needed to switch the magnetization direction. When the magnetization is switched and the field is applied in the opposite direction, again a minimum field is required to switch the magnetization back. This results in magnetic behavior that depends on the field sweep direction.
Hard)axis)
Easy)axis) θ)
M) H)
β) KF) x)
y)
Figure 2.3: Schematic of the vectors and angles for the magnetization M, the anisotropy KF and the applied field H as defined for the Stoner-Wohlfarth model.
Figure adapted from [4].
The total magnetic energy per unit area of a magnetic thin film with a uniaxial magnetocrystalline anisotropy in an applied field along the applied field direction is given by
E = −µ0HtFMFcos(θ − β) + KtFsin2(β) (2.1) With H the applied field, MF the saturation magnetization, K the magnetocrys- talline anisotropy of the layer and tF the thickness of the FM layer. The first term in equation (2.1) is the Zeeman energy contribution describing the effect of the applied field on the magnetization, and the second term is the magnetocrystalline anisotropy term. The angles θ and β are defined as the angles between the applied field and the magnetization respectively with the direction of the anistropy of the layer, as depicted in figure 2.3.
The stable configuration of the magnetization corresponds to an energy mini- mization. Equation (2.1) is minimized with respect to β resulting in
−µ0HtFMFsin(θ − β) + KFtFsin(2β) = 0 (2.2)
µ0HtFMFcos(β − θ) + 2KFtFcos(2β) > 0 (2.3)
β = 0. When applying a field along the hard axis (θ = ±π/2), the magnetization rotates gradually towards the field direction, untill it aligns completely and saturation is reached. A hysteresis loop for a field applied along the hard axis is shown in figure 2.1. When applying the magnetic field along the easy axis (θ = −π) for small fields the magnetization remains aligned along β = 0 since the system is in a local energy minimum. Applying a magnetic field changes the energy landscape and in order to remove the local energy minimum state, a minimum field is required. When this field is applied, the magnetization switches and the system reaches the global energy minimum state (β = −π). When the magnetic field is then reversed (θ = 0), the global energy minimum changes to β = 0, and the system needs a minimum applied field again to switch to this global energy minimum. This results in magnetic loops that depend on the field sweep direction and an example is shown in figure 2.2.
The minimum field to switch the magnetization is the coercive field Hc that can be obtained from (2.2) and (2.3) and is equal to
Hc = ±2KF
µ0MF (2.4)
2.2 The exchange bias effect
As described in chapter 1, in order for a GMR/TMR sensor to work, the magneti- zation of one of the FM layers needs to be pinned. One way to do this is by using the so-called EB effect. The EB effect originates from the exchange interaction be- tween an AF layer with an FM layer at the mutual interface. Figure 2.4 shows the location of the exchange biased layer in a GMR/TMR sensor. The AF spins at the interface align with the FM spins at the interface. In order to couple these spins in a preferential uniform direction, the EB needs to be set. When the EB is set, the hysteresis loop is shifted.
Figure 2.5 shows the magnetic loop before (top) and after (bottom) setting the EB. In order to set the EB, the following procedure is used:
• The temperature of the system is increased such that TN < T < TC
• A large magnetic field is applied along a prefered direction to set the magneti- zation direction of the FM layer
• The temperature is cooled down to T < TN
• The applied field is reduced to zero
for which TN is the Neél temperature at which there is no more macroscopic magnetic ordering in the AF layer and TC is the Curie temperature at which there is no more macroscopic magnetic ordering in the FM layer.
SAF)
Exchange)) bias)
Ferromagne5c)layer) Spacer)layer)
Ferromagne5c)layer) An5ferromagne5c)layer) Non)magne5c)layer) Free)layer)
Pinned) mul5layer)
Figure 2.4: Simplified illustration of a GMR/TMR stack with the exchange biased layers highlighted.
By setting the temperature to TN < T < TC, the spins in the AF layer are disor- dered. By then setting the direction of the FM layer and decreasing the temperature to T < TN, the AF spins at the interface with the FM layer will experience an ex- change interaction and will align with the FM spins at the interface. By decreasing the temperature even further, the orientation of the AF spins is frozen in. If then the field is reduced to zero, the FM spins will remain aligned with the AF spins due to the exchange interaction. This direction along which the AF spins at the interface are aligned is called the ’field cooling direction’.
The coupling at the interface will induce an additional anisotropy in the FM layer. Switching the magnetization of the FM layer antiparallel to the field cooling direction requires a larger field, as both the magnetocrystalline anistropy as well as the exchange coupling needs to be overcome. When the magnetization of the FM layer switches and the field is applied in the opposite direction, a smaller field is required to switch the magnetization back as the exchange coupling direction did not change and is contributing to the switching of the magnetization. This results in a shift of the hysteresis loop of the FM layer. The shift of the loop is the so-called
’exchange bias field Heb’.
EB behavior was first observed by Meiklejohn and Bean[7]. They proposed a model to describe the EB effect, which is described in the next section.
M)
H) Hc)
+2KF/μ0MF) D2KF/μ0MF)
+MF)
DMF)
An5ferromagne5c)) layer)
An5ferromagne5c)) layer)
No)magne5c)) ordering)
No)magne5c)) ordering)
No)magne5c)) ordering)
Ferromagne5c)layer)
M)
HC) H) +MF)
DMF) H)
Ferromagne5c)layer) An5ferromagne5c)) layer)
An5ferromagne5c)) layer)
HEB) Field)cooling)direc5on)
+)
+) +) T)<)TN)
HC,2)
HC,1) 2KFtF+ Jeb
µ0MFtF
−2KFtF+ Jeb µ0MFtF
Figure 2.5: Schematic view of the hysteresis loops during the annealing process to set the EB for an ideal magnetic thin film. When the temperature is above the Neél temperature (top), there is no order in the spins of the AF layer and a regular FM hysteresis loop is found. After setting the EB, at T < TN (bottom), the AF spin orientation is frozen in. When a magnetic loop is measured, the EB results in a shift of the hysteresis loop.
2.2.1 An idealistic model
In the Meiklejohn and Bean model, a number of important simplifying assumptions are made[7][8][9]:
• Both the FM and AF layer are considered single domain layers (macrospins)
• The F layer has a uniaxial anisotropy
• The AF layer is blocked and is considered to have an infinite magnetocrystalline anisotropy
• The spins at the interface have a net magnetic moment equal to the magnetic moment of one sublattice
• The spins of the AF layer are completely uncompensated at the interface:
• The interface is atomically smooth: no imperfections or roughness is present
• The FM and AF layers are coupled by an interfacial exchange coupling due to the exchange interaction at the mutual interface, which is characterized by an interfacial exchange coupling energy per unit area Jeb.
The total magnetic energy per unit area of the system can be writen as:
E = −µ0HMFtFcos(θ − β) + KFtFsin2(β) − Jebcos(β) (2.5) in which H is the external applied field, MF is the magnetization of the FM layer, tF
is the thickness of the FM layer, β is the angle between the magnetization MF and the anisotropy direction of the F layer (KF) and Jeb is the exchange energy per unit area. The first term in equation (2.5) is the Zeeman energy. The second term is the magnetocrystalline anisotropy of the FM layer and the third term is the exchange coupling resulting from the coupling at the interface of the FM and AF layers. The angles used in the Meiklejohn-Bean model are shown in figure 2.6.
By minimizing the total magnetic energy with respect to β behavior of the mag- netization of the FM layer versus an applied field can be calculated. The coervive field values HC,1 and HC,2 can be extracted and are equal to
Hc,1 = −2KFtF + Jeb
µ0MFtF (2.6)
Hc,2 = 2KFtF + Jeb
µ0MFtF (2.7)
The coercive field of the loop Hc and the displacement Heb of the magnetic loop can be calculated and are equal to
Hc= −Hc,1+ Hc,2
2 = 2KF
µ0MF (2.8)
Hard)axis)
Easy)axis) θ)
M) H)
β) KF) x)
y)
Figure 2.6: Schematic overview of the vectors and angles used in the Meiklejohn- Bean model.
and
Heb = Hc,1+ Hc,2
2 = − Jeb
µ0MFtF (2.9)
which shows that EB results in a shift of the loop equal to Heb and does not change the coercivity field Hc of the magnetic loop. The shift of the magnetic loop can be understood by considering the applied field direction as compared to the field cooling direction. When a field is applied opposite to the field cooling direction, the addional coupling has to be broken in order to rotate the FM layer and extra energy is needed. This behavior is asymmetric: when the field is applied along the field cooling direction a smaller field is required to reallign the FM layer along the field cooling direction since the coupling term favours this allignment. This asymmetric behavior leads to the shift of the hysteresis loop.
2.2.2 Crystalline AF layers
The assumtions made in the previous section are idealistic. In reality, the AF layer does not have an infinite anisotropy, and experimentally it was found that the spins in the AF layer can rotate when an external field is applied, as illustrated in figure 2.7. An extra energy term for the magnetocrystalline anisotropy of the AF layer has to be introduced. The total magnetic energy per unit area is then equal to
E = −µ0HMFtFcos(θ − β) + KFtFsin2(β) + KAFtAFsin2(α) − Jebcos(β − α) (2.10)
Figure 2.7: Illustration of the angles of the different magnetizations and anisotropies when a (small) rotation of the spins in the AF layer is taken into account. A rotation of α of the spins in the AF layer with respect to the initial orientation is allowed.
where KAF is the anisotropy of the AF layer, tAF is the thickness of the AF layer and α is the angle between the spins of the AF layer and the anisotropy direction of the AF layer. Again, an energy minimization can be conducted with respect to α and β, leading to
H
Heb∞sin(θ − β) + sin(β − α) = 0 (2.11)
Rsin(2α) − sin(β − α) = 0 (2.12)
in which
Heb∞ ≡ − Jeb
µ0MFtF (2.13)
is the EB field value for an infinitely large anisotropy for the AF layer, and R ≡ KAFtAF
Jeb (2.14)
is describing the ratio between the anisotropy of the AF layer and the interfacial exchange energy Jeb. This ratio determines the properties of the EB system. A plot of the coercive field versus R and the EB field is shown in figure 2.8.
In order to clarify the influence of this ratio, the anisotropy of the FM layer (KF) is assumed to be zero. There are three regions of interest.
Figure 2.8: Plot of the ratio between the anisotropy of the AF layer and the inter- facial exchange energy versus the coercive field and EB field. Figure adopted from [10].
• R ≥ 1. In this region (I), the AF anisotropy is larger than the interfacial exchange coupling. This results in a limited angle over which the spins of the AF layer can rotate. A small rotation of the spins leads to a small decrease of the EB as can be seen when R=1. When R>>1, the anisotropy of the AF layer is so strong that the AF spins do not rotate anymore. At this point, the maximum EB field Heb∞ is found.
• 0.5 ≤ R ≤ 1. In this region (II), the AF spins start to rotate along with the FM spins. This results in a loss of the EB. Depending on the field sweep direction, at a critical angle β of the FM spins, the AF spins will switch. This results in a hysteresis like behavior, and a coercive field is observed.
• R < 0.5. In this region (III), the anisotropy of the AF spins is much lower than the exchange coupling. The spins of the AF layer follow the direction of the FM spins without any jumps. For the limit if R= 0 both the coercive field and the EB field are zero.
In order to pin the FM layer, it is important that EB is present and the AF spins rotate as little as possible. For this, the anisotropy of the AF layer must be large.
2.2.3 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
EAF = tgKAFsin2(α) − cJebcos(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 = −µ0HMFtFcos(θ − β) + KFtFsin2(β) +
N
X
i=1
KAFtigsin2(αi) − Jebcicos(β − αi) (2.16) 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
∆E = KAFAtg 1 +
JebAc 2KAFAtg
2!
− JebAc (2.17)
in which A is the area of a grain.
Ferromagne5c)layer)
An5ferromagne5c)grains) MF)
Easy)axis)
β)
MAF) 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) ΔE) EAF)
½)π) 1½)π) α)
E1) E2)
DE2)
β=π)
μ0H)
Field)cooling)direc5on)
Ferromagne5c)layer) )An5ferromagne5c)grain) )
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.
TB)
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 = TB. 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 temperature and field
As described in Chapter 1, EB is used to pin a FM layer in a GMR/TMR sensor. In order for the GMR/TMR sensors to be applicable in the automotive industry, the GMR/TMR sensors need to function properly at high temperatures and external fields. This requires the EB to be stable at these temperatures and fields. The influence of temperature was adressed in the previous section and it was shown that thermal fluctuations can lead to the switching of spins in the AF grains, resulting in a loss of EB. In this section, the stability versus field will be discussed and two
The)relaxa5on)effect)
Figure 2.13: Illustration of the switching of the AF grains over time when the magnetization of the FM layer is reversed. The switching of an AF grain is a ther- mally activated process, and the variation of size and shape of the samples results in a gradual switching of all the grains over time. When the time is sufficiently long, all grains will have switched, and the original EB is lost. This phenomenon is the so-called ’relaxation effect’.
effects will be discussed: the relaxation effect and the training effect.
2.3.1 The relaxation effect
When a field is applied opposite to the setting field of the annealed bilayer, the magnetization of the FM layer switches. The torque that is then applied on the AF grains can switch the grains, as depicted in figure 2.13. As the grains vary in size and shape, the time it takes to switch an AF grain varies. Small grains will switch sooner than larger grains. However, when the field is applied for a sufficient long time, the larger grains will switch as well, and a significant reduction of EB is found.
As described in the previous section, this effect is a thermal relaxation effect. The probability of grains overcoming this energy barier 4E increases over time. If all grains are identical, this would result in a 4E which is equal for all grains. In real- ity, the grains vary, for instance in volume. A different volume leads to a different relaxation, resulting in a different 4E. When a grain switches, it reaches the global energy minimum, as shown in figure 2.11. When a grain is switched and the mag- netization versus applied field is measured, the grain will remain switched, resulting in a loss of EB. The switched grains will not rotate along with the magnetization of the FM layer, and thus do not contribute to a coercive field. When the experimental time is large enough, enough grains will switch and the EB is lost.
Figure 2.14: Simplified view of a GMR/TMR stack with the SAF illustrated in color. The SAF is part of the pinned multilayer in the GMR/TMR stack. By imple- menting an SAF in the pinned multilayer, the net magnetic moment of the pinned multilayer is zero.
2.3.2 The training effect
The training effect is an effect that occurs when hysteresis loops of an exchange biased system are measured for the first time after annealing. The effect results in a change of the hysteresis loop when consecutive measurements are performed. The coercive fields and the EB decrease for increasing number of measurements. The training effect is related to the unstable state of the AF layer and the F/AF interface as prepared by the field cooling procedure. However, the exact mechanisms behind the training effect are not yet understood. It has been shown that the training effect mainly influences the first hysteresis loop[14][15]and therefore to prevent the influence of this effect, a first field sweep is conducted after the annealing process. This first field sweep is not taken into account when doing measurements.
2.4 Interlayer exchange coupling
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
Edip = µ0
4π
"
−
−→ m1· −m→2
r3 + 3
−−→(m1· −→r )−−→
(m2· −→r ) r5
#
(2.19)
In which −→r is the vector between the two magnetic moments m1 and m2, as shown in figure 2.15.
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 distribution 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 coupling of the FM layers.
2.5 Theoretical magnetization curves of an SAF
In order to understand the magnetization curves of an SAF, consider the total energy per unit area of an RKKY-coupled system which is given by[16]
E = −µ0HMFtF(cos(θ−β)+cos(θ−α))+KFtF(sin2(α)+sin2(β))−JRKKYcos(β−α) In which the first term is the Zeeman energy term for both FM layers, the second term is the magnetocrystalline energy term of the FM layers and the third term is the RKKY-coupling term. JRKKY is the RKKY coupling coefficient which oscillates with increasing spacer layer thickness as described in the previous sections, and when the FM layers are coupled antiferromagnetically to create an SAF, JRKKY < 0. In this thesis, polycrystalline Co layers are used and the anisotropy can be considered ne- glegible. Minimizing the total energy with respect to α and β result in magnetization curves shown in figure 2.20. The angles used in this model are shown in figure 2.21.
As an increasing field is applied, the magnetization will slowly align with the field.
The applied field is fighting the RKKY-couping. When the applied is high enough to completely break the RKKY-coupling, the magnetizations are aligned parallel with the field. The field at which this occurs is the saturation field. The RKKY-coupling coëfficient is proportional to the saturation field and is equal to
JRKKY = −µ0HsatMst
2 (2.30)
2.6 Combining the SAF and EB
As explained in Chapter 1, EB is combined with RKKY-coupling to create a pinned multilayer with no net magnetic moment as part of the GMR/TMR. This pinned multilayer is shown in figure 2.22. When the SAF and EB are combined, the total energy of the system becomes
EA= −µ0HMFtF(cos(β) + cos(α)) + KFtF(sin2(α) + sin2(β))
−JRKKYcos(β − α) − JEBcos(β) (2.31) The first term is the Zeeman energy term of the two FM layers, the second term is the anisotropy term of the FM layers, the third term is the RKKY coupling coëfficient
μ0H)
x) M)
H) Hsat)
Hsat) z)
Figure 2.20: Example of a magnetic loop of an SAF for KF=0. At zero field, the FM layers are coupled antiferromagnetically. When a field is applied, the magne- tizations align gradually with the field direction, untill they are completely aligned.
The field at which the magnetizations completely align is the saturation field, which is proportional to the RKKY-coupling strength.
Easy)axis) Hard)axis)
MF) H) θ) KF)
Ferromagne5c)layer)
α)
MF) β) KF) Spacer)layer)
Ferromagne5c)layer)
Figure 2.21: Angles of the magnetizations and the field directions in an SAF sys- tem.
coupling the two FM layers and the fourth term is the EB term, coupling the AF layer with the bottom FM layer. Assuming that there is no anisotropy and minimizing the total energy with respect to the angles of the magnetizations α and β results in the magnetic loops as shown in figure 2.23.
Combining EB with the AF RKKY-coupling results in a plateau when a small field is applied. The EB pins the magnetization of the bottom FM layer, which is
