Magnetoresistance effects and magnetoelastic behavior of ferromagnetic materials 

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Anisotropic Magnetoresistance (AMR) based sensors

The anisotropic magneto-resistive effect, abbreviated AMR effect, was already discovered in 1857 by William Thomson. Thomson found out that magnetic field can influence the electric resistance of ferromagnetic materials (change of 1-2%). Finally, 120 years later and by applying the thin-film technology, this knowledge was first used for a technological product – the MR sensor.
Thomson’s experiments are an example of AMR, property of a material in which a dependence of electrical resistance on the angle between the direction of electric current and direction of magnetization is observed [Bozorth, 1993]. The physical origin of the magnetoresistance effect lies in spin orbit coupling. The electron cloud about each nucleus deforms slightly as the direction of the magnetization rotates, and this deformation changes the amount of scattering undergone by the conduction electrons when traversing the lattice. A heuristic explanation is that the magne-tization direction rotates the closed orbit orientation with respect to the current direction. If the field and magnetization are oriented transverse to the current, then the electronic orbits are in the plane of the current, and there is a small cross-section for scattering, giving a low resistance state. Conversely for fields applied parallel to the current, the electronic orbits are oriented perpendicular to the current, and the cross-section for scattering is increased, giving a high resistance state. The net effect (in most materials) is that the electrical resistance has maximum value when the direction of current is parallel to the applied magnetic field (Fig. 1.2). AMR of new materials is being investigated and magnitudes up to 50% have been observed in some ferromagnetic uranium compounds [Wiśniewski, 2007].
The AMR effect is used in a wide array of sensors for measurement of Earth’s magnetic field (electronic compass), for electric current measuring (by measuring the magnetic field created around the conductor), for traffic detection and for linear position and angle sensing.
The biggest AMR sensor manufacturers are Honeywell, NXP Semiconductors, and Sensitec GmbH.

Giant, Colossal and Tunnel Magnetoresistance (GMR, CMR and TMR) based sensors

The Giant Magneto Resistive (GMR) effect was first discovered in 1988 by Fert and Grunberg [Grünberg et al., 1986] [Baibich et al., 1988], who were awarded with the Nobel Prize for Physics in 2007 for this achievement. This effect occurs in layer systems with at least two ferromagnetic layers and a single non-magnetic, metallic intermediate layer. If the magnetization in these layers is non-parallel, the resistance is larger than if the magnetization is parallel. The difference may reach up to 50%, thus the name giant.
The key structure in GMR materials is a spacer layer of a non-magnetic metal between two magnetic metals (Fig. 1.3). Magnetic materials tend to align their magnetization in the same direction. So if the spacer layer is thin enough, changing the orientation of one of the magnetic layers can cause the next one to align itself in the same direction. But an oscillation in the coupling strength can be observed as a function of the thickness of the non-magnetic layer. The magnetic alignment of the magnetic layers periodically alternate back and forth from being aligned in the same magnetic direction (parallel alignment) to being aligned in opposite magnetic directions (anti-parallel alignment).
The chief source of GMR effect is spin-dependent scattering of electrons. Elec-trical resistance is due to scattering of electrons within a material. Depending on its magnetic direction, a single-domain magnetic material will scatter electrons with « up » or « down » spin differently (it can be described as in the case of the AMR ef-fect). When the magnetic layers in GMR structures are aligned anti-parallel, the resistance is high because « up » electrons that are not scattered in one layer can be scattered in the other. When the layers are aligned in parallel, all of the « up » elec-trons will not scatter much, regardless of which layer they pass through, yielding a lower resistance (Fig. 1.3).
GMR sensors offer high sensitivity, wide frequency range, small size, low power consumption and they are compatible with many other state-of-the-art technologies. However, as the linearity range of GMR sensors is narrow compared with other sensors such as Hall sensors and AMR sensors, GMR sensors are not suitable for large current sensing [Jedlicska et al., 2010]. In addition, the output of some of GMR sensors is unipolar, which limits its application in AC measurements [McNeill et al., 2008]. The main suppliers for GMR sensors are Sensitec GmbH, NVE Corporation and Hitachi.

Colossal MagnetoResistance

In 1993 von Helmholtz et al. discovered the Colossal MagnetoResistive (CMR) effect [von Helmolt et al., 1993]. This effect occurs in perowskitic, manganese based oxides, that change their resistance in the presence of a magnetic field. Of all the known physical effects, by which a solid changes its properties due to magnetism, MR technology has particularly interesting and convincing advantages. However, a fully quantitative understanding of the CMR effect has been elusive and it is still the subject of current research activities. Early prospects of great opportunities for the development of new technologies have not yet come to fruition.

Tunnel MagnetoResistance

The Tunnel MagnetoResistive (TMR) effect [Julliere, 1975], discovered by Jul-liere in 1975, occurs in layer systems consisting of at least two ferromagnetic layers and a thin insulation layer. The tunnel resistance between both layers depends on the angle of both magnetization directions. If the insulating layer is thin enough (typically a few nanometers), electrons can tunnel from one ferromagnet into the other. Since this process is forbidden in classical physics, the tunnel magnetoresis-tance is a strictly quantum mechanics phenomenon.

Comparison with other magnetoresistive elements

Spin-valve sensors obtain larger output signals than AMR sensors (but for larger magnetic fields). Therefore they are useful for reading information from magnetic tapes or disks (in magnetoresistive heads) where small dimensions of the sensor are the most important parameter. Due to large magnetoresistivity GMR heads allow reading of information with the highest density reported to date. The most frequently used are AMR and GMR/SV sensors – they are also available as com-mercial products (Philips, Siemens, Honeywell, Nonvolatile Electronics). The semi-conductor magnetoresistors are of smaller technical importance due to rather strong temperature dependence and high non-linearity [Popovic et al., 1996].
It is highlighted on Figure 1.4 that for detecting small magnetic fields (for ex-ample Earth’s magnetic field) the permalloy magnetoresistive sensors are the most suitable. In the following our study is concentrated on the investigation of thin film permalloy AMR sensors.

Microscopic magneto-elastic model

The first step of our magneto-mechanical modelization is to get an accurate description of the single crystal behavior. A single crystal is seen as an assembly of magnetic domains. The distribution of magnetization (domain configuration) within a single crystal can be very heterogeneous. As its electrical resistance is strongly dependent on the local magnetization (AMR effect), modelization at this microscopic scale (domains or group of atoms) is used in order to investigate the AMR behavior of the whole single crystal.
The external magnetic field or the mechanical stress applied to a magnetic media can change the distribution of domain orientations. As a consequence it modifies the local resistivity, and thus the overall resistivity.
In this chapter a microscopic magneto-elastic model for AMR is proposed based on the magneto-elastic model derived from [Daniel and Galopin, 2008, Daniel et al., 2008]. It can be noticed that the same model can be applied for a grain embedded in a polycrystal (presented in Chapter 3.). At the scale of a group of atoms, the magnetic equilibrium state can be described by the sum of several energy terms. The free energy W p of a group of atom can be written [Cullity and Graham, 2011]: W p = W ex + W K + W σ + W H (2.1).

Magnetocrystalline anisotropy energy

Magnetocrystalline anisotropy energy is the energy necessary to deflect the mag-netic moment in a single crystal from the easy to the hard direction. The easy and hard directions arise from the interaction of the spin magnetic moment with the crystal lattice (spin-orbit coupling). The magnetocrystalline energy is minimal when the magnetization is aligned with an easy axis and maximal when it is aligned with a hard axis of the single crystal.
The magnetocrystalline anisotropy energy is generally represented as a spherical expansion in powers of the direction cosines of the magnetization. In cubic crystals (like Iron or Nickel) it is sufficient to represent the anisotropy energy in an arbitrary direction by just the first two terms in the series expansion. These two terms each have an empirical constant associated with them called the first- and second order anisotropy constants, K1 and K2 respectively.
In the case of cubic crystallographic structure the magnetocrystalline energy can be written [Cullity and Graham, 2011]: WK=K1 (α2 α2 + α2 α2 + α2 α2) + K2 (α2 α2 α2) (2.3). If the second term can be neglected, the easy axes are the <100> axes for K1 > 0 (for example Iron) and the <111> directions for K1 < 0 (for example Nickel).
The spatial magnetocrystalline energy distribution of an Iron single crystal is presented in Figure 2.1. As the anisotropy constants of Iron are K1 = 42.7 kJ/m3 and K2 = 15 kJ/m3 [Bozorth, 1993] its easy magnetization directions (where the energy W K is minimal) are <100>.

Calculation at the single crystal scale

The intermediate calculation scale in this model – called mesoscopic scale – is the single crystal or grain, that is seen as a collection of magnetic domains (α) with given magnetization orientation α. The microscopic magnetoelastic model allows us defining in a statistical way the domain configuration, introducing as an internal variable the volume fraction fα of domains with orientation α in a grain g. In each domain, depending on the considered orientation α, the magnetization Mα and magnetostriction strain εµα are known.

selection and calculation of state variables

The state variables chosen to describe the magnetization of a single crystal are the volume fraction fα of the domain family α in the single crystal [Néel, 1944, Chikazumi and Graham, 1997, Buiron, N. et al., 1999, Daniel et al., 2008, Daniel and Galopin, 2008]. The fα variables are obtained through the numerical integration of the following Boltzmann-type relation [Daniel and Galopin, 2008] over all the possible orientations for the magnetization in the single crystal : exp(−As.Wα) fα = R (2.18) α exp(−As.Wα)dα.
where As is an adjustment parameter accounting for the non uniformity of the ex-change energy, the magnetic field and the stress tensor within the single crystal. It can be deduced from low field measurement of the anhysteretic magnetization curve (As = 3χ02 [Daniel et al., 2008] where χ0 is the initial anhysteretic susceptibility of µ0Ms the material).

Table of contents :

Introduction
1 Magnetoresistance effects and magnetoelastic behavior of ferromagnetic materials 
1.1 Magnetoresistive elements for magnetic field sensing
1.1.1 Hall sensors
1.1.2 Anisotropic Magnetoresistance (AMR) based sensors
1.1.3 Giant, Colossal and Tunnel Magnetoresistance (GMR, CMR and TMR) based sensors
1.1.4 Comparison with other magnetoresistive elements
1.2 Basic notions of material behavior
1.2.1 Different types of magnetic behavior
1.2.2 Ferromagnetism
1.2.3 Mechanical behavior
1.2.4 Magnetoelastic coupling
1.3 AMR models in the literature
2 Single crystal behavior 
2.1 Microscopic magneto-elastic model
2.1.1 Exchange energy
2.1.2 Magnetocrystalline anisotropy energy
2.1.3 Magnetostatic energy
2.1.4 Elastic energy
2.2 Calculation at the magnetic domain scale
2.2.1 Potential energy of a magnetic domain
2.2.2 Single domain model of AMR
2.3 Calculation at the single crystal scale
2.3.1 Selection and calculation of state variables
2.3.2 Homogenization
2.4 Results and comparison to experimental data
2.4.1 Magnetoelastic properties
2.4.2 Magnetoresistive properties
2.5 Conclusion
3 Polycrystal behavior 
3.1 Macroscopic model
3.1.1 Modeling strategy
3.1.2 Localization step
3.1.3 Calculation of the effective properties
3.1.4 Calculation algorithm and model parameters
3.2 Prediction of the AMR effect on ferromagnetic polycrystals
3.2.1 Effect of stress on the magnetoresistive behavior
3.2.2 Effect of crystallographic texture
3.3 Conclusion
4 Modeling of thin film AMR sensor properties 
4.1 Design and contruction of AMR sensors
4.2 Modeling thin film properties
4.2.1 Introduction of surface effect
4.2.2 Textured AMR thin film sensor properties
4.3 Effect of biasing magnetic field
4.3.1 Definition of the sensitivity of an AMR sensor element
4.4 Influence of the film thickness
4.5 Effect of stress on the properties of AMR thin film sensors
4.6 Conclusion
General conclusion
Bibliography 

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