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 describes the fact that when an electrical current flows in a ferromagnetic metallic wire, an electrical current flows perpendicularly to the wire due to the magnetization of the ferromagnetic wire. The effect is originated from a magnetic interaction of conduction electrons and localized electrons (mainly spin orbit interaction).Note: All data of this page represent only my personal point of view, which are based on my experimental and theoretical research. The description of the common origin of the Spin Hall effect and the Inverse Spin Hall Hall, which is based on features of spin dependent scatterings induced by the magnetic field H_{SO}, I have developed in 20142019. I have realized that there are two substantial contribution to the Hall effect in a ferromagnetic metal (AHE + ISHE) in 2019.Contentclick on the chapter for the shortcut(1). Three contributions to the Hall effect(1a) (1st contribution): Ordinary Hall effect (OHE)(1b) (2nd contribution): Anomalous Hall effect (AHE)(1c) (3rd contribution): Inverse Spin Hall effect (ISHE)(2). History of different views on origin of AHE .(3) AHE Origin. Symmetry(3b) AHE Origin. Uniqueness(4) Origin of AHE: Direction depend scattering of conduction electrons due direction dependence of their orbital moment(4a). Possible Origin of AHE: Spin orbit interaction(4b). Possible Origin of AHE. spd exchange interaction(5). Measurement of the OHE, AHE and ISHE Hall effects. The Hall angle α_{Hall}_{}(6).Hall effect in nonmagnetic metal under spininjection. Experiment(6a) Spin injection from ferromagnetic metal to nonmagnetic metal. Experiment(6b). Hall effect induced by photo excited spin polarized current. Experiment(7) spM type Hall effect(8). Anisotropic magnetoresistance (AMR)(9) Why do scatterings become spindependent?(10). Fitting of experimental and theoretical data(12). Questions & Answers.........
Three contributions to the Hall effect
The general Hall effect describes the fact that when an electrical current flows in a ferromagnetic metallic wire, an electrical flows perpendicularly to the wire either due to a magnetic field H or/and due to the magnetic moments of localized electrons (magnetization M) or/due to the magnetic moments of conduction electrons (which are described by the spin polarization)(1st contribution): (Ordinary Hall effect) due to magnetic field H inside of the metallic wire (origin of effect): The Lorentz force, which conduction electrons experience when moving in a magnetic field (See details origin of the Ordinary Hall effect here) (proportionality to external magnetic field): The Hall angle α_{OHE} linearly depends on external magnetic field H. (affected group of conduction electrons): all conduction electrons (spin polarized + spin unpolarized) (strength of contribution): weak . E.g. α_{OHE}~ 0.20.8 mdeg/kG for FeCoB nanomagnets, which I have studied 
(2nd contribution):(Anomalous Hall effect) due to magnetic moments of localized electrons or magnetization M (origin of effect): spinorbit interaction (plus maybe the exchange interaction???). The origin is not fully clear yet. (proportionality to external magnetic field): The Hall angle α_{AHE} does not depends on external magnetic field H. It is a constant vs H. (affected group of conduction electrons): all conduction electrons (spin polarized + spin unpolarized) (strength of contribution): strong . E.g. α_{AHE}~ 3001500 mdeg for FeCoB nanomagnets, which I have studied  (3d contribution): (Inverse Spin Hall effect) due to magnetic moments of conduction electrons or the spin polarization of conduction electrons (origin of effect): Spinorbit interaction. The Inverse Spin Hall effect (ISHE) describes the fact that a current of spin polarized electrons creates an electrical current, which flows perpendicularly to the spin current. Conduction electrons in a ferromagnetic metal are spin polarized. Asa result, there is a charge current perpendicular to a current of spin polarized conduction electrons, which flows along a ferromagnetic wire. See details about Spin Hall effect and ISHE here. (proportionality to external magnetic field): The Hall angle α_{ISHE} non linearly depends on external magnetic field H. (See details here). (affected group of conduction electrons): only spinpolarized electrons!!! spin unpolarized conduction electrons do not contribute to this effect (See here) (strength of contribution): moderate . E.g. α_{ISHE}~ 530 mdeg for FeCoB nanomagnets, which I have studied 
History of different views on origin of AHE .(Origin 1): Spin orbit interaction (SO):
All following origins, which are based on the SO interaction, are proposed and calculated for spinpolarized electrons. Therefore, they describe the Inverse Spin Hall effect (ISHE), but not AHE. The AHE is due to spins of localized electrons, but not spins of conduction electrons !!!(1st proposal) of the SO as the origin of AHE: Karplus,Luttinger (1954). They used the same formalism as the formalism of the Ordinary Hall effect, but replacing the Lorentz force by the SO. It an incorrect approach since the SO cannot break the timeinverse symmetry and cannot contribute in the similar way. The used approximation was too rough. The error was correctly noticed by Smit (1955), but the error was not accepted by Luttinger (1958). (2d proposal): skew scatterings: Smit (1955), Smit(1958) It correctly describes the fact that the SO makes the scattering probability dependent on the electron movement direction and electron spin. See my explanation on the skew scatterings here.(3d proposal): sidejump scatterings Berger (1970) It correctly describes the fact that the SO makes the scattering probability different on whether a scattered conduction electron shifted to left or to the right with respect to its initial position See my explanation on the sidejump scatterings here.
 (Origin 2): spd exchange interaction: This origin is proportional to the spins of the localized electrons (magnetization). Therefore, the Origin 2 is the origin of AHE, but is not the origin of ISHE (as in the case of Origin 1) !(1st proposal) by Kondo (1962), which is based on 3 assumptions: assumption 1: AHE is originated from the spd exchange interaction, but not from the spin orbit interaction assumption 2: the spd exchange interaction depends on velocity (wave vector) of a conduction electron. assumption 3 (toughest) : the spd exchange interaction is different for opposite movement directions of a conduction electron (2nd modification) by Giovannini (1973) An additional contribution to the spd exchange interaction was assumed in order to explain why the spd exchange interaction is different for opposite movement directions of a conduction electron  There were many science fiction proposals on the origin of AHE. One example is that the AHE is originated by the super conductivity. Such proposals are clearly unrealistic and they will not be discussed here Origin of Anomalous Hall effect. Symmetry
Asymmetry α_{Hall} with respect to reversal of magnetic field H and magnetization M:The Hall and the Hall angle is measured by a pair of electrodes (See Fig.). Even a slight spacial misalignment of electrodes with respect to each other causes a voltage between electrodes due to the voltage gradient along the wire. This voltage is independent on H and M. In contrast, the Hall voltage and the Hall angle α_{Hall} is asymmetrical with respect to H reversal and M reversal. Therefore, it defines as a difference of α_{Hall} measured at opposite H: From Eq. (3.1), the measured Hall angle can be only proportional to material parameters, which reverse their sign when the H is reversed. Additionally to H itself, there are two more such parameters. The first parameter is the total magnetic moment M_{d} of localized delectrons. The second parameter is the total magnetic moment M_{cond} of conduction electrons. Therefore, the Hall angle α_{Hall} can be calculated as where the 1st term β_{OHE}·H describes proportionality of α_{Hall} to external magnetic field ( ordinary Hall effect.); the 2nd term β_{AHE}·M describes proportionality of α_{Hall} to magnetization of the ferromagnetic metal (Anomalous Hall effect); the 3d term β_{ISHE}· describes proportionality of α_{Hall} to the spin polarization of the conduction electrons ( Inverse Spin Hall effect) (symmetry 1): The Hall angle α_{AHE} due Anomalous Hall effect (AHE) is proportional to the magnetization direction:Since the magnetization is defined as the total spin of localized electrons , the AHE should be proportional to magnetic properties of localized electrons (symmetry 2): The Hall voltage is linearly proportional to the electrical current. The Hall voltage increases as the electrical current increasesSince the electrical current is a flow of conduction electrons, the AHE should be proportional to magnetic properties of conduction electrons
Symmetry Forbidden relations for Hall angle α_{Hall}α_{Hall}, magnetization M_{d} (total spin of localized electrons), M_{cond} (total spin of conduction electrons) have the same polarity. They all change their sign when a large external magnetic field H is reversed. As a result, both the M_{d} , M_{cond} are reversed and the following relations are forbidden: α_{Hall} ≠k · (M_{d}) ^{2} >α_{Hall} cannot be linearly proportional to the square of magnetizationα_{Hall} ≠k · (M_{cond}) ^{2}α_{Hall} cannot be linearly proportional to the square of spin polarizationα_{Hall} ≠ k ·M_{d} ·M_{cond}α_{Hall} cannot be linearly proportional to the product of magnetization and spin polarizationwhere k is a constant independent on H. M_{cond} is linearly proportional to the number of spin polarized electrons or spin polarization sp E.g. in the case if α_{Hall} is linearly proportional to the magnetization M_{d} , the α_{Hall} cannot linearly depend on the magnetization M_{cond} of conduction electrons (and therefore the number of spin polarized electrons or the spin polarization). It means that the AHE is induced by all conduction electrons (not only spinpolarized electrons). The Hall effect, which is induced only by spin polarized electrons is called the Inverse Spin Hall effect (ISHE). Details about ISHE are here.Origin of Anomalous Hall effect. UniquenessThe Uniqueness of the Anomalous Hall effect (AHE) is that is determined by the magnetic properties of two different groups of electrons: the group of conduction electrons and group of localized d electrons. As a result, the AHE should be originated from a magnetic interaction between localized d electrons and conduction electrons. There are three possible interactions between localized and conduction electrons: (interaction 1 between localized and conduction electrons): Magneto static interaction (interaction 2 between localized and conduction electrons): spd exchange interaction (interaction 3 between localized and conduction electrons): Spinorbit interaction
Origin of Anomalous Hall effect. Direction depend scattering of conduction electrons due direction dependence of their orbital momentZayets 2020.03
Origin of AHE: Dependence of electron scattering probability on spins of localized electron and the scattering direction. Explanation in short:A conduction electron has a orbital moment, which depends on a movement direction and position of the electron. Magnetic field, which created by the spin of localized electrons, interacts with the orbital moment of a conduction electron. The energy is higher when the spin M of localized electron is along the orbital moment of conduction electron and the energy is smaller when M is opposite to the orbital moment. The energy difference makes the scattering probability dependent on M and scattering direction, because of the directional dependence of the orbital moment. The difference in the scattering probabilities creates an electron current (Hall current) perpendicularly to the bias current.
Origin of AHE. How the spin of localized d electrons creates an electrical current flowing perpendicularly to a ferromagnetic wire. Step by step explanation (step 1) Why electrical current flow across a ferromagnetic wire?The scattering probability of a conduction electron is different to the left and to the right direction. As a result, more electrons are scattered in one direction and an electrical current flows from the left to the right. (step 2) Why the scattering probabilities to the left and to the right are different for a conduction electron?The scattering probability is different because the magnetic energy of a conduction electron is different when its moves to the left and the right directions.. (step 3) Why does the magnetic energy of a conduction electron depend on its movement direction?The magnetic energy is proportional to the magnetic field, which is induced by the spins of the localized d electrons, and the magnetic moment of a conduction electron and therefore the orbital moment of the conduction electron. The orbital moment of a conduction electron is different for its opposite movement direction (step 4) Why does the orbital moment of a conduction electron depend on the electron movement direction?It is an unique feature of conduction electrons. Details see here.
Enhancement of AHE effect due to spinorbit interaction
Possible Origin of Anomalous Hall effect. Spin orbit interaction
Even though the spin orbit interaction and spindependent scattering has been considered as the possible main origin of AHE for about 70 years (See AHE history), as 2020.03 I do see the AHE mechanism, which could be related to the spin dependent scatterings. The spin dependent scatterings originate another type of Hall effect: Inverse Spin Hall effect (ISHE), but not AHE. The spin orbit interaction enhances the AHE, but not originates it (See here)(problem of Spin orbit interaction as the origin of AHE): The Spin Orbit interaction does induce the Hall effect, but this Hall effect is proportional to the number of the spinpolarized conduction electrons and such effect is called the Inverse Spin Hall effect. The Hall effect due to AHE should be proportional to the total number of the conduction electrons (See forbidden relations here). (note) The Hall effect due to the spin orbit interaction is simple and understandable (see here) , but it is proportional to the number of spin polarized electrons and therefore called the Inverse Spin Hall effect. The Anomalous Hall effect (AHE) is proportional to the number of all conduction electrons and the magnetization (the total spin of localized electrons)Hall effects due to the Spin orbit interaction: (source 1): due to a nonzero orbital moment of conduction electrons; (source 2): Skew scatterings (source 3): Sidejump scatterings at defect (source 4): Sidejump scatterings across an interface. All these origin are due to spindependent scatterings and and they are proportional to the number of spin polarized electrons. Origin of the Inverse Spin Hall effect (in short)ISHE is due to the Spin Orbit Interaction and is proportional to the number of spin polarized conduction electrons(step 1) The conduction electrons experience the magnetic field H_{SO} of spin orbit interaction (See here) due to several possible sources: (source 1): A nonzero orbital moment of a conduction electron (Details are here); (source 2): Skew scatterings (Details are here); (source 3): Side jump scatterings in electrical field of a defect (Details are here); (source 4): Side jump scatterings across an interface (Details are here); . Important: Each source makes the direction, polarity and magnitude of magnetic field H_{SO} dependent on either moving direction (k_{x}, k_{y}, k_{z}) or spacial coordinates (x, y, z) of a conduction electron (step 2) Due to the magnetic interaction of the magnetic field H_{SO} with spin of a conduction electrons, the scatterings of conduction electrons becomes direction and spin dependent. The scattering probability is larger when the H_{SO} is along the spin of a conduction electron and it is smaller when the H_{SO} is opposite to the electron spin. E.g. the spinup electrons are scattered more to the left and the spinup electrons are scattered more to the right. (step 3) The difference in scatterings probabilities creates an electrical current flowing perpendicularly to the main electrical current. Only spin polarized conduction electrons contribute to this current. E.g. the spin direction of spin polarized electrons is up. Since there are more spinup electrons scattered to the left than to the right, the electron current (charge current) flows into the left direction. Therefore, only spin polarized electrons contribute to the Hall effect. Note: (about influence of spin dependent scatterings on spin unpolarized conduction electrons and the Spin Hall effect). In spins of the spin unpolarized conduction electrons are distributed equally in all directions (See here). Due to the spin dependent scatterings, more spinup electrons are scattered to the left (for example) and more spin down electrons are scattered to the right. However, in the group of spin unpolarized electrons there are equal amounts of spinup and spin down electrons. As a result, the spin dependent scatterings of spin unpolarized electrons induce no charge current from the left to the right, but only the spin current . This effect is called the Spin Hall effectYou have divided the Hall effect in a ferromagnetic metal into two contributions: the first AHE contribution due to the total spin M_{d} of localized electrons and the second ISHE contribution due to the total spin M_{cond} of conduction electrons. However, the conduction electrons in a ferromagnet metal are spin polarized, because of their magnetic interaction with localized electrons. Therefore, M_{cond} is proportional to M_{d} and both the AHE and ISHE can be combined in one Hall effect, which is only proportional to M_{d}? Do we really need to split the Hall effect in a ferromagnetic metal into the AHE part and ISHE part?Yes. You are correct. The conduction electrons are spin polarized, because the spins of the localized d electrons are aligned in one direction. The mechanism of spin alignment of conduction electrons is called the spin pumping (See here). There are two mechanisms, which aligns the spins of conduction electrons along the spins of localized electrons. (mechanism 1 (usually weaker) ) the exchange interaction between conduction and localized electrons (spd exchange interaction); (mechanism 2 (usually stronger) ) the scattering between localized and conduction electrons. The localized electrons, which spins are aligned in one direction, are constantly scattered into the gas of conduction electrons. Therefore, they are constantly injecting the spin into the group of the conduction electrons. Therefore, it is true that the total spin of conduction electrons is proportional to the total spin of localized electrons and the total spin of conduction electrons is aligned to the spins of localized electrons. However, the localized and conduction electrons are two very different groups of electrons (see here) and their total spins are two independent magnetic parameters. For example, the number of spin polarized conduction electrons can be changed without affecting the localized electrons (the spin injection etc.). In fact, the individual contributions of the AHE and ISHE are measured by changing the number of spin polarized electrons without influence of the localized electrons. The part of Hall angle, which is changing, is the ISHE contribution and the part of Hall angle, which is staying a constant, is the AHE contribution. The spin polarization of conduction electrons can be changed by a spin injection from a neighbor region or by illumination by a circularly polarized light. The simplest method is the applying an external magnetic field along the magnetization of a nanomagnet. The conduction electrons are aligned along the magnetic field, therefore the spin polarization increases. The spins of localized electron initially aligned along magnetic field, therefore spins of localized electron are not affected by the magnetic field (almost). More details about this measurements are herefinal answer: the Hall effect in a ferromagnetic metal independently proportional to the total total spin of localized electrons and the total spin of conduction electrons. As a result, the Hall effect is the sum of the AHE contribution and the ISHE contribution. Each of contribution should be measured individually. (conclusion) Skew scatterings; Sidejump scatterings at defect ; Sidejump scatterings across an interface. are contribute to the Hall effect, but they contribute to the ISHE effect, but not to the AHE effect. It is because these contributions are proportional to the number of spin polarized conduction electrons, but not to the total number of conduction electrons as it should be in the case of AHE effect.Possible Origin of Anomalous Hall effect. spd exchange interaction
Even though the spd exchange interaction has been considered as the possible main origin of AHE for about 70 years (See AHE history), as 2020.03 I do see the AHE mechanism, which could be related to the spd exchange interaction. The spd exchange interaction originates another type of Hall effect: spM type Hall effect , but no AHE. The contribution from the spM type Hall effect is rather small. In contrast to other mechanisms of Hall effect, the spM mechanism does not have a hysteresis loop (See here)(problem of spd exchange interaction as the origin of AHE): In order for the spd exchange interaction to be the AHE origin, the exchange interaction between the localized d electrons and conduction electrons should have some unique properties. E.g. the strength of the exchange interaction should change significantly when the movement direction of a conduction electron is reversed. The exchange interaction should not depend on the spin of the conduction electron. The physical mechanisms, which might cause these unique properties, are still unclear. In order for the spd exchange several requirements should be satisfied (requirement 1) The overlap between wave functions of a conduction electron, which is moved along the metal, and a localized electron, which stays at the same place, should be a constant intime.(possible solution): Simultaneously with the movement along the metal, the conduction electron rotates around many atomic nuclears. The rotation is described by the Bloch function. The rotation is local and is stationary (does not move along metal). The wavefunction overlap of this stationary part with the nonmoving localized d electron determines the exchange interaction. Even though the magnitude of this nonmoving stationary wavefunction part of the conduction electron is changing in time, in the average over many localized electrons the overlap does not change in time (see Fig.11) and the spd exchange interaction is a nonzero and does not change in time (possible problem): Additionally to magnitude, the phase of a conduction electrons is continuously changing as the electron moves along the metal. The exchange interaction does depend on the phase. When averaged over the phase, the strength of the averaged exchange interaction might be small (requirement 2) In order to contribute to the AHE, the spd exchange interaction should change significantly its strength when the moving speed of the conduction electron is changed. However, the exchange interaction depends on the stationary part, which is not moving. So why it should depend on the electron moving speed?(possible solution): The symmetry of the stationary part depends on the electron speed. This fact is described by the kp model (Luttinger, Kohn (1955)) and is well verified in semiconductors. The strength of the exchange interaction does depend on the symmetry of the stationary part (Bloch function) of the conduction electron and therefore on its speed. (possible problem): The change of the symmetry vs the electron speed is very small. As a result, the change of strength of the exchange interaction is small as well. It is difficult to explain the strong AHE effect by this small change of the exchange interaction (requirement 3) In order to contribute to the AHE, the spd exchange interaction should change significantly when the electron movement direction is reversed. The standard kp model cannot explain such property.(possible solution): Both Kondo (1962) and Giovannini (1973) have used some mathematical tricks in order to include such unexpected dependence into the spd exchange interaction. It is unclear how realistically these mathematical tricks correspond to the reality. (possible problem): The exchange interaction is one of most complex effects of the nature. At present, not all its features are fully understood (See here) (requirement 4) In order to contribute to the AHE, the spd exchange interaction should not depend on the spins of conduction electrons. It should depend only on spins of the localized electrons. It wellknown and well verified fact that the exchange interaction strongly depends on mutual spin directions of two interacting electrons.(possible solution): The spins of spin unpolarized electrons is distributed equally in all directions. The dependence of exchange interaction on the spin of the conduction electrons might be averaged out over such distribution. (possible problem): The spin polarized electrons contribute to ISHE (not to AHE)? (Conclusion) At present, it is difficult to clarify whether the spd exchange interaction is the main origin of AHE. There are still too many unanswered questions
Measurement of the OHE, AHE and ISHE 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. or the intristic Hall angle α_{Hall,ferro} in a ferromagnetic metal is calculated from the measured Hall angle α_{Hall,measured} in a double layer nanowire as
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
OHE effect + ISHE effect without AHE effect. Spin injection in nonmagnetic metal(fact) In a nonmagnetic metal there is only one type of Hall effect: OHE. There are no AHE effect (since there are no localized d electrons) and no ISHE effect (since conduction electrons are not spin polarized)Why a Hall measurement in a nonmagnetic conductor under spin injection is important?(importance 1): It is possible to separate contributions to Hall effect from localized and conduction electrons. For about 70 years it has been believed that there is only one contribution to the Hall effect in a ferromagnetic metal (the AHE contribution) and both the conduction and localized electrons jointly contribute to the AHE effect. The experimental observation of the Hall effect in a non magnetic conductor under the spin injection clearly indicates that each localized and conduction electrons contribute individually to the Hall effect and the contributions are very different from each other ( dependence on an external magnetic field etc.) (importance 2): Since the Hall effect (the ISHE) exists in a nonmagnetic metal, which does not have a localized electrons with aligned spins, it clearly indicates that the existence of Hall effect (additional to OHE) does not require the existence of localized electrons. The conduction electrons by themselves are able to produce the Hall effect. (importance 3 (main)): The substantial difference in contributions into the Hall effect from localized and conduction electrons clearly indicates that the spin distributions of the conduction and localized electrons are very different. The spin distribution in a ferromagnetic metal is the classical spinup/ spin down distribution. In contrast, the spin distribution of conduction electrons is the sum of two distributions of groups of spin polarized and spin unpolarized electrons. (Details see here)
Hall effect in nonmagnetic metal under spininjection When spin polarized conduction electrons are injected into a nonmagnetic metal, the ISHE effect exists additionally to the OHE effect.
Why this study is interesting?A. The properties of the ISHE effect without disturbance of the AHE effect can be clarified and studied.. There are no localized d electrons in a nonmagnetic metal. As a result, the AHE effect does not exists in a nonmagnetic metal at any conditions. (one exception is an interface with a ferromagnetic metal) The conduction electrons are not spin polarized in a ferromagnetic metal and there is no ISHE effect in an equilibrium. However, the spinpolarized can be injected in a nonmagnetic metal. In this case, the ISHE effect starts to exist in the non magnetic metal (Note) The ISHE effect in a ferromagnetic metal is strong, but the similar AHE effect also exists in the ferromagnetic metal and it is difficult to separate the AHE and the ISHE effects (See details here). As a result, it is difficult to study features of the ISHE in the ferromagnetic metal, because they can be originated from ISHE.(experiment 1) Spin injection from a ferromagnetic metal + Hall measurements in a nonmagnetic metal this experiment I did in 20162018 in Au: FeTbB samples( Main idea): To inject spin polarized electrons from a ferromagnetic metal into a nonmagnetic metal, while measuring the Hall effect in the nonmagnetic metal. To use the ferromagnetic and nonmagnetic metals with opposite polarity of the ISHE. (Main challenge): The Hall effect in the ferromagnetic metal should not contribute to the measured the Hall angle (solution 1): To use the ferromagnetic and nonmagnetic metals with opposite polarity of the ISHE. As a result, the contributions from each metal can be distinguished by the polarity of the hysteresis loop. (solution 2): the use of the ferromagnetic metal with a small conductivity and the non magnetic metal with a high conductivity. As a result, nearall current flows in the non magnetic metal, nearly no current flows ferromagnetic metal and therefore the main contribution to the Hall angle would be from the non magnetic metal and only a little contribution would be from ferromagnetic metal.
(possibility 2). Classical spin injection. Spin injection from top FeTbB electrode In this case, the electrical voltage is applied between the FeTbB strips and the Au nanowire. There is a spin injection from the FeTbB
(experiment 2) Spin injection into a semiconductor by a circular polarized light + Hall measurements in the semiconductor
See a similar experiment: Wunderlich et.al. Nat. Phys. (2009)( Main idea): The circular polarized light creates spin polarized conduction electrons in a non magnetic semiconductor. As a result, the the ISHE effect starts to exist under illuminations of a circular polarized light.
5 types of the Hall effect:
(type 1 of Hall effect): Anomalous Hall effect (occurs only in a ferromagnetic metal) 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 electrical current. . (type 2 of Hall effect): Inverse Spin Hall effect (may occur in both a ferromagnetic metal and a nonmagnetic metal) The Inverse Spin Hall effect describes the fact that that a change can be accumulated at the edges of a metallic wire when a magnetic field is applied perpendicularly to the electrical current. (type 3 of Hall effect): spM Hall effect (occurs only in a ferromagnetic metal) The spM Hall effect occurs due to the interaction between the spin of conduction electrons and spins of localized electrons. (type 4 of Hall effect): Ordinary Hall effect (occurs in both a ferromagnetic metal and a nonmagnetic metal) The Ordinary Hall effect occurs due to the Lorentz force (change of electron movement direction in a magnetic field) (type 5 of Hall effect ???): 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.
Q. The spin Hall effect (SHE), anomalous Hall effect(AHE) and inverse spin Hall (ISHE) effect are very similar. How to distinguish between these effects? A. Q. Since all effects: SHE, AHE and ISHE are related to scatterings, does it mean that 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 having 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). In the case when the density of defect is very large, the electron wave function overlaps several defect simultaneously, it would make all side jump scatterings spinindependent and they would not contribute to the spin Hall effect, and inverse spin Hall. Other important condition for existence of spindependent side jump scatterings is that the electron meanfree path should be shorter than the effective radius of electrical field around a defect.
Anomalous Hall effect Q. For the anomalous Hall effect, is it essential for electron gas to be spinpolarized? A. Q. Why there are no anomalous Hall effect and no Inverse Spin Hall effect in case when the magnetization and spin polarization is inplane and along the main current (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? The role of the spin orbit interaction
Notes:
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.
spM type Hall effect
It is a join contribution of spins of localized and conduction electrons
The Hall angle α_{Hall,spM} from this contribution is proportional to both the magnetization M_{d} of localized electrons and the magnetization M_{cond} of conduction electrons α_{Hall,spM} ~ M_{d} · M _{cond} Since the magnetization M_{cond} of conduction electrons is proportional to the number of spin polarized electrons and therefore to the spin polarization, α_{Hall,spM} is calculated as α_{Hall,spM} ~ M_{d} · sp where sp is the spin polarization of the conduction electrons
Except a few exceptions, the direction of the spin polarization (sp) of spin polarized conduction electrons is along the spin direction of localized electrons (magnetization). As a result, α_{Hall,spM} (~ M_{d} · sp) does not on direction of M and therefore does not have a hysteresis loop.
In contrast to any other types of Hall effect, the spM type Hall effect does not have a hysteresis loop. α_{Hall,spM} is only slightly depends on an external magnetic field H due to the dependence of spin polarization sp on H
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.
AMR effect due to orbital moment of conduction electrons
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).
Fitting of experimental and theoretical data
I am very surprised that any theoretical model for the origin of the AMR, which has been proposed for the last 70 years, always has a "perfect" fit with with the experimental data. From the present view it is clear that each of the proposed model describes either only one of many contributions to the AMR and/or to the ISHE or completely incorrect.
I am not surprised about the present time (2000 2020), when there is a burst of a fake and "highlight" research and when publishing of "anything" in a research paper is possible. I am surprised about the time of 19601970, when the research was flourished and there were many excellent ideas, models and research results. Still every researcher had to show a "perfect" fit of theory and experiment, even in the cases when it was very clear that the full understanding of the effect is still far away. For example, Smit (1955), (1958) has described correctly the skew scattering mechanism and Berger (1970) has described correctly the sidejump scattering mechanism having a very poor and primitive tools. There were no computers. The understanding of the spin orbit interaction was poor and primitive. Only mathematical method, they have used, was the minimization of Hamiltonian (which itself was not fully correct). It is not the best tool to study the features of the spin dependent scatterings. Even the term "Inverse Spin Hall effect" did not exists at that time. Therefore, the Hall effect was not divided into the AHE and ISHE contributions. Still they have described correctly and precisely the main essence and the main tendencies of effects. They are very amazing and talented scientists!
Questions & Answers
I am strongly against a fake and "highlight" research

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