Micromagnetic Simulations of Spin Valve devices Manuel Jo˜ ao de Moura Dias Mendes No 49542
SENIOR THESIS Licenciatura em Engenharia F´ısica Tecnol´ ogica Instituto Superior T´ecnico Universidade T´ecnica de Lisboa
27th July 2005
Abstract Since the discovery of giant magnetoresistance (GMR) effect in magnetic multilayers, there has been considerable effort in applying this effect in magnetoresistive sensors based on spin valve GMR sandwiches with artificial antiferromagnetic reference layers. These structures prove to be of great interest because of their low-field magnetoresistive behaviour. Especially spin valves with crossed anisotropies of the sensitive and exchange-biased magnetic layers are very favorable for applications, because they exhibit a high sensitivity combined with an extremely low coercivity. This work was performed in the Materials Simulations group of Prof. Jos´e Lu´ıs Martins in colaboration with the experimental group of INESC-MN. The group of Prof. Jos´e Lu´ıs Martins has developed a fully three-dimensional code to simulate devices that is now able to study the response of real multi-layer structures to external applied fields. The multilayers can have arbitrary shapes, and number of layers, and the simulations can be static or dynamic. In this work I have used this micromagnetic program to study the use of both synthetic antiferromagnetic pinned (SAF) and synthetic free (SF) layers in spin valves for sensor applications. There has been a growing interest in using these structures since they lower the magnetostatic fields present at the free layer, allowing even higher sensitivity but at the expense of an increased offset field H0 . The experimental group of INESC-MN produced patterned 2×6 µm2 bottom pinned SAF+SF spin valves, with a SAF structure slightly unbalanced and a SF physical thickness of 55 ˚ A. Several effective thicknesses were performed on the SF trilayer from 50 down to 5 ˚ A. Micromagnetic simulations of these structures confirmed the experimental results, specially in what concerns the offset field H0 dependence on the SF effective thickness (teff ). In addition, simulations were also used to optimize the mentioned structures by showing that it is possible to considerably reduce the offset field H0 (maintaining the value of teff ) either by increasing the demagnetizing fields acting on the SF layers to compensate the coupling fields (reducing sensor dimensions, increasing SAF moment), or by using a separate biasing layer. Lastly, a few spin valves dynamic simulations were made to study the free layer magnetization switching in terms of the damping parameter and frequency of 1
precession. The implementation of a spin-polarized current was also studied and simulated, and it has been verified that this current may increase the effective magnetic damping parameter, thus accelerating the free layer switching mechanism.
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Contents 1 Theoretical Introduction 1.1 Basic Principles of Ferromagnetism . . . . 1.2 Micromagnetism for Ferromagnets . . . . 1.2.1 Zeeman Energy . . . . . . . . . . . 1.2.2 Exchange Energy . . . . . . . . . . 1.2.3 Anisotropy Energy . . . . . . . . . 1.2.4 Demagnetizing Energy . . . . . . . 1.3 Energy Minimization - Brown’s Equations 1.4 LLG Equation of Motion . . . . . . . . . .
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2 Numerical Model 2.1 Calculation of Energy and Heff terms . . . . . . . . 2.1.1 Numerical Zeeman Energy . . . . . . . . . . . 2.1.2 Numerical Exchange Energy . . . . . . . . . . 2.1.3 Numerical Anisotropy Energy . . . . . . . . . 2.1.4 Numerical Demagnetizing Energy . . . . . . . 2.2 Algorithms for Energy Minimization . . . . . . . . . 2.3 Integration methods for the Dynamic LLG equation
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3 Micromagnetic Simulation of Magnetic Sensor devices 3.1 Basic Spin Valves . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Pinning field from Antiferromagnets . . . . . . . . 3.1.2 Interlayer Coupling . . . . . . . . . . . . . . . . . . 3.1.3 Magnetoresistance and Spin Valve operation . . . 3.1.4 Simulation of a Basic Spin Valve . . . . . . . . . . 3.2 SAFs for Pinned Layers . . . . . . . . . . . . . . . . . . . 3.2.1 Simulation Results for a SAF structure . . . . . . 3.2.2 SAF pinned layers in Spin Valves . . . . . . . . . . 3.3 Synthetic Free structure (SF) . . . . . . . . . . . . . . . . 3.4 SAF-SF Spin Valve structures . . . . . . . . . . . . . . . . 3.4.1 Effect of Hcoup on Offset . . . . . . . . . . . . . . . 3.4.2 Effect of Hd on Offset . . . . . . . . . . . . . . . . 3.4.3 Effect of Hk on Offset . . . . . . . . . . . . . . . .
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3.4.4
Comparison with Experimental results and structure optimization . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Dynamic time resolved simulations of magnetization switching in Spin Valves 4.1 A simple switch test in a Bulk sample . . . . . . . . . . . . . . . 4.1.1 Underdamped oscillator . . . . . . . . . . . . . . . . . . . 4.1.2 Critically damped oscillator . . . . . . . . . . . . . . . . . 4.2 Switch test in a patterned 0.25 × 1 µm2 sample . . . . . . . . . . 4.3 Spin Polarized Current triggered Magnetization switching . . . . 4.3.1 Spin Valve simulations with SPC . . . . . . . . . . . . . .
55 56 56 58 58 60 62
5 Acknowledgments
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Chapter 1
Theoretical Introduction 1.1
Basic Principles of Ferromagnetism
Micromagnetism is concerned with the study of magnetic fields in space that is occupied by matter. For that purpose let’s start by analysing the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields in the presence of matter: ∇·D=ρ
∇×E=−
∂B ∂t
∂D ∂t In the usual quasistatic formulation of micromagnetics [4], it is assumed that the magnetic fields present are quasi-static(1 ), that there are no currents passing through the materials and no electric fields (only magnetic). Hence, it follows that all the time derivatives, currents and electric fields can be eliminated from Maxwell’s equations: ∇·B=0
∇×H=J+
∂ → 0, J → 0, E → 0. ∂t Thus the set of four equations reduces to only two, for the magnetic induction vector B and the total magnetic field H: ∇·B=0
∇×H=0
(1.1)
The magnetic induction B is given by B = µ0 (H + M) , 1 Meaning
(1.2)
that any changes in magnetization take much longer than the time required for light to cross the sample
5
where µ0 is the vaccum magnetic permeability (µ0 = 4π × 10−7 N/A2 ), and M is the magnetization of the material. In turn, the total magnetic field H can be written as the sum of the external applied field Ha , which is the field that would exist in the absence of the magnetizable medium, plus the demagnetizing field Hd which is due to the presence of the material: H = Ha + Hd .
(1.3)
Latter on it will be described how the demagnetizing field Hd is calculated. According to their magnetic properties, all the substances that can be found in nature are commonly divided into 4 main groups: • Diamagnetic • Paramagnetic • Ferromagnetic • Antiferromagnetic and other complex magnetic behaviors Substances are called Diamagnets and Paramagnets when they respond linearly to the applied field H. The intrinsic magnetic moment M created in these materials is proportional to the applied field, M = χH ,
(1.4)
where χ is called the susceptibility of the material. For these materials the total magnetic induction that will be measured upon the substance will be proportional to H: B = µ0 (1 + χ)H .
(1.5)
In the case of: • Paramagnets: χ > 0 - the intrinsic magnetization created will add to the applied field • Diamagnets: χ < 0 - field M will subtract to the external field H However, for the very special case of ferromagnets their magnetic behavior cannot be so simply described by (1.4), because they exhibit a residual magnetic moment even for zero applied field. Ferromagnetism can only be fully described according to quantum mechanical principles [1]. The simplest model assumes that the atoms or molecules that constitute the basis of a ferromagnetic material exhibit a certain permanent magnetic moment µ, just as if they were punctual dipoles. In the majority of substances every atomic or molecular orbital has an equal energy level available for spin up and spin down electrons, thus, in a common non-valence orbital normally there’s always an equal number of spin up and 6
spin down electrons. However, in ferromagnets some of their outter orbitals have a certain gap between the energy levels available for spin up and spin down electrons, which originates an excess of electrons with a certain spin in these orbitals. And it is this spin excess, averaged over the entire ferromagnetic boby, that will be the responsible for the existance of a spontaneous magnetization even at zero applied field. So, for ferromagnets, instead of (1.4) if we measure the total magnetic moment M as a function of Ha we will get an history dependent curve like the one plotted in fig. 1.1. Ms Mr
M
-Hc
Hc
Ha
Figure 1.1: The figure shows a typical hysteresis curve of a ferromagnetic sample. Ms is the saturization magnetization, Mr is the remanent magnetization and Hc is the coercivity field. As one can see in fig. 1.1 if the magnetization of the ferromagnet is saturated with a suficiently large field Ha and then this field is reduced to Ha = 0, the material will still have a remanent magnetization Mr that remained just as if the material could “memorise” the disturbance that was provoqued in it.
1.2
Micromagnetism for Ferromagnets
It is a good approach to consider the magnetization inside a ferromagnetic sample of volume V as a continuous Maxwell field defined by (per unit volume) [3]: M s · m(r) , inside V M(r) = (1.6) 0 , outside V
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Where m(r) is a unitary vector that has the direction of the average magnetic moment: |m(r)| = 1
(1.7)
To describe the magnetic properties of a ferromagnet we will use a popular thermodynamic model that takes into consideration the different effects of 4 fundamental energy terms: 1. Zeeman Energy that gives the interaction with the external field 2. Exchange Energy which results from the quantum mechanical interaction between the magnetic dipoles of the material 3. Anisotropic Energy defined by the crystalline anisotropy of the material 4. Demagnetizing or Magnetostatic Energy created by the magnetic charges induced by M in the borders of the sample In the next section it will be given a detailed explanation of each one of these forces that act in metalic ferromagnetic materials, like Ni, Fe, Co and their alloys.
1.2.1
Zeeman Energy
The energy of interaction between the applied field Ha and the magnetization M(r) of the material (defined in (1.6)) is given by the thermodynamic relation of the electrodynamics of continuous media described in sections 31 and 32 of Landau and Lifshitz book [5]: Z EZ = −µ0 M · Ha dV (1.8) V
Meaning that this energy term will be minimum when M(r) is fully aligned with Ha . Expression (1.8) can be illustrated using the theorical atomic dipoles model for ferromagnets referred in the previous section. According to this model the dipole moment of a certain ferromagnetic atom can be calculated by: µ = g · µb · S
[Am2 ]
(1.9)
Where g, the Land´ee factor, is the ratio between the electrons that exist in the spin up orbital and the ones in the spin down orbital next to the Fermi level (for Ni g=0.54), µb is the Bohr magneton, and S is the unitary spin vector that defines the orientation of our magnetic dipole [1]. If we now apply an external Ha field to this dipole then it will align its spin vector with the direction of Ha . Then, in order to rotate that particle to a position where its dipole vector µ makes an angle θ with the direction of the applied field we will have to spend an amount of work equal to: 8
eZ = −µ · Ha = −µHa cos(θ)
(1.10)
This eZ , called the Zeeman energy, is no more than the energy of interaction between the dipole rotated θ degrees from the field direction and this magnetic field. To get the overall Zeeman energy (EZ ) of the sample with volume V one needs only to sum (1.10) over all the Nat atoms present: EZ = −
Nat X
µi · Ha
(1.11)
i=1
We can generalize this expression to the continuum approximation, assuming the thermodynamic limit of huge Nat : Z Z EZ = eZ dV = −µ0 M · Ha dV (1.12) V
V
And so we recover the general expression (1.8).
1.2.2
Exchange Energy
The formation of a ferromagnetic structure in a certain material is due mainly to the exchange interaction of its atoms, which is independent of the direction of the total magnetic moment of the sample. This interaction is a quantum effect resulting from the symmetry of the wave functions of the system with respect to interchanges of the particles. Macroscopically, this energy can be expressed in terms of the derivatives of M with respect to the coordinates [5]: Z Ex = A V
∂mi 2 ∂rj dV i,j=x,y,z X
(1.13)
Where A is a phenomenological exchange parameter with units of J/m. Expression (1.13) can be illustrated using the theorical atomic dipoles model for ferromagnets. According to this simple model, the greatest difference between the ferromagnets and the paramagnets is that the permanent magnetic dipoles of the ferromagnetic materials interact strongly between each other, even without an applied field present. So, each ferromagnetic atom will tend to align with its neighbours due to the spin interaction between nearest neighbours. The interaction energy between a pair of neighbour atoms with spin Si and Sj is often described by the Heisenberg model [1]: ex = −2Jij Si · Sj
(1.14)
This energy is proporcional to the dot product of the dipole moments of both atoms, and Jij is the coupling constant between the atomic spins. This coupling constant is given by the value of the quantum mechanic exchange integral between the wave functions associated with atoms i and j [2], hence it has units of energy. 9
Usually it is a good approach to consider Jij a constant throughout the material, so we can just place J, instead of Jij , in (1.14). For every pair of atoms in a ferromagnetic material J > 0, whereas in Antiferromagnetic substances J < 0. So, in antiferromagnets the spins will tend to align antiparalelly to each other. To calculate the total exchange energy of the material (Ex ) we will have to sum (1.14) over all pairs of nearest neighbours. The continuum generalization of this [4] gives us the general expression for this energy term (1.13): Z Ex =
Z ex dV = A
V
V
∂mi 2 ∂rj dV i,j=x,y,z X
(1.15)
A is a material constant that can be determined by: JS 2 [J/m] (1.16) a Where a is the lattice constant. It can easily be seen that Ex will have the sign of A. If the magnetization varies too rapidly in a short distance the Ex will be very high. So physically, this energy term will have a smoothing effect on the dipoles orientation, introducing a preference for the atoms to remain aligned with each other. The exchange interaction will only dominate in a short range, between atoms that are at a distance of the order of the Exchange length Lex . This exchange length is the distance over which the magnetization will be roughly constant, and is approximately determined by [2]: r A Lex = (1.17) Km A=
Where Km is an energy density given by Km = 12 µ0 Ms2 . For the typical case of permalloy (NiFe), Ms = 8 × 105 A/m and A = 1.3 × 10−11 J/m, which gives an exchange length of 5.7 nm corresponding to about 17 unit cells [6].
1.2.3
Anisotropy Energy
Crystalline materials are magnetically anisotropic because there is a preferential direction for the orientation of the dipoles lying along the main crystallographic axis of the structure. Which means that there exists an internal field that will force the net magnetization to align with certain axis of the crystalline structure - that direction will be denoted by vector K, the anisotropic vector that defines the so called easy-direction. For instance, in a hcp crystalline structure K would be parallel to the c axis of the hexagonal cell. The magnetocrystalline anisotropy energy (Ek ) is defined as the work needed to rotate the sample magnetization to a certain direction out of the easydirection, and it is determined by a series expansion of trigonometric functions of 10
the angles that the magnetization vector makes with the main crystallographic axis. This is discussed in detail in section 40 of Landau and Lifshitz book [5]. For the substances treated in this work it is sufficient to consider only one easy axis, hence Ek is calculated as follows: Z Ek = (K sin2 α)dV (1.18) V
Where α is the angle between m(r) and the easy axis, and K is an energy density (J/m3 ). We could also include higher order terms like sin4 α and sin6 α, but it has been verified that the terms superior to sin2 α are unimportant. The minimization of this energy term will cause the magnetization to prefer to align with the easy axis; and it will also contribute to the “memory effect” of ferromagnets, called hysteresis, that enables us to store information on our computers even after they are turned off. This hysteresis effect is due to the fact that there will be a certain resistance, created by Ek , when the magnetization tries to switch to directions outside the easy-axis. Meaning that not all angles θ are equally probable for the orientation of the magnetic dipoles of the crystalline substance. So, the higher is our anisotropic field the higher will be the coercivity (Hc ) of our M(H) graphic - see fig. 1.1.
1.2.4
Demagnetizing Energy
The demagnetizing energy Ed corresponds to the interaction between the magnetization of the material and the demagnetizing field Hd introduced in (1.3). This energy is given by a similar thermodynamic relation to the Zeeman term [5]: Z µ0 Hd · MdV (1.19) Ed = − 2 V The factor of 21 that appears in this equation (and not in the Zeeman energy (1.8)) has to do with the fact that the source of the Hd is M, it’s an intrinsic energy term. The same thing happens in electrostatic when we have to insert a 1 2 factor to calculate the energy of a distribution of charge, so that the energy of charge i due to charge j does not accumulate with the energy of charge j due to charge i [7]. Demagnetizing Field The magnetostatic or demagnetizing field Hd , originated by M, can be calculated applying the equations deduced in (1.1): ∇ × Hd = 0
and ∇ · Hd = −∇ · M
(1.20)
Now, since the rotational of Hd is zero this field can be given by the gradient of a certain scalar potencial - the demagnetizing potencial Vd :
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Hd = −∇Vd
(1.21)
To understand the source of this demagnetizing field we can think of the electric analogue. If we have a material with a certain polarization P inside it, that polarization will create electric charges on the surfaces of the material normal to its vector direction. These charges will be responsible for the creation of an electrostatic potencial, as described by the Poisson equation [7]. In a ferromagnet a similar case happens with the magnetization (instead of the polarization). Since M is continuous inside the material, there will be an effective magnetic charge density ρ, defined by the Poisson equation: ∇2 Vd = ∇ · M = −ρ
(1.22)
However, on the surfaces M is no longer continuous since this function jumps abruptly from Ms to 0 (1.6). Because of that, surface magnetic charges will appear at the material boundaries with a density σ given by: ∂Vd ∂Vd out in − = −M · n = −σ (1.23) V d = Vd , ∂n out ∂n in Where n is the unitary vector normal to the surface. Physically, magnetic charges emerge on a surface whenever the magnetization has a component normal to that surface. These charges are the sources of the demagnetizing field that will be opposite to the normal magnetization of the material. Hence, this energy term will like to make the magnetization become parallel to the surfaces. After computing the charge distributions ρ(r) and σ(r) the solution to our Poisson problem, the demagnetizing potencial, is calculated as follows [3]: Z Z ρ(r0 ) σ(r0 ) 0 Vd (r) = dV + dA0 (1.24) 0 0 V |r − r | ∂V |r − r | The demagnetizing field will then be given by the symmetric of the gradient of this solution (1.21), and finally we integrate to get our Ed term [4]: Z Z Z µ0 µ0 Ed = ed dV = − Hd · MdV = + H2 dV (1.25) 2 V 2 ∞ d V Where the last integral can easily be obtained by parts. This energy is called “demagnetizing” because the higher is our M the more magnetic charges will be created on the surface, and consequently the greater this energy term will become. Minimizing this energy Ed corresponds to rotating the magnetic dipoles of the sample so that they create a minimum of magnetic charges on the surfaces, and that causes the material to be divided into different magnetic domains oriented in opposite directions (as previously proposed by Weiss) - see fig. 1.2. This way, the magnetic charges formed by a certain domain will cancel the charges of the adjacent domains, reducing Ed [2, 5].
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Figure 1.2: This is a schematic illustration of the break up of magnetization into domains - (a) single domain, (b) two domains, (c) four domains and (d) closure domains. [8]
Hypothetically, if the demagnetizing energy was the only term present, the material would break itself into smaller and smaller domains resulting in a zero total magnetization. The only thing that stops ferromagnets to do like so is their particular exchange energy Ex . This energy term has the opposite effect of the Ed . As it was previously described, the atomic spins tend to align parallel to the orientation of their nearest neighbours, however, this exchange interaction has a very short range, and so only the atoms within a distance of Lex (1.17) will be forced to remain parallel. On the other hand, the magnetostatic forces are of long range, hence the demagnetizing energy has a more significant effect over greater distances, whereas the exchange energy dominates over short distances. It is the balance between these two energies that will be responsible for the formation of our magnetic domains, which are going to be separated by walls with a thickness of the order of the Lex [2, 5]. These walls are of particular interest because they are made of atomic dipoles that are rotated about 180o from one side of the wall to the other, that is, from the orientation of one domain to the orientation of the adjacent one, as represented in fig. 1.3. The thickness of these domain walls, however, will also be influenced by the Anisotropy energy. This energy Ek forces the magnetic domains to align along the easy direction. So, supposing we have two antiparallel domains next to each other, lying along the easy axis of the sample, then the magnetic dipoles that remain in the separating wall between them will have to rotate 180o (as can be seen in fig. 1.3). Thus, these separating dipoles will take directions outside the easy one and, for that reason the Ek “prefers” to create a very thin wall, while Ex will want to enlarge it [2].
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Figure 1.3: This illustration is a schematic representation of a 180o domain wall. [8]
1.3
Energy Minimization - Brown’s Equations
Let’s consider a ferromagnetic body of any shape, in which the magnetization is any function of space of the type defined in (1.6), immersed in an external field Ha . Now that we know how to write all the main energy terms of a ferromagnet we are able to determine the total energy for this particular magnetic configuration m(r), which is: E(m(r)) = Ex + Ek + Ed + EZ
(1.26)
The next step is to find the particular m(r) that minimizes (1.26) in order to determine the equilibrium magnetic configuration for this problem. By doing so [2, 5], we will get Brown’s Static Equations that can be written in the simple form: ∂E = m × Heff = 0 ∂m
(1.27)
∂E ∂E Where ∂m is a notation for a vector whose Cartesian coordinates are ∂m , x ∂E ∂E ∂my and ∂mz . These equations mean that, in equilibrium, the torque applied in M(r) is zero everywhere, and the magnetization becomes parallel to an effective field Heff = Ha + Hd + Hx + Hk . Each one of these 4 terms in Heff is a functional derivative of the corresponding energy density term in order to m(r) [3]:
Hi =
1 ∂ei µ0 Ms ∂m
Which results in: 14
(1.28)
Hx =
2A ∇2 m µ0 Ms
,
Hk = −
1 ∂ek µ0 Ms ∂m
(1.29)
The demagnetizing field Hd and the applied field Ha have been defined earlier. If we then replace the calculated fields Hi into the Heff of equation (1.27) we will get Brown’s differential Equations which can be written in vector notation as [2]: m × µ0 Ms Ha − ∇Vd + 2A∇2 m − 2Kk × (m × k) = 0
(1.30)
Where k is an unitary vector with the direction of K.
1.4
LLG Equation of Motion
As described in the previous section, every magnetic dipole inside a ferromagnetic body will tend to align with an effective Heff (given by the sum of all the fields inside the body); and in equilibrium it will be reached a configuration Meq (r) parallel to Heff (r). So, the dynamical evolution of the magnetization can be obtained from the quantum-mechanical expression for a damped precession of the magnetization vector immersed in the effective magnetic field Heff [5]. This leads us to the well-known LLG (Landau-Lifshitz-Gilbert) equation of motion [3, 4]: γα dM = −γ(M × Heff ) + M × (M × Heff ) dt Ms
(1.31)
Where γ is the gyromagnetic ratio given by: γ=
µ0 e me
(= 2.21 × 105 A−1 ms−1 )
(1.32)
And α is an adimensional phenomenological damping parameter (for NiFe α = 0.021). The first term in eq. (1.31) is orthogonal to M and Heff , so it does not change the total energy. This term corresponds to an uniform (undamped ) precession around the Heff vector, and gives us the ferromagnetic ressonance. The second term in eq. (1.31) is the one that forces M to move closer to Heff as time goes by, reducing the total energy, because it is proportional to the orthogonal-to-M component of Heff . If the damping parameter is rather low the magnetization vector precesses many times before it reaches its equilibrium direction - see plots in fig. 1.4. On the other hand, if α = 1 the precession is critically damped and the magnetization turns directly into the direction of the effective field.
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a)
a=0.1 b)
a=1
Figure 1.4: This picture shows how the switching time depends on the value of the damping parameter [9]. (a) Top-view of the precession of vector M for α = 0.1. (b) Critically damped precession for α = 1
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Chapter 2
Numerical Model 2.1
Calculation of Energy and Heff terms
In the first chapter of this thesis it has been seen that inside a ferromagnetic body the magnetization direction remains approximately constant within a distance of less than the exchange length (Lex ), due to the strong Exchange interaction between neighbour atoms. Therefore, the approach used in computer simulations consists in dividing our material in rectangular prisms, with volume δV and edge lenght smaller than Lex , and assuming that the magnetization is practically constant inside these cells. To auxiliate calculations it is usefull to adopt a natural system of units (n.u.) where the magnitude of the magnetization Ms inside each cell is equal to 1 in that system of units. This micromagnetic approach allow us to work on a mesoscopic scale of dimensions (10 nm), which is at least 1003 times less memory and time consuming than if we worked on a scale of the order of the lattice constante (0.1 nm). The micromagnetic program used in this work allows the choice between several geometries for the ferromagnetic samples, however for the simulations performed in this thesis all the ferromagnetic layers have a simple orthorhombic shape (rectangular prism) with volume V = Lx × Ly × Lz - see sketch in fig. 2.1. This sample is divided in a lattice of N = Nx × Ny × Nz identical cells of dimensions δV = δx × δy × δz; with δx, δy and δz ≤ Lex . In the center of each cell we define the discretized magnetization:
Mi,j,k = M(ri,j,k ) with
i, j, k =
1 3 , , ..., 2 2
and
ri,j,k = [i δx, j δy, k δz]
(2.1) L and i = 12 , ..., Lδxx − 21 , j = 12 , ..., δyy − 21 , k = 12 , ..., Lδzz − 12 . As defined, Mi,j,k = Ms mi,j,k where m2 = 1, and Ms (A/m) is a constant of the material being studied.
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X
Y Z
Lz
Ly
Lx
M
Figure 2.1: This is a sketch of the axis labels on the orthorhombic ferromagnet
Of course this micromagnetic approach is not entirely realistic because even within a distance of Lex the magnetization direction is not, in fact, totally constant. Therefore the normalization constraint |m(ri,j,k )| = 1 that we apply isn’t totally physically correct, since it does not account for what is observed in reality. Thus, a future task could be to replace the holonomic constraint: |M| = Ms By an additional energy term of the form: Z C (|M(r)| − Ms )2 dV
(2.2)
(2.3)
That could take into account the error introduced by the constraint, which also makes the dynamic calculations slower. The discretized equations of the several energy terms that were implemented in the computacional routines will be presented in the following subsections.
2.1.1
Numerical Zeeman Energy
In a discrete space the definition of the Zeeman energy (1.8) is given by: X EZ = −µ0 Ha |i,j,k · Mi,j,k δV (2.4) i,j,k= 21 , 23 ,...
If we assume a uniform external Ha field throughout the material, then this can simply be reduced to the summation: X EZ = −µ0 Ms Ha dV mi,j,k (2.5) i,j,k= 12 , 23 ,...
Which is very fast to calculate since the time it takes to go through all the cells in the sum is just proportional to N, the number of lattice cells. 18
2.1.2
Numerical Exchange Energy
After some careful manipulation, the numerical approximation of the integral in eq. 1.13 resumes to [3]:
X
Ex = 2A
i,j,k= 12 , 32 ,...
1 − mi,j,k · mi+1,j,k 1 − mi,j,k · mi,j+1,k 1 − mi,j,k · mi,j,k+1 + + δV δx2 δy 2 δz 2
(2.6) And the exchange field is just the functional derivative of the Ex with respect to mi,j,k 1.29, numerically speaking we have: mi−1,j,k + mi+1,j,k mi,j−1,k + mi,j+1,k mi,j,k−1 + mi,j,k+1 + + δV δx2 δy 2 δz 2 (2.7) As can be seen, the computational time needed to calculate Ex and Hx is also proportional to N, so these terms are not time consuming. Hx |i,j,k = −
2.1.3
2A µ0 Ms
Numerical Anisotropy Energy
Discretizing expression (1.18) leads to: X X Ek = K |mi,j,k × k|2 dV ≡ K [1 − (mi,j,k · k)2 ]δV i,j,k
(2.8)
i,j,k
Where k is a unit vector with the direction of the easy axis. The effective magnetic field for this energy term, given by 1.29, is: Hk |i,j,k = −
2K ∇m Ek |i,j,k = 2K × (mi,j,k × k)δV µ0 Ms
(2.9)
Like the previous two energy terms, the computational time necessary to calculate this one also scales with N.
2.1.4
Numerical Demagnetizing Energy
The Exchange, Zeeman and Anisotropic energies are very fast to calculate. The time consuming term is always the demagnetizing energy. In contrast with the previous terms, in order to calculate the demagnetizing energy it is first necessary to determine the demagnetizing field Hd . Once this field is known the difficult part is over, and one only needs to use it in eq. 1.25 to get the demagnetizing energy with our finite differences schem: X µ0 Hdi,j,k · Mi,j,k δV (2.10) Ed = − 2 1 3 i,j,k= 2 , 2 ,...
However, to know the demagnetizing field Hd one must previously compute the demagnetizing potencial Vd and then calculate its gradient: Hd = −∇Vd . 19
As stated in subsection 1.2.4, potencial Vd is created by two sources: The volume charge density ρ = −∇M and the surface charge density σ = M · n. Using a finite difference scheme one can easily compute the surface charge density in exactly the same way as the volume charge ρ if we extend the magnetization to the outside of the body and set it as zero. Just to give an example, in the one dimensional case the surface charge at the boundary i = j is: σj = mj− 21 = −
mj+ 12 − mj− 21
δx = ρj δx (2.11) δx Where mj+ 12 = 0 because the body ends at site i = j. Thus, the discrete version of Vd , given by eq. (1.24) can be written in the form: Vd (ri ) =
X ρ(ri0 ) |ri − ri0 | 0
(2.12)
i
Where i stands for i, j, k. The biggest problem that emerges from eq. (2.12) comes from the fact that the computational time needed to calculate it is proportional to N 2 , because if we want to know Vd in each of the N cells we must consider the magnetization at all the other N-1 cells. Looking at the equations of the other energy terms, it can be noted that the program just has to solve a triple integral in the discretized space, however, for the Vd it must solve a sextuple integral. This is the main source of delay in micromagnetic simulations, and several studies have been made in order to lower the exponent of N 2 . The exponent of N must be lowered significantly or three dimensional simulations in micromagnetism will be restricted to very small problems. In our program we use a very efficient method that consists in calculating Vd by solving the finite difference Poisson equation directly [6]: (∇2 Vd )i,j,k = −ρi,j,k
(2.13)
2
Where ∇ is the finite difference equivalent of the Laplacian operator. The explicit expression is: Vd |i+1,j,k − 2Vd |i,j,k + Vd |i−1,j,k Vd |i,j+1,k − 2Vd |i,j,k + Vd |i,j−1,k + δx2 δy 2 Vd |i,j,k+1 − 2Vd |i,j,k + Vd |i,j,k−1 = −ρi,j,k δz 2 To solve this partial differential equation an iterative relaxation method is used, but we must first define our boundary conditions since the Poisson equation is a Boundary Value problem. The formal solution (2.12) considers the boundaries at infinity with Vd (∞) = 0. However, what we do is that we divide the sample in a few larger parallelepiped cells and calculate the monopole, dipole and quadrupole values of the charge distribution [7] within these large +
20
cells. Then the Vd at the chosen boundaries is determined by the sum of the multipole terms (truncated after the quadrupole term). We have to consider a volume larger than the sample involving it, in order to place the surface of integration sufficiently far from our material, so that the multipoles expansion has converged. Therefore, we will have to expand the mesh covering the sample, and consider a total expanded volume NV , as depicted in fig. 2.2.
a a b
a
Figure 2.2: Dividing the sample volume in a mesh of large cells of dimensions a and b and calculating the multipole moments at the center. The volume for the calculus of Vd is the volume of the sample expanded by the diameter a of the large cells [3].
The diameter of the multipole cells is the dimension that must be added to the sides of the sample in order to have a good boundary condition approximation. The computer time to calculate the multipoles is proportional to N . The time to obtain the boundary conditions by summing the multipoles is roughly 2/3 proportional to Nmp NV , where Nmp is the number of multipoles considered. Using the multipoles method, the solution of (2.13) is then determined with any of the following iterative relaxation methods available in the program: • Sucessive Overrelaxation (SOR) - A detailed description of this method can be found in [10, 11]. The necessary time to reach a solution is roughly 4/3 proportional to NV . Typically, the number of relaxation iterations varies from 10 to 100, depending on the precision required. • Multigrid Method (MG) - This method is particularly faster for a high resolution mesh, since it finds a solution do the Poisson problem in a time proportional to NV log(NV ) [11, 12].
21
2.2
Algorithms for Energy Minimization
After computing all the energy terms that affect the magnetization of our ferromagnetic body, our goal now is to find a magnetic distribution m(r) for which the total energy E = Ex + Ek + EZ + Ed is a minimum, and so determine the equilibrium configuration. It is also important to emphasize that when we talk about finding an energy minimum we are referring to the local minimum closest to the initial magnetic configuration, because physically that’s what happens. The system is trapped in a local minimum and when the external field is changed the magnetization distribution changes continuously to the adjacent minimum. Otherwise we would not have hysteresis curves! Several methods could be used to reach the minimum of E(m) from an initial configuration. The two methods that are implemented are: Molecular Dynamics with a dissipative component and the Conjugate Gradient Method (CG). The first one is easy to implement [13] and was the method implemented in the early version of the program, but the CG is more efficient and was the method used for all the simulations performed in this work. The Conjugate Gradient algorithm takes the gradient of a function, minimizes the function in the gradient direction, and calculates from this point the gradient and a new (conjugate) direction in which to minimize. Notice that the gradient of the energy (given by eq. (1.27)) is just m × Heff . The routines used are based in the ones presented in [10] where can also be found a complete description of the algorithm.
2.3
Integration methods for the Dynamic LLG equation
The algorithms presented in the previous section are implemented in static simulations whose purpose is just to find an equilibrium magnetic configuration m(r) that satisfies Brown’s static equation (1.30) by minimizing the total energy E. In a dynamical simulation we want to study the time evolution of the magnetic configuration in the transition between the equilibrium states. For that purpose, we will be mainly interested in the effective field Heff that is introduced in the LLG equation (1.31) and must be recalculated at every time step, since it changes with the magnetic distribution m(r). Our program performs the numerical integration of the LLG equation using any of the following methods available [11]: • Euler • 2nd Order Runge Kutta (RK2) • 4th Order Runge Kutta (RK4) • Explicit Gauss-Seidel (EGS)
22
The EGS method was implemented in this work, but it is an explicit integrator like all the other three, and so these methods are not unconditionaly stable for every time step used. For a given mesh resolution there’s a minimum time step required to achieve a stable solution to the laplacian calculation of the exchange energy term. It was confirmed by [14] that the maximum time step required for stability is proportional to l2 , where l is the dimension of the cell side. Thus, the higher is the mesh resolution the more time must the program take to reach a physical solution. The severe time step constraint introduced by the exchange field is one of the main difficulties in dynamic micromagnetic simulations. Using standard explicit integrators leads to a physical time step of pico seconds, which is often two orders of magnitude smaller than the fastest physical time scales. For instance, table 2.1 shows the values of the maximum times steps TM ax required to achieve a stable solution on the dynamic simulation of a 1 × 2µm2 ferromagnetic body 50 ˚ A thick, using a 1 × 50 × 100 grid. Method EGS RK2 RK4
TM ax (pico seconds) 1 5 10
Table 2.1: Approximated maximum time steps required by our program to reach a stable solution To overcome this difficulty an implicit method was developed by Wang, Garc´ıa-Cervera and Weinan [15], the Gauss-Seidel projection method, which is supposed to be unconditionaly stable for every cell size. A good task for a future work on this topic would be to implement and study the stability of this method on our micromagnetic program.
23
Chapter 3
Micromagnetic Simulation of Magnetic Sensor devices The real devices simulated are especially designed for magnetoresistive sensor applications, such as read heads or the ones used in biochips. These biochip sensors are used at INESC-MN for biomolecular detection (DNA, enzymes) using magnetically tagged biomolecules that produce fields of about 2 Oe. These small fields can be detected by a variation of the sensor resistance that is caused by the induced rotation of the magnetization on some of its layers. Since we are dealing with very small fields, it is quite important to simulate with good precision the effects that each field term will have on the magnetic distribution of the sensor ferromagnetic layers. The program enables the recording of greyscale bitmap images (PGM format) or arrow diagrams, at every step of the simulation, that show the magnetic configuration of each cell layer - see fig. 3.1. A pixel in those images corresponds to a cell, and its grey color value is equal to the normalized value of the magnetization on that cell (m(r)). In every simulation the initial state is the vortex configuration, as shown in fig. 3.1. Then we apply the minimization procedures refered in the previous chapter to determine the equilibrium configuration for a given set of conditions [6].
3.1 3.1.1
Basic Spin Valves Pinning field from Antiferromagnets
The difference between an antiferromagnet (AF) and a ferromagnet (F) is that, as explained in subsection 1.2.2, the exchange constant J is negative for AF. So, according to the atomic dipoles model, each magnetic dipole inside an AF will tend to align antiparalelly to it’s nearest neighbour. Obviously, we cannot perform a micromagnetic simulation, on a mesoscopic scale, to study the nanoscopic interactions of these materials. However, here we will only be in-
24
Figure 3.1: This picture shows the vortex configuration for the initial magnetic distribution, and corresponding PGM images built by the program. For each magnetization component (Mx , My , Mz ) we represent the corresponding normalized value at each cell site using a greyscale. The greyscale color values are shown at the bottom.
terested in the effect of the exchange between a F and an AF layer coupled together (AF/F) [16]. Meiklejohn and Bean discovered in 1956 [17] that a ferromagnet, in contact with an antiferromagnet, exhibits an unidirectional exchange anisotropy, manifested in a magnetization versus field loop that is shifted away from zero applied field. This extra magnetic field, created in the ferromagnetic dipoles due to this coupling, can be modelled as an additional Zeeman field, having the direction of the alignment of the dipoles in the AF, and with an intensity that is equal to the value of the hysteresis loop shift caused by the coupling.
25
Thus, the AF/F coupling can be easily simulated in our micromagnetic program by introducing an extra energy term, called the Pinning Energy. This is an energy term completely similar to the Zeeman one (1.8), but instead of the external applied field Ha we introduce the value of the pinning field Hpin . For a given AF/F coupling the intensity of this field can only be known from an experimental measurement of the hysteresis loop shift in a bulk sample. This shift from zero Ha is usually called the Exchange or Offset field. So, although not being possible to simulate directly an AF layer, the effect that this layer has in on a F layer can be taken into account. In Spin Valves, this effect is used to pin the magnetization direction of one of the ferromagnetic layers by exchange coupling it to an antiferromagnet - see fig. 3.2. In that case the ferromagnet is usually called a Pinned layer.
Figure 3.2: The illustrations show the structure and the usual materials used in a simple Magnetoresistive Spin Valve. The F1 magnetization is pinned because it is coupled to an AF. Since this AF/F coupling doesn’t exist in F2 , its magnetization can rotate more or less freely. The metallic Cu layer is called the spacer.
3.1.2
Interlayer Coupling
Looking at the squematic of a basic Spin Valve structure (F1 /N M/F2 ) in fig. 3.2, we realize that the F2 layer is not coupled to an AF like the F1 pinned layer. Since there is no pinning field in ferromagnet F2 it is usually called a Free layer. Nevertheless, F2 magnetization will not be able to rotate entirely freely because there is a coupling field, through the Cu layer, from the pinned to the free layer. If this coupling is not too strong, the free layer magnetization will be free to rotate in response to an external field, while the pinned layer magnetization is “fixed” by the exchange coupling to the AF adjacent to it.
26
The coupling strength observed between the pinned and free layers is the cumulative effect of a number of mechanisms. The most important ones are the N´eel-type topological coupling [18] and the indirect oscillatory exchange interaction (usually called the RKKY interaction - Ruderman-Kittel-KasuyaYodsida) [20]. In the case of a Cu spacer the dominating term will be the N´eel coupling, which decreases exponentially with increasing Cu thickness (tCu ) [16]. Concerning the RKKY term, it will originate an oscillatory coupling constant (Ac ) dependence on tCu , according to the function plotted in fig. 3.3. For a Cu spacer this term will only cause some minor fluctuations on the interlayer coupling constant given by the dominating N´eel term.
Figure 3.3: The graphic is a plot of the RKKY exchange coupling dependence on spacer thickness. The maximums correspond to ferromagnetic coupling (parallel alignment) between free and pinned layers, and the minimums correspond to antiferromagnetic coupling (anti-parallel alignment). Courtesy of Susana Freitas. So, since the interlayer coupling field diminishes as the spacer thickness increases, for sufficiently high tCu the free layer would be completely uncoupled to the pinned layer and would then rotate absolutely freely. In that case, the transfer curve of the free layer would be centered at Ha = 0, meaning zero offset field. However, as we will see in the next subsection, we can only have a good response in our Spin Valve for very small Cu thicknesses. The range of Cu thicknesses most commonly used in Spin Valves is 20-25 ˚ A, and for tCu = 22 ˚ A the exchange coupling constant is Ac = 2 × 10−3 erg/cm2 [21]. This interlayer coupling is implemented in the Exchange Energy calculations
27
of our program simply by changing the constant A → Ac on the interaction between the top cells of the pinned layer and the bottom cells of the free layer (in the direction normal to the layers plane - ~x).
3.1.3
Magnetoresistance and Spin Valve operation
Magnetoresistance (MR) is the effect by which the electrical resistance of a magnetic material changes depending on the relative direction of the current and the magnetization. The level of magnetoresistance shown by a material is usually expressed in terms of the percentage change in resistance from the highest to the lowest resistance, and is usually of the order of a few percent. The particular magnetoresistive properties of spin valves are known by Giant Magnetoresistive (GMR) effects. A complete description of the giant magnetoresistive properties of Spin Valves is given in [16], so it will only be made here a very brief note on this subject. If a current is applied in the Spin Valve structure depicted in fig. 3.2, flowing in the direction of the layers plane (see fig. 3.4), this current will pass mainly through the Cu layer since this is the material with lowest resistivity. However, the total resistance that this current will “feel” will not only be due to the scattering effects inside the Cu layer, but also due to the scattering of the electrons on the interfaces between the two ferromagnetic layers adjacent to it.
Figure 3.4: The picture is a microscope image of a Spin Valve sensor. The contact lines make the current flow through the sensor in the direction of the layers plane (current in plane geometry - CIP). The tapered magnetic field generating lines (MFG) are used to atract magnetically tagged nanoparticles. Courtesy of Hugo Ferreira. The main thing to bear in mind is that the value of the resistance due to the interface scattering will vary according to the relative orientation of the magnetizations of the two ferromagnets. If their magnetizations are parallel 28
we will have a mininum in resistance (low-resistance state), on the other hand if they are anti-parallel we will measure a high-resistance state. The reason has to do with differences in the density of states at the Fermi level [19] which cause a spin-dependent scattering of the current electrons, resulting in distinct mean free paths for spin-up and spin-down electrons. Mean free path is the average distance an electron will travel between scattering events - the longer it is the lower the resistance. It is possible for electrons of appropriate spin to pass through many aligned magnetic layers and have a very long mean free path. This means that the distance between scatters is increased the most when the layers become magnetically parallel. So, the magnetoresistance ratio of a spin-valve is defined by the difference in resistance between the high and low-resistance states divided by the minimum resistance: M R = (Rhigh − Rlow )/Rlow . It is important to refer that we can only accuratelly measure the spin valve MR for a suficiently low Cu thickness, so that the resistance due to the interface scattering may be of the same order of the resistance inside the Cu layer. Hence, the MR of a spin valve is crucially dependent on the Cu thickness, and a compromise must be assumed because the lower is tCu the higher is our MR, but also the greater will be the coupling field between the pinned and free layers, as referred in the previous subsection. For a certain equilibrium configuration of both pinned and free average magnetization vectors, making angles θpin and θf ree with the ~z axis, the response of the spin valve sensor is given by [21]: 1 ∆R W < cos(θpin − θf ree ) > (3.1) RS 2 R L Where W is the sensor width and L is the length, as in fig. 3.2. The angles are averaged over the thicknesses of the free and pinned layers, and <> represents an average over all the active area of the sensor. RS is the sensor square resistance. If we set a pinning direction along the ~y axis, for small applied field norms (bellow Hpin ) the pinned layer magnetization will always remain aligned with this direction (θpin = 90o ). However, the free layer will be able to align with an external field; for instance if a magnetically tagged nanoparticle reaches the top of our sensor it’s magnetic field changes θf ree and that is measured by a difference in ∆R. ∆R = −
3.1.4
Simulation of a Basic Spin Valve
Without Demagnetizing Energy In order to illustrate the points referred in the previous sections I will now show some results of simple Spin Valve simulations. The main purpose of this study is to understand what is the role of each energy term on the magnetic switching
29
mechanism of these structures. So, it’s a good idea to analyse the effects of each field one at a time. First of all, let’s start without considering the demagnetizing field: Hd = 0. If we neglect the magnetization component perpendicular to the layer plane (Mx ), this is similar to what we would measure for a bulk spin valve. In a bulk sample the borders are too far away, and so the magnetic charges formed in those distant limits will create a demagnetizing potencial Vd that will have almost no effect on the overral magnetization. Hence it can be assumed that there is no interlayer magnetostatic interaction between the different layers. In that situation, if we apply an external magnetic field along the short axis (y direction) of the 1 × 5 µm2 structure shown in fig. 3.5, with intensity varying from -400 to 300 Oe, the magnetization hysteresis loops obtained are the ones plotted in fig. 3.6.
Figure 3.5: This is a squematic picture of the 1 × 5 µm2 Spin Valve structure simulated by our 3D program. The easy-direction was set along the ~y axis, parallel to the applied field, to both pinned and free layer.
From this hysteresis curves there’s a lot of information that can be extracted. For the sake of simplicity, it will be defined as the state -1 when the magnetization of the layer is aligned with the negative direction of ~y , and the state +1 when it is aligned with the positive direction of ~y - as depicted in fig. 3.6.(b). The transitions between the states are very sharp because it is not taken into account the Hd , which has the effect of smoothing the slope of the transitions. Starting with the free layer (fig. 3.6.(b)), it can be seen that the curve is slightly shifted to the left. The shifted value is called the offset field H0 . The value of the H0 is equal, in this case of no Hd , to the symmetric of the coupling field −Hcoup , which is the exchange field that exits between the Pinned and Free layer through the Cu spacer [22]. The coupling field acting on the free layer of the structure can be calculated as follows [16]: Hcoup =
4Ac mpin µ0 Mf tf
30
[SI units]
(3.2)
Figure 3.6: The graphics show the simulation results of structure depicted in fig. 3.5 for Hd = 0. The values of the magnetization are always normalized by the Ms of the bottom layer (Pinned). (a) Transfer curve of the total magnetization M = Mf + Mp . (b) Hysteresis loop of the free layer Mf only. (c) Hysteresis loop for the pinned layer Mp
Resulting in Hcoup = 20 · mpin Oe. Where mpin = Mpin /Mpin and Ac = 2 × 10−3 erg/cm2 , which is the value typically used for a Cu thickness of 22 ˚ A. For the pinned layer (fig. 3.6.(c)) the hysteresis curve is also shifted, but of an higher value of 200 Oe which is equal to the pinning field that was introduced on our program (based in the experimental results) for this AF/F coupling. As can be seen, the Pinned layer will always have a fixed orientation in the direction of +~y if Ha > −300 Oe. Another aspect that should be referred concerns the coercivity field Hc . Since the external field is being applied in the direction of the anisotropy axis K of the layers and there is no demagnetizing field, the coercivity Hc will simply be equal to the anisotropic field Hk inside each layer. It is notourious that the pinned layer has a lot of coercivity (Hc = 100 Oe) when compared with the free layer (Hc = 5 Oe). This is because it has been set an anisotropic constant Kpin = 4 × 104 erg/cm3 , which is 20 times greater than the Kf ree = 2 × 103 erg/cm3 . A simple way to calculate the anisotropic field Hk inside a layer, with a saturation magnetization Ms and an uniaxial anisotry K, is using: Hk =
2K µ0 Ms
(3.3)
It is easy to understand the origin of this expression. Since K is, by definition, the energy density necessary to supply by the external field to align the magnetization of a sample from an initial random state (M=0) to the direction of the anisotropy axis (M = Ms k).
31
With Demagnetizing Energy Now let’s simulate the patterned 1 × 5 µm2 sample of fig. 3.5 with the self and interlayer demagnetizing field - see fig. 3.7.
Figure 3.7: The graphics show the simulation results of the patterned 1 × 5 µm2 structure depicted in 3.5. The materials constants used in this simulation are the same as the one before. (a) Transfer curve of the total magnetization M = Mf + Mp . (b) Hysteresis loop of the free layer Mf only. The offset field is H0 ≈ 30 Oe. (c) Hysteresis loop for the pinned layer Mp . The Hd favours the -1 → +1 switch but not the +1 → -1, that’s why this last transition seems to remain sharp.
As can be seen, the unphysical sharp transitions have disappeared and the slope of the switching of the free layer is notoriously reduced when compared with the one for Hd = 0. This is an advantage for our magnetic sensor because this way it’s easier to measure a variation of magnetoresistance (given by 3.1) for fields in the interval −2 < Ha < 2 Oe, which are the typicall fields produced by the magnetically tagged nanoparticles that must be detected. Notice that, for a sensor, one of the most important things is to control the slope of the free layer transition, and for that we must work with its geometry. The aspectio ratio (k ) of a given sample is defined as the ratio between length and width k = L/W , or k = Lz /Ly using the notation introduced in chapter 2. The higher is our aspect ratio the more “rectangular” is our sample and the lower will be the slope of the transition. That’s because for high aspect ratios the average demagnetizing field will be higher along the long axis (~z direction), so the magnetic dipoles will have more difficulty in switching to directions perpendicular to ~z. The switching field HS of a ferromagnetic sample is defined as represented in fig. 3.8 [22], and it is given by the difference of the average demagnetizing field inside the sample and its anisotropy field: HS =< Hd > −Hk [16]. In fact, for a fixed length (Lz ) and varying width (Ly ) the value of HS decreases almost exponentially with increasing Ly . This is because our aspect 32
M -Hk
-Hs +Hs
Ha
-Hk
Figure 3.8: This figure shows how the switching field Hs is defined. Hs is the field at the point of intersection between the dashed line tangent to the hysteresis curve at M=0 and the horizontal line at ±Ms .
ratio is decreasing, and thus the lower becomes < Hd >. In what concerns the offset field H0 of the free layer, attending to fig. 3.7.(b) it has a value of H0 ≈ 30 Oe. In a patterned sample (interlayer Hd 6= 0) this field can be calculated as follows: H0 = −Hcoup + Hdpin
(3.4)
Where Hcoup is the exchange field through the Cu spacer, which is the same as the one calculated in (3.2), and Hdpin is the demagnetizing field coming from the fixed magnetization of the Pinned layer. As it was explained in the theoretical introduction, the exchange and the demagnetizing field have completely different effects on the magnetization. The exchange field will prefer the magnetization of the free layer to remain parallel to the one of the pinned layer, in the -1 state. On the other hand, the demagnetizing field will want these layers to become antiparalelly oriented in order to reduce the total magnetization, and thereby the magnetic charges created by it. So, the Hd will force the transition to the +1 state, whereas the Hcoup will force it the other way. Thus, it is the balance between these two fields that will give us the value of the shift of the transfer curve, calculated by eq. (3.4). It was already determined, in (3.2), that Hcoup = 20 Oe, so Hdpin = H0 + Hcoup ≈ 50 Oe. This Hdpin field is proportional to the net magnetization of the pinned layer, and the constant of proportionality is a certain demagnetizing factor, here denoted generally by N : Hpin = N · Mpin d eff 33
(3.5)
It is very difficult to find a simple expression able to calculate analitically (as it was done with Hcoup in (3.2)) this demagnetizing factor N with considerable precision. Some approximations can be used [23], however they seldom give us an accurate value. But since we need to be able to predict the offset fields with the least error possible the best way to determine Hdpin is with a 3D micromagnetic simulation like this one, and using a procedure that will be described in subsection 3.4.2. Now taking a look at the pinned layer on fig. 3.7.(c), it can also be seen that its coercivity field Hc was reduced in comparasion with the case of no demagnetizing field. This happens because the +1 → -1 state transition of the pinned layer is favoured by the Hd but the -1 → +1 switching is not. So, comparing graphic 3.7.(c) with graphic 3.6.(c) we realise that the +1 → -1 transition is exactly in the same place (at Ha = Hpin + Hk ), however the right side switching -1 → +1 was shifted (and smoothed) because it was supported by the effect of the Hd . Therefore the coercivity was reduced on this right side, and consequently the offset H0 of the pinned layer increased due to that. To support these arguments it is a good idea to analyse the graphic of the total demagnetizing energy Ed (Ha ) of the structure, plotted in fig. 3.9.
0,006
0,005
0,004
0,003
0,002
0,001
0,000 -400
-300
-200
-100
0
100
200
Figure 3.9: The figure shows the total Ed curve as a function of the applied field. The Ed is minimum when the total magnetization of the Spin Valve is also minimum. The minimum of this function corresponds exactly to the configuration where both layer are antiparallel, minimizing the total magnetic charges formed on the borders of the structure. So the demagnetizing fields “prefer” the spin valve to 34
remain in that configuration. The quantity of interest here is, of course, the magnetoresistance observed, which is proportional to < cos(θpin − θf ree ) > (see eq. (3.1)). For this sample this function is plotted in fig. 3.10.
1,0
0,5
0,0
-0,5
-1,0
-400
-300
-200
-100
0
100
200
Figure 3.10: The graphic is a plot of the average quantity < cos(θpin − θf ree ) >, which should be proportional to the measured MR, in function of the applied field. θ is the angle that the magnetization vector makes with the ~z axis. The main objective is to have the maximum ∆R variation for Ha near 0 Oe (within [-2;2] Oe) to detect the small fields coming from the magnetically tagged nanoparticles that reach the top of the spin valve. For that purpose the ideal is to have the transition of the free layer centered in Ha = 0 and with a high slope. So, attending to fig. 3.10, we must shift the curve H0 Oe to the left. Fig. 3.11 is an arrow diagram of the free and pinned layers equilibrium magnetic configurations, for no applied field. It can be seen that Mp is completely aligned with the pinning field, as would be expected. However, the offset field acting on the free layer forces Mf to remain almost aligned with −~y , except for a small region near the left edge where the demagnetizing field from the surface magnetic charges originates a domain wall (vortex). If we now apply an external field Ha on the +~y direction this vortex will be shifted to the right side, increasing the small domain on the left parallel to Ha and decreasing the big domain on the right pointing downwards.
35
Eliminating the offset field can be easily accomplished by applying a bias current, following in the same direction of the long axis (~z) of the spin valve, able to produce a magnetic field Hj on the free layer that shifts its hysteresis curve to the left on the amount required (≈ 30 Oe) [21]. In our micromagnetic program we can simulate the effect of this biasing Hj (x, y), simply by performing a tridimensional integration of the Biot-Savart law to calculate the field at every cell site created by a current density Je distributed uniformly throughout the sample.
Free Layer H0
Ly=1 mm
Hk
Ha = 0
y z
Pinned Layer Ly=1 mm
Hpin
Lz=5 mm
Figure 3.11: The pictures show a representation of the magnetic distributions of both Spin Valve ferromagnets. At zero applied field Mf is still almost aligned with −~y , as plotted in fig. 3.7.(b), due to the offset field H0 effect. In an ideal case, if there were no offset field the vortex on the left would be at the center of the free layer, resulting in Mf = 0 (zero shift). The pinned layer magnetization remains always like this for Ha approximately higher than -300 Oe. A better idea would be to lower the effective magnetization of the pinned pin layer Mpin ≈ 50 Oe making it equal to eff , so that we could reduce the value of Hd Hcoup = 20 Oe (by eq. (3.4)), compensating completely the coupling field. This can be accomplished by reducing the pinned layer thickness or, even better, by using a synthetic antiferromagnetic pinned layer (SAF). This type of structure will be studied in the next section.
3.2
SAFs for Pinned Layers
Synthetic antiferromagnetic coupled layers have recently been object of considerable study because they can be used either as better pinned layers or as better free layers for our devices [25]. An alternative pinning mechanism to the traditional AF/F pair in spinvalves is obtained by replacing this F layer by a AF/F1 /Ru/F2 structure, called SAF. For a Ru (or Re) spacer the interlayer exchange coupling, between the two 36
F layers, will be quite different from the one previously studied (in subsection 3.1.2) for a Cu spacer. In the case of Ru the N´eel-type coupling is very small and the dominating term is the RKKY oscillatory exchange interaction, so the coupling constant Ac dependence on tRu will be mainly given by the function plotted in fig. 3.3. Furthermore, in these SAF structures it is used a thickness of Ru (or Re) of 8-10 ˚ A, so there will be a very strong antiferromagnetic coupling between F1 and F2 since, for this very small thicknesses, we will be placed in the first minimum of the RKKY coupling constant curve Ac (t). For tRu = 8 ˚ A this value was measured to be Ac = −1.3 erg/cm2 . In the applied field range of interest (Ha ∈ [-300;300] Oe) the two F layers always remain anti-parallel, so we can define an effective moment of this SAF that is given by the diference of the magnetizations of F1 and F2 weighted by their thicknesses: Mpin eff =
M1 t1 − M2 t2 t1 + t2
(3.6)
This solution has several advantages compared to the traditional mechanism [16, 25]: • There is a reduced magnetostatic coupling between free and pinned layers, coming from the descreased net moment of the SAF structure (Mpin eff ) • The pinning fields Hpin of this structure are almost one order of magnitude larger (about 1000-3000 Oe) than the ones of the simple AF/F pinned layer • It shows a slow decrease of pinning field with increasing temperature
3.2.1
Simulation Results for a SAF structure
A 1 × 3 µm2 SAF sample was simulated with the structure shown in fig. 3.12. Generally for the structures involving SAFs it is quite important to consider on the simulations the slight thickness differences between the layers, since that difference is the sole responsible for the existance of a Mpin eff . However, it is quite difficult for the program to have that kind of precision in the cell sizes. For instance, in this simple case, to simulate rigorously the 10 ˚ A difference between the 2 CoFe layers we would need to consider 5 discretization points in the thickness direction (nmx = 5), in order to have a cell thickness of 10 ˚ A. Then the three bottom cell layers (in the ~x direction) would be assigned to F1 material, and the two top cell layers would go to F2 . The spacer is simulated just by introducing the corresponding coupling constant Ac between the F1 and F2 cells. In this case it would take about 3 days to simulate the complete hysteresis curve of this structure, since we are using a fine mesh. Three days may not be too much, nevertheless in the future much more complicated structures will have to be simulated, with even slighter thickness differences between the layers, and 37
Figure 3.12: The illustration is a squematic of the 1 × 3 µm2 SAF structure simulated by our 3D program. The values chosen for Ms (CoF e) and Ac (Ru) were taken from experimental measurements performed by Andr´e Guedes.
these would require a very long time to be simulated properly in this way. So, a rough alternative is to implement in the program always the same thickness for each layer and use compensated values for the corresponding Ms (weighted with the real thicknesses). For example, in this case we could use a mesh with nmx = 2 (dx = 20 ˚ A) assigning F1 to the bottom cell layer and F2 to the top one. Then the fact that the F1 layer has an higher thickness is taken into account just by setting Ms (F1 ) = 1360 × 30/20 = 1700, and Ms (F2 ) = 1360 remains unchanged. The use of compensated Ms is ok for the calculation of the demagnetizing energy but introduces a certain error in the determination of the correct values of the exchange energy. Using this alternative the SAF hysteresis loop is ready within 1 day, but it must be taken into consideration that the exchange coupling fields between the layers will present values a little higher than the correct ones. Nevertheless, it has been seen that for this analysis that doesn’t cause relevant discrepancies. The total hysteresis loop for the SAF depicted in fig. 3.12 is shown in fig. 3.13. Since the coupling between the 2 layers is so antiferromagnetically strong it is extremelly difficult to make them orientate parallel to each other, that can only be possible for applied fields of the order of the exchange fields Hex that are created between adjacent cells of the same material. The coupling field that F2 feels from the F1 layer is given by: F1 Hcoup =−
4Ac (Ru) = 1.91 × 104 Oe µ0 Ms (F2 )t2
(3.7)
In the same way we can calculate the coupling field that F1 feels from the F2 , replacing the corresponding terms in (3.7). In the plot of fig. 3.13 the 38
0,4 0,3 0,2 0,1 0,0 -9000
-6000
-3000
0
3000
6000
9000
-0,1 -0,2 -0,3 -0,4 -0,5
Figure 3.13: The graphic shows the total magnetization M = M1 + M2 as a function of the applied field Ha , for an unbalanced SAF. The angle between M1 and M2 only starts to decrease from 180o for applied fields above 3000 Oe or bellow -1200 Oe.
magnetization was not calculated for such high fields because the region of interest here is just the plateau between -1200 and 3000 Oe, meaning that the magnetization of both SAF layers remains fixed (pinned) for such high fields. That plateau of the SAF will be at a magnetization equal to the average of SAF the Ms of both layers, weighted by their thickenesses (Meff given by (3.6)) as can be checked in fig. 3.13. ˚, If a totally balanced SAF is used, with equal layer thicknesses t1 = t2 = 20 A SAF = 0. In this case the plateau will be at M=0, as sketched in fig. then Meff 3.14, and it also becomes more centered between -2000 and 1500 Oe. In this situation the pinned SAF has zero net moment, so it would create no magnetostatic coupling on a free layer placed on top of it.
3.2.2
SAF pinned layers in Spin Valves
The basic spin valve structure studied before can now be enhanced replacing the ferromagnetic pinned layer by a SAF structure, as in fig. 3.15. The hysteresis loop for the patterned 1 × 3 µm2 structure shown in fig. 3.15 is the one represented in fig. 3.16.(a). As can be seen in 3.16.(a) the plateau becomes divided at Ha = 0 when the free layer rotates to the +1 state. Then for Ha > 3000 Oe it’s the top layer of the SAF that starts to rotate, as sketched with the arrows. 39
0,4
0,2
0,0 -7500
-5000
-2500
0
2500
5000
7500
-0,2
-0,4
Figure 3.14: The graphic shows the total magnetization M = M1 + M2 as a function of Ha , for a totally balanced SAF. The angle between M1 and M2 only starts to decrease from 180o for applied fields above 1500 Oe or bellow -2000 Oe.
Figure 3.15: This is a squematic picture of the 1 × 3 µm2 SAF+F spin valve simulated.
From the Ed (Ha ) graph, in 3.16.(b), we realise that the mininum plateu is in the configuration where all 3 layers are anti-paralell to each other (as can be seen by the squematic arrows). Then, when the free layer rotates there is another plateu of higher Ed since the free layer has become parallel to the bottom layer of the SAF, increasing the overral magnetization of the structure.
40
a)
b)
|M| minimum
Figure 3.16: The figure shows the simulation results for the patterned 1 × 3 µm2 structure shown in fig. 3.15. (a) Transfer curve of the total magnetization component parallel to the applied field. (b) Graphic of the overral demagnetizing energy for each hysteresis step.
3.3
Synthetic Free structure (SF)
A synthetic free structure (SF) is just like a SAF but none of its ferromagnetic layers is exchange coupled with an antiferromagnetic pinning layer (F1 /Ru/F2 ). Consequently, the F layers will rotate freely but antiferromagnetically coupled to each other. As calculated in the previous section (3.7), for a Ru thickness of 8 ˚ A there’s an exchange field between these two layers of the order of 1 × 104 Oe. This is exactly the same order of magnitude of the Hex between two adjacent cells on a single ferromagnetic layer. Meaning that the coupling between the dipoles of the upper F2 layer and the bottom F1 layer is as strong as if they were on the same material, but, with the crucial difference that it’s a negative coupling (antiferromagnetic) due to the negative value of the exchange constant Ac (Ru). So, i.e., when the magnetization of a certain cell on F1 rotates θ degrees, the magnetization of the immediately upper cell on F2 also rotates θ degrees on the same direction but always pointing to the opposite side. Because of this tight antiferromagnetic coupling the behavior of a SF structure can be modelled as if it was a single free layer with the thickness of F1 plus F2 but an effective magnetization given by: MSF eff =
M1 t1 − M2 t2 t1 + t 2
(3.8)
SAF Which is equal to the expression of Mef f given by (3.6).
As explained in subsection 3.1.4, the switching field of a certain free layer, when 41
the external field is applied along its easy-axis, is given by: HS =< Hd > −Hk - see fig. 3.8. Where < Hd > is the average self-demagnetizing field felt by the free layer dipoles. In order to reduce this switching field and increase the sensitivity(1 ) of our sensors it is convenient that the < Hd > is as low as possible. The advantages of using an SF, instead of a single free layer, are mainly related to the fact that this structure allows the magnetic flux to close creating less demagnetizing field, which reduces the magnetostatic coupling with close neighbouring layers [24, 25, 27]. The strong torque created by the antiferromagnetic coupling on the magnetizations of the two ferromagnetic layers of the SF favours a coherent rotation of the magnetization in detriment of a mechanism based on nucleation/propagation of walls in the SF layers. Therefore, it is expected that SFs facilitate to form single domain structure even in the lower aspect ratio (k = L/W ), resulting in reduced switching field Hs . A large physical free layer thickness can therefore be maintained (tSF = t1 + tRu + t2 ) while decreasing its effective magnetic thickness, defined as [16]: teff =
M 1 t1 − M 2 t2 Ms (N iF e)
(3.10)
The greatest difficulty in the implementation of this alternative is due to the high exchange coupling Hcoup between the SF structure and the pinned layer, which is much higher than for a single free layer. Since both F layers are highly antiferromagnetically coupled the exchange field between the SF and the pinned layer (given by 3.2) will be inversely proportional not to the magnetization of the bottom F1 layer but to the effective magnetization of all the SF. That’s why larger offset values (H0 - given by (3.4)) are measured for these structures.
1 The sensitivity is defined as the steepness of the resistance (R) versus applied field (Ha) curve [22]: 1 ∂R S(Hop ) = (3.9) R ∂Ha Ha =Hop Where Hop is the field of operation, i. e., the static field around which the applied field is varied.
42
3.4
SAF-SF Spin Valve structures
The combined advantages of the SAFs for pinned layers and the SFs for free layers motivate us to think of a better spin valve structure: the SAF+SF. Although this structures have several advantages there are a couple of difficulties in their implementation, because experimentally it was found that they have high offset field H0 . Anabela Veloso [16] studied the dependence of H0 with the effective thickness teff of the SF (given by (3.10)), and it was seen that the offset field measured varied with 1/teff . So, the more compensated is the SF the higher is H0 . This behaviour seemed strange at first sight, because it was initially thought that the exchange coupling field between the pinned SAF and the free SF was only related with both layers adjacent to the Cu spacer. However it is not like so, and the layers that are antiferromagnetically coupled with these two will also influence the switching offset. Micromagnetic simulations were performed to explain this behaviour and confirm the experimental results obtained by Andr´e Guedes, who built several SAF+SF structures like the one shown in fig. 3.17.
Figure 3.17: This is a squematic picture of a SAF+SF with teff = 15 ˚ A. The F3 is constituted by two layers of CoFe and NiFe but, to simplify the simulations, it can be considered as a single layer with a total thickness equal to the sum of the thicknesses and an effective magnetivation given by the average (weighted by the thicknesses) of Ms (N iF e) and Ms (CoF e). Note that in fig. 3.17 the SAF is unbalanced but on the opposite direction of the one studied before in fig. 3.12, since now it’s the F2 that has an higher thickness than the F1 . The maximum stability of these spin valve structures 43
with respect to arbitrarily positive or negative perturbation fields is obtained when the pinned F1 layer is slightly thicker than the F2 layer. But this is at the expense of the MR signal, so we prefer an F1 layer that is slightly thicker than the F2 , with the resulting net moment of the SAF preventing the magnetization flop at small fields [16]. Fig. 3.18.(a) shows the total magnetization hysteresis loop of the SAF+SF of fig. 3.17 for a patterned 1 × 3 µm2 sample.
a)
b)
Figure 3.18: The graphics show the simulation results for the SAF+SF with teff = 15 ˚ A. (a) Total magnetization in function of Ha , the orientation of the layers is represented by arrows. (b) Total Ed plot. Looking at the arrows squetched in fig. 3.18.(a), it can be seen that the two layers of the SF always switch antiparalelly, that’s why the minimum plateau of the Ed is now at Ha > 0, contrary to what happenned in fig. 3.16.(b). Because this is the interval where the total magnetization of the structure is minimum. As before, the region of interest here is the one closer to the switching of the SF, so let’s take a closer at the hysteresis curve of the free F3 layer for −150 < Ha < 150 Oe and compare it with the experimentally measured one see fig. 3.19. For this range of applied field it was chosen to plot the simulated hysteresis loop of the F3 magnetization, since it’s the orientation of this layer the one that will cause the variation of magnetoresistance according to formula (3.1). The curves of the experimental and simulated samples have very similar values of coercivity and sensitivity. However the H0 simulated value is superior in 13 Oe to the experimental one, probably due to our lack of knowledge of the exact values of: • Ac (Cu) - The exchange coupling constant varies with temperature, surface roughness and Ms of adjacent layers. The value used for this study is the 44
a)
b)
Figure 3.19: The picture compares the simulated and experimental results for the structure depicted in fig. 3.17. (a) Simulated hysteresis loop of the F3 magnetization component (see fig. 3.17) parallel to the applied field - H0 = 38 Oe. (b) Experimental measurement of the magnetoresistance of the same structure - H0 = 25 Oe.
typical one of Ac (tCu = 22 ˚ A) = 2 × 10−3 erg/cm2 [21], however the real value could be different. • Ms - It has been used compensated values for the materials Ms (as explained in subsection 3.2.1) to simulate the slight thickness differences between the layers. This may increase a little the values of the exchange coupling between layers, and thus the offset field. To study what exactly causes the offset field it must be taken into account the combined effects of 3 fields: Hd , Hcoup (or interlayer He x) and Hk , because they have more or less the same order of magnitude when the transition takes place. In order to see what’s the role of each one of these three fields in the switching mechanism of the SF layers we shall study them separately in the following subsections.
3.4.1
Effect of Hcoup on Offset
In this subsection simulations will now be performed just using the Exchange Energy, eliminating Hd and Hk from the calculations. The resulting curves are similar to the ones that would be obtained for bulk samples without anisotropy. Before simulating the complete structure in fig. 3.17 it’s better to first simulate just the three bottom layers, the SAF plus the F3 free layer (see fig. 3.20), excluding the top F4 layer.
45
Figure 3.20: This is a squematic picture of the SAF+F structure simulated. It was introduced Ms (F3 ) = 1080 emu/cm3 which is the average between Ms (N iF e) and Ms (CoF e) since these materials have the same thickness in F3 - see fig. 3.17
Neglecting the demagnetizing and anisotropic fields, the offset H0 that should be observed on the transfer curve of F3 magnetization will simply be equal to the symmetric of the exchange coupling field through the Cu spacer. Using expression (3.2), with Mf = 1080 emu/cm3 , the symmetric of the coupling field is −Hcoup = H0 = 24.7 Oe. Which is the value that was indeed measured for the shift of the free layer transition in fig. 3.21.(a), appart from a certain precision error. Of course that the offset value measured from the simulated hysteresis curve is never 100% accurate because the plotted points of Mf y (Ha ) are separated from about 5 Oe in Ha , so there’s always an uncertainty of that order in the determination of the simulated H0 . Now placing the upper layers of Ru/NiFe on top of this SAF+F structure we get the SAF+SF previously shown in fig. 3.17. Without demagnetizing nor anisotropic fields the SF switching observed for this spin valve is the one plotted in fig. 3.21.(b). As can be seen, by placing F4 layer antiferromagnetically coupled with our free layer the offset H0 increases a lot, from 26 Oe to 65 Oe! This increase can only be caused by a difference in the Hcoup since we are running a simulation considering only 2 fields: the Hx and HZ . As explained in subsection 3.3, when using a SF structure instead of a single Free layer one must consider the effective moment, equal to the Ms difference of SF the two SF layers weighted by their thickenesses (Meff given by (3.8)). So, in the 46
a)
b)
Figure 3.21: The figure compares the offset field values of the SAF+F and SAF+SF structures. (a) SAF+F - Graphic of F3 switching for Hd = Hk = 0; the transition takes place at H0 = 26 Oe. (b) SAF+SF with teff = 15 ˚ A - Graphic of F3 switching for Hd = Hk = 0; the transition takes place at H0 = 65 Oe
calculation of the Hcoup , by exp. (3.2), instead of using the Ms of the free layer SF it must be used the Meff and the total thickness of both layers (tSF = t3 + t4 ). Hence, a general expression for the coupling field on a SF free layer is: Hcoup (SF ) =
4Ac µ0 (Mb tb − Mt tt )
(3.11)
Where the subscript b corresponds to the bottom layer of the SF (F3 ) and t to the top one (F4 ). Applying this formula (3.11) to the structure given in this example (fig. 3.17), we obtain Hcoup = −64.6 Oe, which justifies the offset value H0 = 65 Oe on the simulation of fig. 3.21.(b). The important thing to remember here is that a SF structure behaves as a SF single free layer with Ms = Meff . SF For the SAF+SF sample studied in this example with teff = 15 ˚ A the Meff = 413 Oe. Thereby, to test the assumptions made, a SAF+Free structure was simulated, like the one in fig. 3.20, with Ms (F3 ) = 413 Oe. In principle if formula (3.11) is correct, using a free layer with moment equal to the effective moment of the SF should give the same offset value H0 on the hysteresis curve (without considering Hd and Hk ). And it does. The H0 of this SAF+F structure is 65 Oe, equal to the H0 of the SAF+SF as expected. Expression (3.11) is also in accordance with the results of Anabela Veloso thesis. According to her experimental results she confirms that the offset field
47
varies with 1/teff . This is the main reason why the observed H0 of the SAF+SF structure is much higher than the one of the simple Spin Valves.
3.4.2
Effect of Hd on Offset
In this subsection it will be added the demagnetizing field effect on the simulation of the SAF+SF structure with teff = 15 Oe (depicted in fig. 3.17, but still ignoring the anisotropy on the calculations (no coercivity). For external fields in the range of interest for sensor operation, the pinned SAF layers remain with their magnetic moments fixed during the switching of the SF. In the present case, the bottom F1 layer remains aligned with +~y (+1 state) and the top F2 layer oriented to −~y (-1 state). This gives an effective SAF magnetization of Meff = −151 emu/cm3 , according to expression (3.6). The graphic in fig. 3.22.(a) is a plot of the magnetization of the bottom and top layers of the SF in function of Ha . Now the offset is H0 = 50 Oe, which is lower than the 65 Oe obtained for the case without Hd .
a)
b)
Figure 3.22: (a) The graphic is a plot of both SF layers magnetization switching for Hk = 0; the transition takes place at H0 = 50 Oe. (b) This is a plot of the Demagnetizing field curve inside these SF layers in function of Ha - the graphic is shifted Hd = 17.5 Oe upwards on F3 , meaning that the demagnetizing field coming from the pinned SAF is HdSAF = −17.5 Oe In this situation the offset field can be calculated using the formula previously applied for the basic spin-valve: H0 = −Hcoup + HdSAF . Where the Hcoup is given by formula (3.11) and HdSAF is the demagnetizing field coming from the effective magnetic moment of the SAF layers: HSAF = N · MSAF d eff
(3.12)
Where N is a certain negative demagnetizing factor that depends on the geometry of the structure and the tSF eff . As it was refered in subsection 3.1.4 we 48
do not have a good theoretical approximation for the N factor, the best way that we have to predict the HdSAF for a given structure is using the micromagnetic calculations. In fig. 3.22.(b) there is a graphic of Hd (Ha ) on the top and bottom layers of the SF for the switching interval in study. The Hd on the bottom SF layer is the one that matters since this is the layer adjacent to the Cu spacer, and therefore it’s the one that will give us the sensor magnetoresistance response (given by (3.1)). As can be seen, the Hd field stands between two plateaus at Hd = 45 Oe and at Hd = −10 Oe, so there is a shift of 17.5 Oe upward. This vertical shift can only be due to the extra demagnetizing field coming from the SAF pinned layers, which is exactly the symmetric of the HdSAF value that we want to know. So, applying formula (3.4): H0 = 64.6 − 17.5 = 47.1 Oe. This is very near to the value of H0 ≈ 50 Oe that was observed on the simulation, considering that there are always uncertainties on the simulated graphic (of about 5 Oe) due to lack of precision. To sum up, in a SAF+SF sample without coercivity (Hc = 0), the SF transition offset H0 is approximately given by: H0 = −
3.4.3
4Ac SAF + N · Mef f µ0 (Mb tb − Mt tt )
(3.13)
Effect of Hk on Offset
Every crystalline ferromagnetic sample has an anisotropic field aligned with its easy-direction (k). Repeating the previous simulated hysteresis loops, using the anisotropy field Hk oriented parallel to the applied field in all the layers, will result in the appearance of coercivity Hc that disables the use of our devices for sensor applications. For several SAF+SF simulated with different teff it is observed that Hc increases with decreasing teff , as shown in graphic 3.23. It is quite difficult to predict the value of the Hc for a given SAF+SF structure, some studies have been made for MRAMs [24, 27] considering applied fields along the long-axis (Lz ). For MRAMs the prediction of Hc is quite important because it must be high enough to allow efficient memory storage. However, coercivity is not favourable at all for sensors, since these devices must always have Hc = 0 in order to achieve good linearity in the signal response at small Ha . There are two possible solutions that highly reduce the Hc , and both of them were simulated on the SAF+SF structure with teff = 15 ˚ A of fig. 3.17. The results are shown in fig. 3.24. Solution B, with crossed easy-axis between the SAF and SF, is what is commonly used for typical Spin Valves and it surely eliminates Hc completely. However, solution A, with crossed easy-axis between the SF layers, may be better to use with SF structures because it not only eliminates Hc but also maintains better sensitivity (lower switching field Hs ) than solution B.
49
60
50
40
30
20
10
0 5
10
15
20
25
30
35
40
45
Figure 3.23: The graphic shows a plot of the simulated coercivity field in function of the SF effective thickness, using the same SAF as the one in fig. 3.17. It was introduced an anisotropic constant K = 5 × 103 erg/cm3 in both SF layers.
3.4.4
Comparison with Experimental results and structure optimization
Patterned 2×6 µm2 bottom pinned SAF+SF spin valves were produced on glass substrates by Andr´e Guedes and co-workers from the INESC-MN experimental group. The structure used was the same as the one simulated before and deSAF picted in fig. 3.17, with Meff = −151 emu/cm3 . Several effective thicknesses were performed in the SF trilayer, and the offset field dependence on teff was studied. Figure 3.25 compares these experimental results of Andr´e Guedes with the simulated ones for the same structure. It is important to refer that both the experimental and the simulated hysteresis loops show coercivity because the easy-axis of all ferromagnetic layers were set parallel to the applied field direction. Since Hc 6= 0 the values of H0 simulated in this way will be inferior to the ones calculated with no coercivity (Hk = 0) in subsection 3.4.2. A variation of the type predicted by Anabela (H0 ∝ 1/teff ) is observed [16]. The major discrepancies between simulated and experimental results arise for low teff . This is understandable since, according to (3.13), it is expected that H0 would tend to infinity as teff → 0. However, for not so low effective thicknesses (above 20 ˚ A) the simulated offset is quite close to the experimental values.
50
A)
B)
Figure 3.24: In these simulations the offset fields are the same as the one simulated with Hk = 0. A) Solution with crossed easy-axis only between the SF layers: Hs = 25 Oe. B) Solution with crossed easy-axis between the layers of the SAF and SF: Hs = 55 Oe.
The SAF+SF spin valves produced experimentaly cannot be used in a sensor device because of their large offsets. The easiest way to reduce H0 is increasing teff , but we don’t want to go that way since we would be losing sensivitivity and so there would be no point in using these SAF+SF structures. Therefore, since we cannot lower the coupling field Hcoup on the SF layers the only alternative to eliminate H0 for a given SAF+SF structure is increasing the demagnetizing fields acting of the SF to compensate the Hcoup - see expression (3.13). This can be accomplished using three types of solutions: SAF • Increasing the Meff - The SAF structure used on the experimental samples produced by Andr´e Guedes was slightly unbalanced, with a small SAF Meff = −151 emu/cm3 that creates a demagnetizing field on the SF F3 layer equal to 17.5 Oe. If we unbalance a little more the SAF of fig. 3.17, for instance by raising the F2 thickness to 30 ˚ A and replacing the material SAF on F1 by NiFe, we get Meff = −496 emu/cm3 which will produce an
51
120
100
80
60
40
20
0
-20 5
10
15
20
25
30
35
40
45
Figure 3.25: The figure shows a comparison between experimental and simulated SAF were calculated using expression (3.6). Simulations results. The values of Meff show that if the effective moment of the SAF is increased the H0 is considerably reduced.
higher demagnetizing field of about 52 Oe (for teff = 15 ˚ A). Using this more unbalanced SAF, the values of H0 obtained for the same effective thicknesses of the previous simulations are considerably reduced - see fig. 3.25 • Lowering the patterned Ly dimension - If we reduce the width of the sample there will be more magnetic charges created on the borders, hence the demagnetizing potencial will be higher. The simulation of structure in fig. 3.19, with teff = 15 ˚ A, was repeated but using an aspect ratio two times bigger with a geometry of 0.5 × 3 µm2 , instead of the patterned 1 × 3 µm2 previously used. With this Ly = 0.5 µm an offset of H0 = 27 Oe was determined, which is certainly lower than the value of H0 = 38 Oe obtained for the same SAF+SF with Ly = 1 µm. The resulting curve is shown in fig. 3.26. • Depositing another pinned layer on top of the SF to produce a biasing magnetostatic field - A 15 ˚ A layer of CoFe coupled with an 52
Figure 3.26: The graphic shows the transfer curve of the 0.5 × 3 µm2 SF bottom layer of fig. 3.17. The offset field H0 was reduced 11 Oe in comparison with the H0 of the 1 × 3 µm2 structure, but the switching field Hs increased by 7 Oe.
antiferromagnet should be enough to produce a demagnetizing field high enough to eliminate the offset fields of these samples. However, there must be a zero exchange coupling field between this extra ferromagnet and the top layer of the SF. In addition, the pinning field from the antiferromagnet acting on the extra pinned layer must be on the opposite direction of the Hpin acting on the SAF, so that the demagnetizing field produced by the extra CoFe magnetization compensates the Hcoup on the SF. New patterned SAF+SF structures will be built by Andr´e Guedes and coworkers using the solutions described, to test the predicted offset field reduction. From the three solutions described, the ones that seem most promissing are the first and the third. The second one has two disavantages that will limit its implementation: First of all, the dimensions used in these sensors are already very small, and if we lower them even more we may loose efficiency in the magnetic detection capabilities of our devices. Secondly, it raises the self demagnetizing fields inside the SF free layers, thereby increasing the switching field (Hs - defined in fig. 3.8) and lowering the sensor sensitivity, which is exactly what we don’t want. The first and third solutions are more wise because they just increase the interlayer demagnetizing field, that comes from pinned layers, and acts on the SF. Throughout the SF transition this interlayer Hd remains constant, because
53
the pinned layers magnetization is fixed, and so it does not increase the Hs like the self Hd does. The only purpose of the interlayer Hd in these solutions is to counterbalance the Hcoup and lower the offset. Hence, in principle, the first and third solutions do not reduce the sensor sensitivity. SAF For the simulations plotted in fig. 3.25 for Meff = −496 emu/cm3 the SAF measured values of Hs are very close to the ones of Meff = −151 emu/cm3 . In particular for the structure with teff = 15 ˚ A it was measured Hs ≈ 13 Oe in both situations. According to the simulations performed of basic spin valves (F/NM/F), the Hs values measured were around 35 Oe, for similar dimensions. So it is notorious that the sensitivities of those simple structures are much lower than the ones of the SAF+SFs. However, as previously explained, the simulated transfer curves used in fig. 3.25 have a certain coercivity which increases with decreasing teff . To eliminate this coercivity crossed easy-axis must be used, and this will inevitably increase the Hs . As shown in fig. 3.24.(A), using crossed easy-axis on the SF layers of the structure with teff = 15 ˚ A results in Hs ≈ 25 Oe, which is a value that is not so far from the ones of simple spin valves. It would be interesting for a future work to make a more detailed study of the switching field dependence on SAF+SF structures with crossed easy-axis, and to compare the values obtained with the ones of basic spin vales with just two ferromagnetic layers.
54
Chapter 4
Dynamic time resolved simulations of magnetization switching in Spin Valves Experimental techniques are being developed that can probe magnetization dynamics on a time scale of the order of the nanosecond [26, 30, 31], so it has become important to compare these experimental results with the micromagnetic simulations that use LLG equation (1.31) to describe that very same motion. Furthermore, the raising issue of writing on magnetic storage media ever faster has motivated the study of current-induced magnetization switching (CIMS) [28], provoqued by spin-transfer torque. Spin-polarized electrons traversing a ferromagnet can transfer spin angular momentum to the local magnetization, thereby applying a torque that may produce magnetic reversal or steady-state precession. This spin-transfer mechanism allows nanomagnets to be manipulated without magnetic fields, and it is the subject of extensive research for applications in non-volatile memory, programmable logic and microwave oscillators [31]. In this chapter I present the results of time domain simulations of the free layer switching on a simple spin-valve structure, in order to study the magnetic dissipation and the corresponding damping constant α. Then the effect of an injected spin polarized current (SPC) is introduced on the simulations and it is verified that it allows faster switching since it increases the effective damping parameter.
55
4.1
A simple switch test in a Bulk sample
Consider the basic spin valve structure shown in fig. 4.1. In this section the effect of the demagnetizing field will be ignored in the dynamic simulations (Hd = 0), which is comparable to the case of a bulk sample if we neglect the magnetization component perpendicular to the layers plane (Mx ). In the next section the same simulations will be repeated but including the demagnetizing field (intra and interlayer), and this way we will be able to understand the role of this particular energy term on the switching mechanism of spin valves.
Figure 4.1: This is a squematic picture of the Spin Valve structure used in the dynamic simulations of this chapter. The pinning field on F1 was set to lie along the ~z direction, parallel to the easy-axis of both ferromagnets. The external field is applied perpendicularly to ~z with an intensity of |Ha | = 100 Oe. In equilibrium, for no applied field, the free layer magnetization will be parallel to the pinned layer magnetization (Mf //~z). Then, if (at t=0) a normal 100 Oe field is applied in the ~y direction, Mf will rotate to the ~y axis and then it will precesse in turn of the Heff axis [26], as depicted in fig. 4.2.
4.1.1
Underdamped oscillator
Considering no demagnetizing field (bulk sample), the movement of the magnetization of both pinned and free layers is plotted in fig. 4.3. In this type of experiments the quantity that is directly measured is the magnetoresistance response (given by (3.1)) which is proportional to the value of < cos(θpin − θf ree ) >. So I have chosen to also plot quantity < cos(θpin − θf ree ) > (no units) to facilitate the comparison with experimental results. As we can see from fig. 4.3, if the demagnetizing field isn’t used the magnetization of the free layer will describe a damped oscillatory motion in turn of the Heff direction (at θ = 78.5o ), with damping constant α = 0.08. For this situation the free layer switching time is about 25 ns. 56
y Heff
Mf
z
q
Figure 4.2: The magnetization precesses in turn of Heff and will gradually converge to its direction as time increases.
in
ree
-TF )>
Figure 4.3: The angle θ (Theta - in rad ) that the free and pinned layer magnetization vectors make with the ~z axis, after the application of the external field Ha = 100ˆ y Oe, is plotted as a function of time. Mpin remains aligned with the ~z axis, but Mf rotates to θf = 78.5o . The material damping parameter introduced for this simulation was α = 0.08.
This motion can be more easily understood attending to fig. 4.4. This is the motion of a typicall underdamped oscillator since the magnetization vectors need to spin around their Heff several times before becoming perfectly aligned with the equilibrium direction. The frequency of the free layer magnetization oscillations around Heff can
57
a)
b)
Figure 4.4: The figures show the trajectory of the unit sphere of the tip of the magnetization vector from state just prior to switching until equilibrium is reached at the set applied field. (a) Mf (b) Mpin - remains almost fixed.
be easily calculated doing a Fast Fourier Transform (FFT) to the functions θf and < cos(θpin − θf ree ) > plotted in fig. 4.3. Proceeding like so we get two well defined frequency peaks at Fr = 0.275 GHz.
4.1.2
Critically damped oscillator
As referred in the first chapter of this thesis, if a damping parameter α = 1 is set then we would have a Critically damped oscillator (see fig. 1.4). In this case, the average magnetization vector no longer precesses around Heff but imeadiately converges to its final direction in only one half turn. So, in this situation the Mf sinusoidal oscillations observed in fig. 4.3 will no longer be present, since θf rapidly becomes equal to the equilibrum angle (78.5o ) - see fig. 4.5.
4.2
Switch test in a patterned 0.25×1 µm2 sample
The Spin Valve structure represented in fig. 4.1 was simulated now including the demagnetizing field Hd in the total Heff . We have assumed a patterned sample with dimensions 0.25 × 1 µm2 , where an external field Ha = 100ˆ y Oe was applied (at t=0) in order to switch the free layer magnetization from the ~z to the ~y direction. Although the conditions are the same as the previous problem, now with the demagnetizing field the final equilibrium angle will, of course, be different and we will no longer observe the perfectly damped oscillatory motion of Mf - see fig. 4.6. This is because the free layer Heff has become dependent on Mf , and so it continuously changes every time step. In the previous case, for Hd = 0, the 58
f )>
Figure 4.5: The picture shows the simulation results for a critically damped oscillator. With a critical damping constant α = 1 the free layer switching time is about 2.7 ns.
Heff was a constant vector throughout the complete Mf rotation, since it always remained at a fixed equilibrum angle (θf (eq) = 78.5o ). However, in the present case with the addition of Hd , the equilibruim angle at a certain time step will be different from the one on the next time step if there is change in Mf on that step, due to the fact that Hd dependends on the orientation of Mf . The result of not having sinusoidal oscillations is that the magnetization movement will not waste time ringing around the Heff , so it will converge to the equilibruim position very fast with the demagnetizing field effect, just as a critically damped oscillator forced to remain in the yz plane - see fig. 4.7. As can be seen in fig. 4.6, the switching time (Tsw ) for this patterned sample is just 1.6 ns, which is one order of magnitude less than in the bulk sample. In equilibrium Mf will stabilize at an angle θf (eq) = 64o , which is smaller than the 78.5o of the bulk sample because now we must take into account the shape demagnetizing anisotropy. Since our patterned 0.25 × 1 µm2 sample has the long axis along ~z, the Hd will have a component stronger in that direction, thereby making the θf (eq) smaller. With the addition of the demagnetizing field, the free layer Heff intensity has increased in this case. So, performing the Fast Fourier Transform (FFT) of the functions θf and < cos(θpin −θf ree ) > plotted in fig. 4.6 will give ressonance peaks at higher frequencies than the ones of Hd = 0. For this case the frequency spectrum showed peaks at Fr = 1.26 Ghz.
59
-TF )> ree in
Figure 4.6: The angle θ (Theta - in rad ) that the free and pinned layer magnetization vectors make with the ~z axis is plotted as a function of time. Mf rotates to θf = 64o . The material damping parameter introduced on this simulation was α = 0.03, which is close to the typical value for permalloy.
4.3
Spin Polarized Current triggered Magnetization switching
If a voltage is applied between the top and bottom layer of a spin valve structure we will have a current flowing in the ~x direction from the pinned to the free layer. The effect of this current can be easily described with the atomic dipoles model of ferromagnetism described in the first chapter. The majority of the electrons coming from the pinned layer have their spins oriented parallely to the average Mpin , hence this current is said to be polarized on the pinning direction. When the spin polarized current (SPC) reaches the free layer its electrons will provoque a torque on Mf by exchange interaction with the magnetic dipoles on the ferromagnetic body, forcing them to rotate towards the polarization direction (~ p) [28]. The effect of an SPC passing through a ferromagnetic body was studied in detail by J. C. Slonczewski [29], who derived a theoretical model able to explain with good precision the experimental results observed. This model consists in the addition of an extra term to the LLG equation, that describes the free layer magnetization Mf motion, due to the spin-transfer torque effect:
60
a)
b)
Figure 4.7: The figures show the trajectory of the unit sphere of the tip of the magnetization vector from state just prior to switching until equilibrium is reached at the set applied field. Here the switching is much faster than in the case of Hd = 0 (Tsw = 1.6 ns), because the demagnetizing field forces Mf to oscilate only in 2 dimensions (yz plane) and makes the motion converge as a critically damped oscillator.
dmf = −γMs mf × heff + αmf × (mf × heff ) + τ g(θ)mf × (mf × mpin ) dt (4.1) Where mf and mpin are the unitary vectors of the overral free and pinned layer magnetizations. The constant τ is proportional to the SPC density (Je [A/m2 ]): τ=
¯hγJe µ0 Ms tf e
(4.2)
Where tf is the free layer thickness and e is the electronic charge. The scalar function g(θ = θpin − θf ) in (4.1) is given by: h (1 + P )3 (3 + mf · mpin ) i−1 g(θ) = − 4 + 3 4P 2 Where P is the electrons polarization which is set equal to 0.45 [28].
(4.3)
The extra term introduced by Slonczewski in the LLG equation has a damping effect since it will simply add to the damping term associated with α, and it will be proportional to the current density Je injected. If this term is not strong enough the current won’t be able to reverse the Mf to the polarization direction (~ p//Mpin ), nevertheless it will always contribute to the damping effect forcing Mf to converge faster to its equilibrium position parallel to Heff . But, if the current density is high enough the Slonczewski term will not only allow faster switching but it will also drive the Mf direction to the polarization direction. 61
Experimentally, it was seen that there will be a certain critical current density Jc value which separates both situations [31]: • If Je < Jc : The SPC increases the effective damping parameter (αeff = α + αSP C ) allowing faster switching by supressing the ringing around the Heff axis. • If Je > Jc : The SPC drives coherently the precession axis to the polarization direction.
4.3.1
Spin Valve simulations with SPC
To study the SPC effects we have used the same structure of the previous dynamic simulations, depicted in fig. 4.1, so that we can compare the results obtained before with the ones using an SPC injected in the direction perpendicular to the layers plane (~x). First of all, we start by neglecting the demagnetizing field (Hd = 0), which is similar to the case of a bulk spin valve. If a small current density is used Je = 0.03 A/µm2 it is confirmed that the effective damping constant diminuishes significantly - see fig. 4.8. The effective damping parameter αeff of the simulated function θf (t) can be calculated performing a fit of the resulting curve to the expression of a damped harmonic oscillator [30]: θ(t) = θ(eq) + θ0 e−αeff t cos(ωd t)
(4.4)
Doing like so, the fit result is: αeff = 0.23, which is about 3 times greater than the α material parameter introduced (0.08). Now, introducing the demagnetizing field and considering a patterned 0.25 × 1 µm2 sample like before, it must be used a higher current density in order to observe a measurable increase in the αeff . As can be seen in fig. 4.6, there is almost no ringing on the Mf precession movement since for such a high demagnetizing field (low dimensions) the magnetization rapidly converges to the Heff position. So, the damping parameter here can hardly be increased by using a small SPC density. According to the tests made, in this case only for current densities around 0.2 A/µm2 (or higher) can effectively be measured a decrease in the switching time of the free layer - see fig. 4.9.(a). In fact, if the sample dimensions were increased a lower current density would be enough to accomplish the same effect. In fig. 4.9.(b) an even higher current was used to simulate the free layer magnetization behaviour with Je > Jc . With Je = 1 A/µm2 we have clearly exceeded the critical current and Mf starts to be driven by the current to its direction of polarization (// ~z), thereby reducing the θf (eq) from 64o to 57o . As a last remark, it is important to refer that if a negative SPC is applied (Je < 0) the Sloncweski term on LLG equation (4.1) will not add to the damping 62
,1)
in
ree
-TF )>
Figure 4.8: The graphic shows the simulation results for a bulk sample with SPC. The α damping parameter introduced in this simulation was 0.08, however the αeff was measured to be 0.23. The SPC with Je < Jc considerably reduced the switching time (compare with graphic 4.3), but the equilibruim angle remains at θf (eq) = 78.5o .
α term but, instead, subtract to it. Thereby, if the Sloncweski and the α damping term have opposite signs the effective damping parameter αeff will be lower. So, in the case of Je < 0 the effect of SPC on Mf precession will be opposite to the one studied in this section, since instead of diminuishing the ringing the SPC will increase the free layer magnetic oscillations. This “antidamping” SPC effect has also been confirmed on the simulations performed.
63
a)
b)
in
ree
-TF )>
Figure 4.9: The figures show the simulation results for the patterned 0.25 × 1 µm2 structure with SPC. The α damping parameter introduced in these simulations was 0.03. (a) Here the SPC with Je < Jc slightly reduced the switching time (Tsw ≈ 1.45 ns), but the equilibrium angle remains at θf (eq) = 64o . (b) A SPC density higher than the critical one (Je > Jc ) was used, hence there’s a significant torque from the polarized electrons that force Mf to their polarization direction at θ = 0. Now the equilibrium angle was reduced to about 57o .
64
in
ree
-TF )>
Chapter 5
Acknowledgments In the first place, I am greatly indebted to Prof. Jos´e Lu´ıs Martins, my advisor, for his guidance and assistance during the entire work of this thesis. Thanks to him I have gained a lot of knowledge and practical expertise in computational physics and, particularly, in the field of micromagnetism. I am also very grateful for all the help and excellent advices that Prof. Jos´e Lu´ıs Martins gave me during the entire process of application to PhD studies in the United States, which shall begin next month. Secondly, I wish to thank Prof. Paulo Freitas, the supervisor of INESC-MN experimental group. With him I not only learned a lot about the development and study of nanotechnology devices, but also understood what it takes to manage a group of 7 graduates and 4 final year students with a total of 14 different projects all in the leading areas of magnetic sensors and memory devices. For this huge task I learned that 3 main things are essential: Dedication, organization and a lot of work. Before deciding to join INESC-MN to do the senior thesis with Prof. Jos´e Lu´ıs Martins I was inclined to follow a different field of research. It was only after taking the course “Micro and Nanotechnology”, with Prof. Susana Freitas, that I got really interested in the work of this research group. So, my third acknowledgment goes to Susana Freitas, for the wise advices that she gave me. Then I would like to express my appreciation to my partner Andr´e Guedes, with whom I’m going to publish the article “Study of SAF-SF structures for magnetic sensor application” for the Conference MMM 2005 proceedings, which shall be latter published in the next Journal of Applied Physics edition. He was the responsible for the SAF+SF experimental results shown here, and his help was indispensable to the development of the conclusions presented in this work. I am also grateful to Ricardo Ferreira for the useful discussions and for all the articles and books that he lent me to support the studies made in this thesis. And the last but not the least, I want to thank all my colleagues of INESCMN for the support and motivation they gave me throughout this year. Surely they have not only been great colleagues but also good friends, and to all of them I wish the best luck and success! 65
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