more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11

Anomalous Hall effect (AHE). Anisotropic magnetoresistance (AMR)
Spin and Charge TransportAbstract:The Anomalous Hall effect is the Hall effect, which is originated from the from the spindependent scatterings. The Anomalous Hall effect is proportional to the magnetization of the material (not intrinsic magnetic field).Note: All data of this page represent only my personal point of view, which are based on my experimental and theoretical research.
Ordinary Hall effect (OHE) vs Anomalous Hall effect (AHE)Origin OHE is induced by the Lorentz force AHE is due to the spindependent scatterings of the conduction electrons M or H?  OHE is linearly proportional to the intrinsic magnetic field H (or B)  AHE is linearly proportional to the magnetization M Dependence on spin polarization  OHE does not depend on the spin polarization AHE is linearly proportional to the spin polarization Bulk and interface contributions  OHE has only the bulk contribution AHE has two very different contributions: from bulk and from interface
Origin of Anomalous Hall effect (AHE)
The origin of Anomalous Hall effect is spindependent scatteringsnote: AHE has the same origin as the Spin Hall effect.What is the spin dependent scattering? A. it is a scattering, which probability is different between left and right directions with respect to the electron movement direction and which probability depends depends on the electron spin.
Which effect makes the scatterings of conduction electrons to be direction and spindependent? A. The spinorbit (SO) interaction (details see here) How the SO interaction makes a scattering to be directiondependent and spindependent? There is an electrical field around the defect. It is directed to the left from left side of defect and to right from right side. As a consequence, the effective magnetic field of SO interaction is opposite directions at the left and right sides of the defect. The effective magnetic field of SO interaction affects the energy and symmetry of electron. Since the field is different, the energy and symmetry depends on the direction and the spin. As a result, the scattering probability depends on the direction and spin.
Physical mechanisms of spin depend scatterings
Bulktype spindependent scatterings on defects:The electrical field around a defect induces the effective magnetic field due to the spinorbit interaction (Fig.5(a)). The effective magnetic field is in opposite directions when a delocalized electron moves from the left or from the right side from the defect. Because of the effective magnetic field of the spinorbit interaction, the scattering probability is different whether the electron is scattered from left to the right side or from the right to the left side of the defect. Interfacetype spindependent scatterings across interface:There is a charge accumulation at the contact (the violet balls of Fig. 5(b)). For example, the charge may be accumulated because of the different work functions of the metals. The electrical field of the accumulated charge induces the effective magnetic field of the spinorbit interaction for electrons flowing along the interface. The effective magnetic field is of opposite sign for electrons flowing at each side of the contact. The electron scattering probability across contact interface is different whether an electron is scattered from the left to the right metal or in the opposite direction. Because of this difference, a spin current flows across the contact interface when a charge current flows along the contact.
Measurement of the Ordinary & Anomalous Hall effects. The Hall angle α_{Hall}
The OHE and AHE is measured by the Hall angle α_{Hall}. The contributions from AHE and OHE are independent: α_{Hall}=α_{AHE}+α_{OHE} What is better to use the Hall resistance or Hall angle α_{Hall}_{} A. The Hall angle α_{Hall}. It is only correct parameter characterizing the AHE and OHE. What are merits to use the Hall resistance for measurement of AHE and OHE? None. It is just a number in the Ohm unit. It has no physical meaning. It depends on the device geometry. It is not a material intrinsic parameter. What are merits to use Hall angle α_{Hall} of measurement of AHE and OHE? merit 1: The α_{Hall} has a direct physical meaning. It the angle of deviation of electron movement from a straight line along an applied electrical field. merit 2: The α_{Hall }is an intrinsic parameter of a material. It does not depend on the device geometry or film structure. The magnitude of the OHE and AHE in different devices and different films should be only compared by comparing their α_{Hall}. Measurement of α_{Hall}The Hall angle α_{Hall} is defined as where σ_{xx} and σ_{xy} are diagonal and offdiagonal components of the conductivity tensor
Single layer film: In this case the Hall angle α_{Hall} is calculated as where the Hall voltage V_{Hall}, is the Hall voltage L is wire length and w is wire width. The Hall resistance R_{Hall} can be calculated where R is wire resistance Doublelayer metallic wire: This case when metallic wire consists of two layers: The first layer is made of a ferromagnetic metal. The second layer is made of a nonmagnetic metal. In this case the Hall angle α_{Hall} of the ferromagnetic metal is calculated as where t_{ferro},t_{isot}, σ_{ferro},σ_{isot} are thicknesses and conductivities of ferromagnetic and nonmagnetic metals.
to see how to obtain Eqs.(4.15),(4.24), click here to expand it
Singlelayer metallic wireDue to the Hall effect there is an electrical current current across the metallic wire, which can be calculated as where V is the bias voltage, L is wire length, j_{} and j_{⊥} is current density along and across the wire . The current j_{⊥} makes a charge accumulation at walls of the metallic wire. This charge induces the voltage, which is called the Hall voltage V_{Hall}. The Hall voltage induces the current, which is opposite to j_{⊥}. The current density of this current j_{⊥,comp} can be calculated as where w is the width of the wire. In total, there is no current across the wire, σ=σ_{xx} is conductivity of the wire. Since in total there is no current flow across the wire, we have Substituting Eqs.(4.2),(4.3) into Eq.(4.4) gives or The Hall resistance R_{Hall} is defined as where J_{} is the current flowing throw the nanowire where thick is wire thickness Doublelayer metallic wireThis case when metallic wire consists of two layers: The first layer is made of a ferromagnetic metal. The second layer is made of a nonmagnetic metal. The Hall current is generated only inside ferromagnetic The Hall current J_{⊥}, which flows across wire, can be calculated as where j_{⊥} is the Hall current density , t_{ferro} is the thickness of ferromagnetic layer, w is the wire width, L is the wire length and V is the bias voltage. α_{Hall} is the Hall angle of the ferromagnetic metal. It is assumed that the Hall angle α_{Hall}=0 in the nonmagnetic metal. Due to the charge accumulation, the current flows in the opposite direction in both layers t_{isot} is the thickness of layer of the nonmagnetic metal, σ_{ferro},σ_{isot} are conductivities of ferromagnetic and nonmagnetic metals.. Since in total there is no current flow across the wire, we have Substituting Eqs. (4.21),(4.22) into Eq.(4.22a) we have Simplifying Eq.(4.23) gives the Hall angle in case of wire consisted of two layers as
Notice: Polarities of loops in Au and FeBTb are different.
How to distinguish different contributions to the Anomalous Hall effect? A. From hysteresis loop of measured Hall effect, it can be distinguished which is the major contribution to the Anomalous Hall effect.
Ordinary Hall effect due to the Lorentz force in a ferromagnetic metal
Anomalous Hall effect due to the spindependent scatterings
Anomalous Hall effect & Ordinary Hall effect: The Anomalous Hall effect describes describes the fact that a change can be accumulated at the edges of a metallic wire when a magnetic field is applied perpendicularly to the drift spinpolarized current. Spin Hall effect: The Spin Hall effect describes the fact that spin may be accumulated at the edges of a metallic wire when spinunpolarized drift current flows in the wire. Inverse Spin Hall effect: The Inverse Spin Hall effect describes the fact that the spin may be accumulated at the edges of a metallic wire when diffusive spin current flows in the wire.
Q. Do the spin Hall effect, anomalous Hall effect and inverse spin Hall effect occur because of the electron scattering on magnetized defects (defects with nonzero magnetic moment). Are the defect should be magnetized ? A. It is not necessary. The defect can be either magnetized or nonmagnetized. It is important the electrical field around the defect, which induces spinorbit interaction and which makes a scattering to be spindependent. Q. The spin Hall effect, anomalous Hall effect and inverse spin Hall effect are very similar. How to distinguish between these effects? A. These three effects are very similar. It is difficult to distinguish between them. For example, the inverse Spin Hall effect can be explained as the sum of the Spin Hall effect and the anomalous hall effect. The spinpolarized part of the spin diffusion current experiences the anomalous hall effect, which makes a charge accumulation at edges of a wire. The spinunpolarized part of the spin diffusion current experiences the Spin Hall effect, which makes (changes) a spin accumulation at edges of the wire. Q. The spin Hall effect, anomalous Hall effect and inverse spin Hall effect can occur only in "bad" metals with a large number of defects? A. The magnitude of these effects may be large only in case of metal is a sufficient number of defects. Therefore they can be observed easily in “ bad” metals with a small conductivity. For example, FeBTb, which conductivity is 50 times smaller than the conductivity of gold and 12 times smaller than the conductivity of monocrystal iron, has a large Hall angle (~0.5 deg) of the anomalous Hall effect and a large SpinHall torque. In the case when the density of defect is small, the current the runningwave electrons is large and the contribution of the scattering current into the electron transport is negligible. It makes all scatteringoriginated effects to be screened and magnitude of the will be not large. In the case when the density of defect is larger, the electron wave function overlaps several defect simultaneously, it would make all scatterings spinindependent and there will be no spin Hall effect, anomalous Hall effect and inverse spin Hall. Other important condition for existence of spindependent scatterings is that the electron meanfree path should be shorter than the effective radius of electrical field around a defect. For a larger magnitude of the spin Hall effect, anomalous Hall effect and inverse spin Hall effect, the number of defects should be not very large and not very small, but it should be an optimum.
Anomalous Hall effect Q. For the anomalous Hall effect, is it essential for electron gas to be spinpolarized? A. The anomalous Hall effect, which is originated due to the spindependent scatterings, occurs only when the electron gas is spinpolarized. The Hall angle is linearly proportional to the spin polarization of the electron gas. Q. Why there is no anomalous Hall effect in case when a magnetic field is applied along the wire? A. The spin polarization of electron gas is directed along an applied magnetic field. The defect induces an electric field around itself. An electron, which moves in this field, experiences the effective magnetic field of spinorbit interaction , which is directed either up or down, when for electrons scattered into the left or right. Because of different directions of the effective magnetic field of the spinorbit interaction, the probability of electron scattering toward left and right may be different and it this difference is the spindependent. In the case when the effective magnetic field is along the spin of electron, electron energy is larger. The energy difference for electrons with spin up and spin down is the reason for the spindependent scatterings. When the magnetic field is applied along the wire, the spin direction of the spinpolarized electrons (electrons of TIA assembly) is along the wire as well. The electrons of this spin direction have the same energy in magnetic field directed up and down. Therefore, the effective magnetic field of the spinorbit interaction, which is directed up and down, does not make scattering probabilities into the left and into the right to be different and scattering to be spindependent. This is the reason why there is no anomalous Hall effect when a magnetic field is applied along a wire.
Q. The electrical field around defect is distributed in all directions. Why there is difference of scattering probabilities only between left and right directions as it is shown in above Fig.? The scattering current may flow only if there is a difference in scattering probabilities into two opposite direction. The spinorbit interaction makes this difference. The effective field of the spinorbit interaction is directed up and down for electrons scattered into the left and into the right. Since the electrons with spin directed up and down have different energies in this field, their scattering probability is different. Similar, the electrons, which spin is directed left and right, have different scattering probabilities into up and down directions. Q. In above Figure, spinup electrons are scattered into the right and spindown electrons are scattered into the left. Why not in opposite direction? A. This direction is chosen as an example . In which direction electrons are scattered depends on a material. And specifically it depends whether the density of states in a metal increases or decreases with energy at the Fermi surface. When electron has been scattered, it experiences the electrical field at back side of the defect, which directed from front to back of the wire. This electrical field induces the effective magnetic field of the spinorbit interaction, which is directed upward when the electron moves to the left (See here) and it is directed downward when the electron moves to the right. In the case when the effective magnetic field is along the spin of electron, electron energy is larger. When the density of states in a metal increases with increasing electron energy, the scattering probability is larger when electron spin is along the effective magnetic field. Therefore, the case shown in above Figure corresponds to a metal, in which the density of states decreases with energy. Q. In Figure 1 hit it is only shown that only the electrons contribute into the effects. Is there any contribution of holes into the effects?? The holes (electrons of the energy lower than the Fermi energy) contributes nearly equally as the electrons (electrons of the energy higher than the Fermi energy). Importantly, the polarity of the hole and electron contributions are the same. This is the reason why the spin Hall effect, anomalous Hall effect and inverse spin Hall effect are effects with a large magnitude. It is similar to the case of the ordinary Hall effect
Why do scatterings become spindependent?
Notes: 1. Scattering can be spindependent on both the magnetized and nonmagnetized defect. 2. In electrical field around a defect an electron experiences the spinorbit interaction, which makes the electron scattering on the defect spindependent.
When an electron is scattered on a defect, behind the defect the electrical field of the defect is directed along the electron movement. A moving electron experiences the effective magnetic field of the spinorbit interaction only when electron moves across an electrical field. When the electron is scattered towards left, the effective magnetic field of the spinorbit interaction Hso is directed up and along the electron spin. Therefore, the energy the scattered electron becomes smaller. When the electron is scattered towards right, the Hso is directed down and opposite to the electron spin. Therefore, the energy the scattered electron becomes larger. As can be seen from Figure, for scattering towards left there are many unoccupied quantum states, therefore the scattering probability towards left is higher. In the contrast, there are almost no unoccupied quantum states for scatterings toward right, therefore the scattering probability towards right is lower.
Anisotropic magnetoresistance (AMR)
wiki page is here and classic review paper on the AMR is T.R. Mcguire and R.I. Potter, IEEE Trans. Magn. (1975)The effect of the anisotropy magnetoresistance describes the fact the resistance of a ferromagnetic metal becomes smaller, when the magnetization of the metal changes from parallel to perpendicular direction in the respect to the direction of the electrical current flowing in the metal. Spin dependent scatterings originate this effect.
The magnitude of the AMR effect is linearly proportional to the spinpolarization of the electron gas in a ferromagnetic metal.
Q. What is the difference between cases when magnetic field is applied along electron current or perpendicular to current? Spin polarization of electron gas is along to the applied magnetic field. When magnetic field is perpendicular to the electron current, the scattering is spindependent and scattering probabilities towards left and right are different. When magnetic field is along the electron current, the scattering becomes spinindependent.
Q. Why the resistance of a metallic wire becomes larger when a magnetic field changes from perpendicular to parallel directions with respect to the direction of electrical current in the wire? When magnetic field is perpendicular to electron movement, the scattering probability toward left and right are different. In contrast when magnetic field is along to electron movement the scattering probability toward left and right are same. It is important that the total scattering probability are different for perpendicular and parallel directions of the magnetic field, because of the nonlinear nature the FermiDirac distribution. Usually and the metal resistivity is larger when magnetic field is along the electron current. However, in some rare cases when there is a substantial energy dependence of the density of state at the Fermi surfers, the resistivity may be larger for a perpendicular magnetic field.
Q. The electrons in a metal move in all directions, why only electrons, which move along the electron current, contribute to the AMR effect? A. All electrons contribute to the AMR effect. The contribution of the electrons, which move in the opposite directions, are of opposite signs. Along electron current the number of electrons, which moves along and opposite current, are different and their contributions do not compensate each other.
Q. Is any correlations between the AMR effect and Anomalous Hall effect, the Spin Hall effect and Inverse Spin Hall effect? A. There is some correlation, but it is not very straight and obvious. All effects are originated from the spindependent scatterings. However, if the Anomalous Hall effect, the Spin Hall effect and Inverse Spin Hall effect occur because of a difference of scattering probabilities towards left and right, the AMR effect occurs because average probability for both scattering into left and right changes as well.
Unresolved issueIt is still unclear relation between magnitude of the AMR and magnitude of AHE & OHE. For discussion on experimental data on this topic See T.R. Mcguire and R.I. Potter, IEEE Trans. Magn. (1975).

I will try to answer your questions as soon as possible