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)
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 orbital moment on conduction electrons with spin of localized localized electrons.
Note: All data of this page represent only my personal point of view, which are based on my experimental and theoretical research.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}(7) Families of linear Hall effects & hysteresis loop(8) Why do scatterings become spindependent?() AHE and hot electrons(9). Fitting of experimental and theoretical data(10). Questions & Answers(q1) Planar Hall/ AMR vs. AHE(q3) How to distinguish Planar Hall effect from Anomalous Hall effect experimentally?(q4) about validity of the Barry Phase formalism(q5) about the importance of the scatterings for the charge and spin transport(q6) about relation between spd interaction and AHE() about influence of standingwave electrons on AHE due to defects and interfaces() standingwave electrons & their influence on the AHE() Origin of the standing wave electrons() an interface as the origin of the standingwave electrons() why the standingwave electrons are important for AHE and ISHE?() About "full" states or spininactive states() increase of standing wave electrons at a lower energy. The hole transport() standingwave electron vs localized electron() timescale of AHE; AHE in picosecond timescale() fast nonequilibrium effects: hot electrons() saturation velocity: the fastest speed of the electrons in a solid() definition of the photoinduced Hall effects() timescale of the Kerr and Faraday effects
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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
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.
Families of linear Hall effects & hysteresis loop
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
AHE and hot electronsThe polarity of the Hall voltage due to the hot electrons is usually opposite to that of the equilibrium electrons. The conductivity of Fe and Co is holedominated. The conductivity of the hot electrons is the electrontype. Therefore, the polarity of the Hall voltage is different for these two types of conductivity. The hot electrons can be excited in a metal or semiconductor by a light pulse or due to an electron current above a small tunnel barrier or contact barrier
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 & AnswersIs there any clear or evident relation between AMR/PHE and AHE? In my experience materials with strong AMR/PHE not necessarily exhibit strong AHE and viceversa. But since they are both "magnetization angle dependent phenomena" I was just wondering if from a fundamental point of view they could be related, or if some reasonable theory relates them.There is no relation between AMR/PHE and AHE. From an experiment, this fact is known for a while. See, for example, T.R. Mcguire and R.I. Potter, IEEE Trans. Magn. (1975). From theory point of view, the AMR/PHE and AHE have different symmetries and different physical origins. Therefore, they are two very different effects. Difference 1: difference in the symmetry The AHE is a linear magnetotransport effect. The AMR/PHE is a second order magnetotransport effect. In a nanomagnet there are 3 independent variables, which timeinverse symmetry is broken: (1) externallyapplied magnetic field H; (2) the total spin Sd of localized d electrons (or the magnetization M) and (3) the total spin Scond of the spinpolarized conduction electrons (or the spin polarization). A linear magnetotransport effect is linearly proportional either to H or Sd or Scond. The ordinary Hall effect is proportional to H. The AHE is linearly proportional to Sd. The inverse spin Hall effect (ISHE) is linearly proportional to Scond. A 2nd order magnetotransport effect is proportional to a product of a pair from H, Sd and Scond. Additionally to the AMR/PHE, the inplane GMR is also a 2nd order magnetotransport effect. Difference 2: difference in the physical origin. The AHE is dependent only on the magnetization (Sd) and is independent of spin polarization of the conduction electrons (Scond). In contrast, the AMR/PHE depends on both the magnetization and the spin polarization. The origin of the AHE is the spindependent scatterings of conduction electrons, which depend on the spin of a d electron, but is irrelevant to the spin of a conduction electrons. The origin of the AMR/PHE is also spindependent scatterings of conduction electrons, but of different type, which depend on the angle between spin of a d electron and the spin of a conduction electron.
How to distinguish Planar Hall effect from Anomalous Hall effect experimentally??A 1st order magnetotransport effect (Anomalous Hall effect, Inverse Spin Hall effect & Ordinary Hall effect) can be easily distinguished experimentally from a 2nd order magentotransport effect (AMR/PHE, inplane GMR etc.). Since the 1st order magnetotransport effect is linearly proportional to magnetization M + external magnetic field H, it reverses its polarity when H+M are reversed. In contrast, the 2nd order magentotransport effect is proportional to a square/product of magnetization M + external magnetic field H, it does not reverse its polarity when H+M are reversed. It is a common rule for any magnetotransport measurement that two measurements are always done with the magnetization in the forward and reversed direction. Next, the symmetric and antisymmetric contributions of measurements are calculated. The anisymmerical contribution is associated with the 1st order magnetotransport effects and the symmerical contribution is associated with the 2st order magnetotransport effects. In this way any unwanted contribution of 1st order magnetotransport effects to a measurement of a 2st order magnetotransport effect can be avoided and vice versa. As an example see my AMR/PHE measurement for nanomagnets here. (about Barry phase, ) Why origin of AHE in terms of barry phase is not discussed here, in some articles it has been said the Barry phase originates AHE and skew and side jump scattering further increases the effect ? Here you have written that spd interaction to be the origin of AHE which has made me confused on what is correct ? please clarify?(about validity of the Barry Phase formalism) The formalism of the Barry Phase is valid and can be used only at a very low temperature (e.g. below 1K), when the electron mean free path (or the electron coherence length) is sufficiently long. For example, the formalism of the Barry Phase is useful for the description of the Quantum Hall effect and superconductivity. However, in the case of a metal or a semiconductor at T>1K the electron mean free path becomes shorter, the contribution of the spindependent scattering to AHE, ISHE, AMR/PHE dominates and the contributions related to a long mean free path are negligibly small. For example, at room T the meanfree path is about 1 nm10 nm in a metal and about 10100 nm in a semiconductor. ( about the importance of the scatterings for the charge and spin transport) The scatterings are a major engine mechanism, which creates transport itself (the superconductivity is an exception). There is no transport without scatterings (even in the case of coherent transport like tunneling) . It is no surprise that 90 % of transport properties are related to the scatterings. It is no surprise that properties of spinrelated transports like AHE, ISHE, AMR/PHE are determined by features of the spindependent scatterings, which are the skew and sidejump scatterings. (about spd interaction) The electron transport is determined by features of scatterings of the conduction electrons. The spin of localized electron influences these scatterings. (about influence of standingwave electrons on AHE due to defects and interfaces ) Due to defects and interfaces, a part of the conduction electrons becomes the standingwave electrons. As a result, the conduction of metal decreases, but the strength of the AHE increases. (from Szymon Królak) you frequently mention, that there exist two currents: the current of fullfilled states, as well as halffilled ones. However, during the explanation of the Hall effect, you state, that the fullfilled states form standing waves and therefore do not move. Could you elaborate on that? I am referring to the explanation below, figure 15: The current of "full" states is the normal current. This means that the negatively charged "full" states diffuse from a "" source toward a "+" drain. Due to the Hall effect, they should turn to the left (similar to the electrons shown in Fig. 14). However, most of the electrons, which occupy the "full" states, are the standingwave electron. (See Fig.9 here). The standingwave electrons do not move, therefore they do not contribute to the Hall current You are correct.
(about full state and standing wave electrons): The full state and the sdandingwave electrons are two different states of an electron. The full state is related to the electron spin and the standing wave electrons are related to the electron wave vector. (standingwave electrons & their influence on the Hall effect) (What is the electron current?) The electron current does not mean that the electron starts to move in one direction under applied voltage from "" to "+" potential or the electron speed in this direction is changed. Even in absence of the applied voltage, there are always electrons, which move in both the forward and the backward directions. An electron current means that there are more electrons, which are moving along applied voltage, than electrons, which are moving in the opposite direction. In absence of the voltage the number of electrons moving in any direction is exactly equal to the number of electrons moving in the opposite direction. The applied voltage makes a number of electrons moving in each direction different. In the direction along the applied voltage the number becomes slightly larger and in the opposite direction the number becomes slightly smaller. (Origin of the standing wave electrons): Next, just imagine, that there are two defects close to each other inside a metal, which reflects the electron backward as a mirror reflects light. Then, the electron bounces between these two defects forward and backward. Such a bounced electron is a sum of forward and backwardmoving electrons with exactly the same amplitude of the wave function. It means that the existence of these two defects firmly fixes the ratio between numbers of forward and backwardmoving electrons and the applied voltage does not change this ratio and the numbers of the forward and backwardmoving electrons are always equal independently of the applied voltage. Since the applied voltage does not change the ratio between the numbers of the forward and backward moving of these electrons, these electrons do not contribute to the electron current. These electrons, I call the standingwave electrons. In the above example, I assumed that the reflection of an electron from a defect is 100 %. In reality the reflection is much smaller. It is smaller than 1 %. As a result, only a small part of the conduction electrons are the standing wave electrons, for which the ratio of is fixed and which do not contribute to the electron current. For other conduction the ratio is not fixed and they do contribute to the electron current. (an interface as the origin of the standingwave electrons): A surface or an interface between two metals substantially reflects the conduction electrons. As a result, there are many standingwave electrons in the proximity of the interface and the conductivity near an interface is smaller than the conductivity in the bulk of the metal. Additionally, the conductivity near an interface is anisotropic, because the reflection of an electron from the interface depends on the electron incident angle. (Why should we care about the standing wave electrons and why the standingwave electrons are important for AHE and ISHE?) The standing wave electrons do not contribute to electron current, so what? Why we just cannot forget about them? The localized electrons do not contribute to current as well. There are two reasons: The 1st reason is that there is another kind of the electron current, for which an electron just jumps from one spatial position to another. It is a very inefficient current and usually can be ignored, but when the number of standing wave electrons becomes larger, the contribution from this 2ndtype of current can become dominated. This kind of current has a huge AHE. Therefore, when the number of standingwave electrons becomes larger, the normal current decreases and the 2nd type of current increases, AHE becomes huge. For example, the sidejump scatterings and the spindependent scatterings across the interface belong to this 2nd kind of current. The 2nd reason, why the standingwave electrons are important for AHE, is that a standingwave electron effectively couples the conduction and localized electrons. (localized vs conduction electrons)The localized and the conduction electrons are very different. Their size, symmetry and properties are very different. For example, the spin distributions of localized and conduction electrons are very different. The spins of localized electrons are either parallel or antiparallel to each other. There are 3 groups of conductions electrons with respect to the spin: spinpolarized, spinunpolarized and spininactive conduction electrons (or full state). The different frequency of the scatterings makes the properties of conduction and localized electrons to be so different. (difference in scattering probability) The scatterings of the conduction electrons are frequent and the scatterings of localized electrons are very rare. The scatterings between localized and conduction electrons are very rare as well, because of the difference in size and symmetry of these different types of electrons. Additionally, a conduction electron moves, a localized electron stays at one place. Therefore, there is a difference in momentum. The standingwave electrons have the same symmetry and the same size as the moving conduction electrons, but they stay at the same place. As a result, both types of scatterings between localized and standing wave electrons and standing wave and moving conduction electrons are relatively frequent. Overall, the existence of standingwave electrons makes the probability of scattering between the localized and standing wave electrons higher. Since AHE is originated from a spin dependent scattering of conduction electrons on the localized electrons, the high scattering probability between the conduction and localized electrons makes AHE larger. (About "full" states or spininactive states) A conduction electron also can be distinguished by its spin. There are 3 different groups of conduction electrons, which are distinguished by the spin. (group 1) “Full” state or spin inactive state. It is the case when two electrons occupy one quantum state. The timeinversesymmetry for this state is not broken. The spin direction of each individual electron cannot be distinguished in this state. As a result, such electrons do not contribute to any spin related features. (group 2) Spin polarized electrons. It is the case when one electron occupies one quantum state and one state remains unoccupied. The spins of all electrons are directed in one direction. (group3) Spin unpolarized electrons. It is the case when one electron occupies one quantum state and one state remains unoccupied (similar to group 2). The spins are distributed equally in all directions. Each group has its own distribution of electrons over the electron energy. Since the electron distribution mainly determines features of the electron current, the electron current for each group is very different. The electrons of each group could be either the standing wave conduction electrons or “normal” moving conduction electrons. (increase of standing wave electrons at a lower energy. The hole transport) (following question from Szymon Królak) Why most fullstate electrons are standingwave electrons? In the explanation above, you stated that a single defect can localize an electron. Are most of the fullstate electrons standingwave electrons because of the fact that at energies lower than the Fermi energy there are a lot of electrons and the Pauli pressure is large? In short: is the effect of a large amount of neighboring fullstate electrons similar to a defect, in the sense that it can localize an electron? You understand the case correctly. Even though the full state is a property of the spin and the standingwave electrons are a special property of electrons, at an electron energy substantially lower than the Fermi level nearly all electrons are in the fullstate or the spininactive and are the standing wave electrons. At an energy substantially lower than the Fermi level, almost all states are filled by two electrons and there are only a few, which are filled by one electron and which one space is not filled. The states filled by two electrons are "full" states or spin inactive states. The states filled by one electron are holes. When there are a few empty states, the scattering of an electron from a "full" state into the empty space becomes rare. For this reason, the life time of the "full" state becomes long (e.g. in comparison to the life time of the "full" state at the Fermi energy) and, consequently, the length of the "full" state (or the same the propagation length between two scatterings) becomes longer. For a longer length the reflection from defects and interfaces becomes more probable and, therefore, there are more standingwave electrons. It is because one electron covers many defects and interfaces simultaneously. By the way, the standingwave electrons are not a "defect" electrons, because an interface is more effective in creation of a standingwave electron than a defect due to its better reflection of an electron. (standingwave electron vs localized electron) The standingwave electron is not a localized electron. The localized electron is an electron of the size of an atomic orbital of ~0.1 nm. In Fe or Co the localized electron has the d symmetry. The scatterings of the localized electrons are rare. The standingwave electron is a conduction electron of a longer size of about 10100 nm. The symmetry of the standingwave electron is the symmetry of the conduction electrons (p or s). The scattering of the standingwave electron is frequent. (timescale of AHE; AHE in picosecond timescale) (from Chen Xiao) What about the timescale of AHE? People use ISHE to generate THz electrical field, which indicates that ISHE is in the subpicosecond range or shorter. The magnetooptical effects such as KERR or FARADAY rotation have been popular techniques to probe magnetism. The timescale of MO effects might be a few femtoseconds. Would it be possible to detect the AHE in the dynamical regime? (It is noticed that G. Sala et al. achieve a time resolved halleffect detection in the nanosecond range.) The timescale of AHE is not very fast. AHE relaxation time is moderate. (reason 1: the time to chage distribution function) The AHE, ISHE or any other transport effect cannot be very fast, because any transport effect is related to a change of the electron distribution function and such a change cannot be very fast. In order to reach an equilibrium or a quasi equilibrium distribution, the electron should experience many scatterings and it takes a time. (reason 2: the time to accamulate charge on sides of a metalic wire) the timescale of AHE depends on the sample geometry. The longer timescale is not only because of the timing taking to establish the equilibrium distribution function, but also because the time required to redistribute the charge. The steps to establish an equilibrium AHE are: (step 1) After the bias electron current is on, the perpendiculartobias current starts to flow due to the spindependent scatterings. (step 2) The charge is accumulated at the boundaries of the nanowire due to the perpendiculartobias current. (step 3) The current flows opposite to the perpendiculartobias current due to the AHE voltage created by the charge accumulation. (step 4) The current due to the charge accumulation increases until it fully balances the perpendiculartobias current. The currents, the accumulated charge and AHE voltage are larger for a wider nanowire. (note) Even though the conventional measurement of AHE is a measurement of accumulated charge at sides of a metallic wire (the Hall voltage), it might be a method to measure directly the perpendicular to wire current due to AHE instead of the accumulated charge. In this case, the AHE is faster and independent of the sample geometry. (reason 3: the timing of the spindependent scatterings)) the origin of AHE is spindependent scattering of the conduction electrons (skew scattering, side jump scattering, scatterings across interface). The timescale of AHE cannot be shorter than the timescale of the spin dependent scatterings. ( fast nonequilibrium effects: hot electrons) There are faster effects, which do not require an establishment of a thermal equilibrium. For example, a current of the hot electrons.Even though the properties of such nonequilibrium effects are different from their equilibrium cousins, there are many similarities between both effects. The timescale of AHE due to the hot electrons is still limited by the reason 2 and reason 3. ( saturation velocity: the fastest speed of the electrons in a solid) The moving speed of even the hot electrons is not infinite and, in fact, it is moderate. The moving speed is limited by the saturation velocity of the electron in a solid, which equals to ~75 um/ns. For example, even at the fastest speed of the hot electrons (or of any electrons) it takes 80 ps in order for the hot electrons to pass 6 micrometers. It limits the timescale of the Hall effect. I have measured the saturation velocity (See here). The origin of the saturation velocity is that near the saturation velocity the electrons lose their energy to photons very fast similarity as an airplane flies close to a breaking of the sound barrier. ( correct definition of the photoinduced Hall effects) Many researchers name any photoinduced Hall effect as the Anomalous Hall effect (AHE). It is incorrect. There is a family of the Hall effects (See here). Each type of the Hall effect is distinguished according to its origin. Types of photoinduced Hall effect: (type 1) photoinduced Anomalous Hall effect (AHE). It is the case when light changes the number of the localized d electrons. It can occur only in a ferromagnetic metal. (type 2) photoinduced Inverse spin Hall effect (ISHE). It is the case when light changes the number of the spinpolarized conduction electrons. It is the most common photoinduced Hall effect. See my description of the photoinduced ISHE here. (type 3) photoinduced Planar Hall effect (PHE) or, the same, the photoinduced Anomalous Magnetoresistance (AMR). It is the case when light changes either the number of the spinpolarized conduction electrons or the number of the localized d electrons, which affects the probability of spindependent scatterings of a spinpolarized conduction electron on a localized delectron. The symmetry of the photoinduced AMR/PHE effect is different from the symmetry of AHE and ISHE. (type 4) photoinduced Ordinary Hall effect (PHE). It is the case when light changes or creates the internal magnetic field in a material, which is not related to a change of the number of the spinpolarized conduction electrons or the number of the localized d electrons.
(about timescale of the Kerr and Faraday effects) According to the classical Quantum Mechanic any Magnetooptical effect (Faraday, Kerr and MCD effects) has no timescale. They are infinitely fast. The Faraday, Kerr and MCD effects all originate from a Zeeman splitting of the energy level in a magnetic field. The energy level and the Zeeman splitting exists primarily of any dynamics. Therefore, it is meaningless to assign any dynamics to the energy level. Of course, a change of some external parameter may change the strength of the magnetooptical (MO) effect. However, the effect timescale is determined by the timescale of the external parameter, but not by the timescale of MO effect. For example, the MO strength depends on the strength of the external or internal magnetic field, under a change of the magnetic field the MO strength changes. The timescale of this process is determined only by the timescale of the change of the magnetic field. The timescaly of MO effect itself is instant. Another example is a filling of one of Zeemansplit energy levels by the spin polarized electrons. This event also changes the MO strength. Similarly, only the speed of the filling or emptying of the energy level determines the timescale in this case. Also, there is a timescale of the optically induced electron transition from one level to another. This timescale is very short and equals to a half of the period of a Rabi oscillation. (following question from Chen Xiao) The response speed of AHE is limited by the distance and path that hot electrons travel, while the MO effect is supposed to be "realtime". The charge is accumulated to build the electric field, in which another current flows antiparallel with its counterpart. The PNjunction and ferromagnetic nanowire (or ferromagnetic "gate") have something in common. Very interesting! It's possible to make timeresolved measurements of this nonequilibrium effect and to detect the ferromagnetic state by electrical methods. Roughly, I have had a picture of how the AHE voltage is generated and to be read through your clear description.

I will try to answer your questions as soon as possible