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

Spin Hall effect. Inverse spin Hall effect
Spin and Charge TransportThe Spin Hall effect describes the fact that a spin current is generated perpendicularly to an electrical charge current in a metallic wire. The Inverse Spin effect describes the fact that a charge current is generated perpendicularly to a spinpolarized electrical current. Both effects have the same origin, which is spin dependent scatterings due to the spinorbit interactionUniqueness of the Spin Hall effect: It causes a completely nonmagnetic material to become a magnetic when an electrical current flows through it.Note: The Spin Hall effect is important and substantial in a ferromagnetic metal as well (See SpinOrbit torque). It is not only a feature of a nonmagnetic metal, but a magnetic metal as well.This page describes my personal view on the Spin Hall effect, which is based on my theoretical and experimental findings. All used explanations are based on my description method of spin orbit interaction using the magnetic field H_{SO} (See details here). Some descriptions might be different from an official view. My use of terms "Skew scatterings" and "Sidejump scatterings" is different from that originally proposed for description of AHE effect in 60th and 70th. I believe the terms perfectly describe the meaning of the described effects. The Web page was made in 20142019. Some parts may be different from officially accept point of view of main research stream. It is a relatively recently discovered effect. Therefore, there are several slightly different descriptions.Contentclick on the chapter for the shortcut(1). Common single origin and common mechanisms of Spin Hall effect and Inverse Spin Hall effect:(1a) Complementariness of the Spin Hall effect (SHE) and the Inverse Spin Hall effect (ISHE)(2). How does the spinorbit interaction create the spin current?(3). Mechanism of Spin Hall effect: (Origin 1) due to a nonzero orbital moment of conduction electrons(4). Mechanism of Spin Hall effect: (origin 2) Skew scatterings(5). Mechanism of Spin Hall effect: (origin 4) Sidejump scatterings at defect(6). Mechanism of Spin Hall effect: (origin 5) Sidejump scatterings across an interface(7). Spin dependent scatterings as the common origin of the Spin Hall effect and the Inverse Spin hall effect(8) Orbital moment of conduction electron as origin 1 of Spin Hall effect(9) Condition for generation of a spin current by the Spin Hall effect(10). Effects similar to the Spin Hall effectsimilar effect 1: Electron movement in electrical field (Schottky type) at interfacesimilar effect 2: Ordinary Hall effectsimilar effect 3: Spin polarization induced by the Oersted magnetic field of electrical current (the Ampere's law)similar effect 4: Spin detection(11). Spin Detection using Inverse Spin Hall effect(12). History(12a) Spin Hall effect and the symmetry puzzle
(13). Questions & Answers(q1) How to distinguish between the a sidejump or skew scatterings. Scattering on a defect or a general scattering?skew scattering

Spin Hall effect (SHE) 

(definition) The SHE describes the fact that spin is accumulated at sides of metallic wire, when an electron current flows through the wire 
The conduction electrons (green balls) are scattered on a charged defect (blue ball). The conduction electrons are spin unpolarized (spins are distributed equally in all directions)). Due to the spinorbit interaction, the scattering probability for spinup electrons is higher for a scattering to the right than to the left and in contrast the scattering probability for spindown electrons is lower for a scattering to the right than to the left. As a result, the spinup electrons is accumulated at the right side of wire and the spindown electrons is accumulated at the left side of the wire 
(note) The SHE redistributes the spin unpolarized electrons (all spin directions) into two separated places in which the number of either spinup or spindown electrons are larger. In total, the the number of electron with spin of a specific direction remains the same. It does not re align electron spin (as for example magnetic field(See here and here) 
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Spin Hall effect (SHE) and Inverse Spin Hall effect (ISHE) are fully complementary effect. They have identical origins and in a material they have the same magnitude
The Spin Hall effect describes the fact that a spin current may flow across a flow of electrical charge current. It occurs when there are a spin dependency and direction dependency (spacial dependency) of electron scattering probability. E.g. the scattering probability of spinup electrons is higher to the left than to the right and correspondingly the scattering probability of spindown electrons is higher to the right than to the light. As a result, more spin up electrons flow to the left and more spindown electrons flow to the right. There is a spin current perpendicularly to the charge current.
The spin dependency and direction dependency of electron scattering probability are due to the difference of the magnetic field H_{SO} of the spinorbit interaction, which the electron experience whether it is scattered to the left or to the right. E.g. if the direction of H_{SO} is up in the left direction, the scattering probability of spinup electrons is higher than of spin down electrons for scatterings to the left. Correspondingly, the direction of H_{SO} is down in the right direction and the scattering probability of spindown electrons is higher than of spin up electrons for scatterings to the right.
There are several physical mechanisms, which originate the Spin Hall effect. The mechanism can be divided into groups: (group 1) band current and (group 2) scattering current. For the band current, the H_{SO} depends on the electron movement direction. For the scattering current, the H_{SO} depends on the electron position.
Spin Hall effect. Band current. 

Origin 1. Spin Hall effect due to a nonzero orbital moment of conduction electrons 




(in absence of an electrical current) Electron scattering are different for the left and right directions. However, all electrons experience the same difference. As a result, the scattering difference does not lead to any additional current. E.g. electrons moving along +y direction are more scattered toward +x direction, but in contrast electrons moving along y direction are more scattered toward x direction. Since the number of electrons moving along +y and y directions are equal. There is no any additional current along x direction  
(there is an electrical current along yaxis) There are more electrons moving along +y than along y direction. As a result, there are more electrons toward +x direction than toward x direction. The number of electrons moving along +x direction becomes larger than in x direction. Therefore, a current flows along xdirection. This is origin 1 of the Spin Hall effect.  
(spin dependence) the scattering probability is higher when the spin of the scattered electron is along H_{SO}, because of a higher energy.. As a result, spinup electrons are scattered more to x direction and spindown electrons scattered more to +x direction (spin is referred with respect to yaxis). It makes more spinup electrons moving along x direction and more spindown electrons moving along +x direction. Therefore, a spin current flows along xdirection.  
note: This contribution exists only when a conduction electron has a nonzero orbital (rotational) moment  
(fact):Each election moving in a different direction experience H_{SO} of a different direction. It makes electron scattering spin and direction dependent, which result a flow of spinpolarized current perpendicularly to the electrical current  
(additional fact) In total, average H_{SO}, which experience by all conduction electrons (shown in lefttop corner), is a nonzero. It is because, there are more along ydirection than in opposite direction. As a result, the rotational moment of conduction electrons is not compensated and the whole electron gas experience a nonzero H_{SO} in z direction  
skew scattering is another origin of Spin Hall effect of the band current type (see here)  
See below more details about Origin 1 of the Spin Hall effect 

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(step 1): The electrical current in a nonmagnetic metal creates the magnetic field H_{SO} of the spinorbit interaction.
(step 2): The interaction of H_{SO} with electron spin makes the electron transport spindependent. The electron scattering probability becomes dependent on H_{SO} and electron spin. As a result, the electron transport becomes both direction dependent and spin dependent. E.g. more spinup electrons move, for example, to the left and more spindown electrons move to the right. This spin current, which flows perpendiculars to the electrical (charge) current, creates a spin accumulation at the edges of the metallic wire.
(group 1: Band current or intrinsic origin): Spin dependence and directional dependence of electron scattering direction. E.g. movement direction of the scattered conduction electrons is changed more frequently towards the left than towards the right. This creates a spin polarized current flowing perpendicularly to an electrical current.
(group 2: Scattering current or extrinsic origin): Spin dependence and directional dependence of electron spacial shift after a scattering. E.g. spacial position of the scattered conduction electrons shifted more frequently to the left than towards the right. This creates a spin polarized current perpendicularly to an electrical current.
(Just an example) The result of mechanism 1 +mechanism 2 + origin 1(example): More spinup scattered electrons are moving to the left than to the right with respect to the electrical current direction. In opposite, more spindown scattered electrons are moving to the right than to the left. Therefore, more spinup electrons are moving to the left and more spindown electron are moving to the right.
(origin 1): due to a nonzero orbital moment of a conduction electron
(origin 2): Skew scatterings due to electrical field of defects
(origin 4):sidejump scatterings induced by defects
(origin 5):sidejump scatterings across an interface
(result of effects)result of SHE: a charge current creates a perpendicular spin current. result of ISHE: a spin current creates a perpendicular charge current.
(common origin of effects) Spindepending scatterings. The dependence of magnetic field H_{SO} of spinorbit interaction on the movement direction (k_{x}, k_{y}, k_{z}) and spacial position (x, y, z) of a conduction electrons.
(only difference between effect origins): Which electrons create the effect. Spinpolarized conduction electrons create ISHE effect. Spinunpolarized conduction electrons create ISHE effect.
brief explanation of both effect common origin
Common source: Spin dependent scatterings. E.g. the spin up electrons scattered more to the left and spin down electrons scattered more to the right
(origin of ISHE). For example, spins of all spin polarized electrons is up. Therefore, spinpolarized electrons are scattered more to the left and there is a flow of electrons to the left (a charge current)
(origin of SHE). E.g. more spinup electron scattered to the right and more spindown electrons are scattered to the left. Since the group of spin unpolarized conduction electrons has equal amounts of spinup and spin down electrons, the total number of electrons scattered to the left and to the right are equal and there is no charge current of spin unpolarized electrons from the left to the right. However, more spinup electrons flow to the right and more spindown electrons flow to the left. It means there is a spin current flowing from the left to right.
Spin Hall effect: Scattering current 



The interaction of H_{SO} with the spin of a conduction electron makes the electron transport spindependent. 

note: The direction of H_{SO} depends at which side of the defect or the interface.  
click on image to enlarge it. Zayets 2014 
Yes, they are complementary effects and they have the same physical origins.
(Spin Hall effect) acts on spin unpolarized conduction electrons:
More spinup electrons flows to the left and more spindown electrons flows. As a result, there is a spin current flowing perpendicularly to the charge current. E.g. more spinup electrons flows to the left and the same amount of the spindown electrons flows to the right.
(Inverse Spin Hall effect) acts on spin polarized conduction electrons:
E.g. If a spinpolarized electrical current has more spinup electrons than spindown electrons, the more electrons are moving to the left than to the right. As a result, there is a charge current perpendicularly to the original spinpolarized current as there is a flow of electrons to the left.
Spin Hall effect is linearly proportional to
(1) charge current j;
(2) The number of spinunpolarized electrons or (1sp), where sp is the spin polarization. The effect decreases, when the spin polarization sp increases.
symmetry: spin direction of accumulated spinpolarized electrons is reversed when direction of the current j is reversed.
Inverse Spin Hall effect is linearly proportional to
(1) charge current j;
(2) The number of spinpolarized electrons or sp. The effect increases, when the spin polarization sp increases.
symmetry 1: direction of generated charge current is reversed when direction of the bias current j is reversed.
symmetry 2: direction of generated charge current is reversed when the spin direction of spin polarized electrons is reversed (e.g. when the magnetization M of a ferromagnetic metal is reversed)
Spin Hall effect. Band current. 
Origin 2. skew scatterings 

H_{SO} induced by the bias electrical field along the bias current 
Fig.11.Skew scattering as an origin of the Spin Hall effect. An electron, which moves perpendicularly to the wire and the bias current, experiences the magnetic field H_{SO} of spinorbit interaction. The directions of H_{SO} are opposite for cases when the electron moves from the left to the right and from the right to the left. It makes different the electron scattering probability to the left and right directions. As a result, there is a current of spinpolarized electrons across the bias current. 
For a spin up electron: probability to turn its movement direction after a scattering to the left is higher than the probability to turn to the right. As a result, there is a current of spinup electrons to the left. 
For a spin down electron: probability to turn its movement direction after a scattering to the right is higher than the probability to turn to the left. As a result, there is a current of spindown electrons to the right. 
Electrons, which moves perpendicularly to the applied electrical field E, experience an effective magnetic field of spinorbit interaction H_{SO}. Direction of this magnetic field is different for electrons, which moves parallel and anti parallel to the xaxis. The electrons, which move along the electric field (along the yaxis) do not experience any magnetic filed of the spinorbit interaction. When the electron is scattered, it may change its movement direction. The probability that electron movement direction is changed to the left direction is higher in the right direction. It is because The electrons may be scattered on a defect (shown as a blue ball), when wave functions of two states are overlap (shown in red). Because of the magnetic field H_{SO}, the probability for an electron to be scattered toward and opposite to the xxis direction are different. This causes a flow of spin current along the xaxis, when a drift current flows along yaxis. 
See below more details about Origin 2 of the Spin Hall effect 
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The electrical current induces the magnetic field H_{SO} of the spinorbit interaction (see details here). The interacts differently with conduction electrons, which spin is parallel and opposite to H_{SO}. It causes spindependent current (spin current)
The electrical current breaks the timeinverse symmetry. The spinorbit interaction (SO) only enhances that breaking.
two groups of origins of Spin Hall effect 

p_{spin, up} (1> 2) is scattering probability from quantum state 1 to state 2 for a conduction electron, which spin is in up direction  


Fig.1 Two major origins of Spin Hall effect in short. Both origins are due the spin and directional dependencies of electron scattering probability. 
Spindependent scatterings:Origin of the Spin Hall effect 



The Spin Hall effect occurs in a material, in which there are spindependent scatterings. A scattering is spindependent when the scattering probability between two states depends on the electron spin direction and the direction of the scattering is not reciprocal. This means that the scattering probability from the first state to the second state is different from the probability of the scattering in the opposite direction from the second state to the first state. Example: the scattering probability is higher for a scattering from a state where the effective magnetic field of the spinorbit interaction and the electron spin are parallel into a state, where their directions are opposite (right figure), than the scattering probability for a scattering in the opposite direction. 
Spindependence of scattering probability .That means that the scattering probability of a conduction electron from a quantum state, in which the effective magnetic field H_{SO} of the spin orbitinteraction is parallel to the electron spin, to a quantum state, in which the H_{SO} is antiparallel to the electron spin, is different from probability of scattering in the opposite direction. Reason: An electron scattering to a lower energy state has a lower probability than a scattering to a higher energy state.
When an electron is scattered from one quantum state to other quantum state, it changes its position and its wave vector ( the movement direction)
The scatterings, after which the electron changes only its position, are defined as sidejump scatterings. The
The scatterings, after which the electron changes only its movement direction, are defined as skew scatterings
Both the sidejump scatterings and the skew scatterings may be spindependent and they both may contribute to the Spin Hall effect.
See more about the scattering current here
Direction dependence and spindependence of electron scatterings as the origin of the Spin Hall effect 



Size of the yellow arrow is proportional to the scattering probability of corresponded scattering. 

click on image to enlarge it 
The electron gas of conduction electron is a quantum mechanical object. The electron current is very different from a flow of balls or water in river. Even without any electrical current, the conductions electrons move at a substantial speeds. Additionally, there are many many conduction electrons about 1E21 electrons per cubic centimeter or about a billion in a cubic micrometer. All conduction electrons moves at a high speed in all different directions. In an equilibrium, there is no total electron current, because there is an equal amounts of electrons moving in any directions.
When an electrical or magnetic field is applied to a conductor, the field does not influence much the conduction electrons. Applying the field changes nearly nothing. The conduction electrons are still moving in all directions at a high speeds. However, the field breaks the subtle balance that there is exactly the same number of electrons moving in any direction. The field makes scattering probability between quantum states direction dependent. As a result, there are more electrons moving in one direction than in other directions. Therefore, in average there is an electron current.
In an electron gas of conduction electrons, each quantum states is characterized by 6 parameters: its spacial coordinates (x, y, z) and the electron speed or wave vector (k_{x}, k_{y}, k_{z}). There are only two possible electron currents in a conductor
(current type 1) Band current
This type of an electron current occurs in the case when number of electrons moving in one direction is different from number of electrons moving in the opposite direction. It is the case when external field makes the probability of an electron scattering in one direction different from scattering probability in opposite direction. E.g. electric field applied in xdirection makes scattering probability to a state (k_{x}+ Δk_{x}, k_{y}, k_{z}) larger than to the state (k_{x} Δk_{x}, k_{y}, k_{z}) . The reason, why the scattering probability is direction dependent, is because the conduction electron gaining/losing energy when it accelerating/ decelerating in the electrical field
Examples of band current: (1) current along electrical field (See here); (2) Ordinary Hall effect (See here)
(current type 2)Scattering current
After a scattering, additionally to the movement direction the electron spacial position is changed as well. The scattering type of an electron current occurs when position shift after electron scattering is different for two opposite direction. As a result, the electron is constantly shifted in one direction after consequent scatterings It is the case when an external field makes the probability of an electron scattering in one position different from scattering probability in opposite position. E.g. electric field applied in xdirection makes scattering probability to a state (x+Δx, y, z) larger than to the state (xΔx, y, z) As a result, the electrons are constantly drifted (moving) along xdirection The reason, why the scattering probability is spacial dependent, is because the energy of quantum states gradually becomes large along the electrical field. A scattering to a lower energy state has a lower probability.
Examples of scattering current: (1) tunneling current ; (2) hoping conductivity; (3) current in a lowconductivity metal (See here)
The Spin Hall effect describes a creation of spin polarized current. There are only two origins of electron current in a conductor. Therefore, only two types of spin polarized currents can be created by different physical mechanisms!
Both effects have the same origin: spindependent scattering (or spindependent electron flow) perpendicularly to the flow of a charge current j. The origin consists of two mechanisms:
(effect origin. Mechanism 1): Generation of magnetic field H_{SO} due to the flow of the bias charge current and the spinorbit interaction.
(effect origin. Mechanism 2): Creation of spin polarized current (or spin polarization) perpendicularly to the the flow of the bias charge current by the magnetic field H_{SO}.
(contribution 1) : the electrical field of the nucleus of the host material.
(contribution 2) : the electrical field around defects
( contribution 3): the electrical field at an interface (electrical field of the Schottky barrier or the electrical field of any charge accumulation/depletion at interface) .
( contribution 4): (not related to SO) There is an Oersted magnetic field around an electrical current (see Fig.6)
(contribution 1) (main) : the spin dependent scatterings
(contribution 2) : spin precession around H_{SO} and corresponding spin precession damping ?????
( contribution 3): changing of distribution function similar to the ordinary Hall effect ????
1st experimental observation of Spin Hall Effect. Spin accumulation at sides of nonmagnetic GaAs wire created by an electrical current 
Fig.2 (left) 2D scan of Kerr rotation angle θ_{Kerr} in GaAs 30umwide wire. θ_{Kerr} is proportional to number of spinpolarized electrons n_{S } , when an electrical current flows through the wire. T=30 K; (right) top view optical optical image of same GaAs wire 
The red and blue regions at wire side are regions of spin accumulation. Red and blue colors corresponds to spinup and spindown directions of spin accumulation. White color corresponds to regions without any spin accumulation. 
Effect Origin 2:: There is a Schottky barrier at each edge of the GaAs wire and therefore an electrical field perpendicularly to the edge. Since an electrical current flows in the wire perpendicularly to that electrical field, the electrons experience the SO magnetic field H_{SO} , which creates the spin accumulation. The direction of the electrical field is opposite on opposite sides. As a result, the polarity of and spin accumulation is opposite on opposite sides. 
Y. K. Kato, et. al, "Observation of the Spin Hall Effect in Semiconductors". Science 306, 19101913 (2004) 
click on image to enlarge it 
Even without an electrical current, the conduction electrons move along the metal with a substantial speed. However, amounts of electrons moving in any two opposite directions are always equal. Therefore, they experience the opposite H_{SO} and all spindependent transport is canceled out.
When there is an electron current, there are more electrons moving along the current than opposite to the current. As a result, there are more electrons, which experience +H_{SO} than H_{SO} It causes more spinup electrons moving, for example, to the left and more spindown electrons moving to the right.
Figure 2 shows the experimental observation of spin accumulation in nonmagnetic GaAs wire, when an electrical current flows throw it. There are spinpolarized electrons at sides of the wire. The spinpolarized electrons rotate the polarization of light. As a result, the Kerr rotation in the regions of the spin accumulation is either smaller or larger in comparison with other regions of no spinaccumulation.
Effect Origin 1 (major): There
Effect Origin 2 (minor): There is a Schottky barrier at an edge of the GaAs wire and therefore an electrical field perpendicularly to the edge. Since low doping (n~1E16 cm3), the barrier is wide and the penetration of electrical field is deep. Since an electrical current flows in the wire perpendicularly to that electrical field, the electrons experience the SO magnetic field H_{SO} , which creates the spin accumulation. The direction of the electrical field is opposite on opposite sides. As a result, the polarity of and spin accumulation is opposite on opposite sides.
Without
It is absolutely correct. The electrical field if a defect can contribute only when it is case it is not fully screened. It is often the case, when the defect is near interface. Also, it is one of reasons why the electrical field at interface (Schottky type) contributes more to the Spin Hall effect.
Additionally to defects, the electrical field of the nucleus of the host atoms contributes similarly to the Spin Hall effect.
(fact 1): The electrical current breaks the timeinverse symmetry. The breaking of the timeinverse symmetry is the required condition for the electron gas to be spinpolarized.
(fact 2): The electrical current + the spinorbit interaction create a magnetic field H_{SO}, which interacts with the electron spin. The electrons, which moves in opposite directions, experience the magnetic field H_{SO} in opposite directions.
(fact 3): The electron scattering probability depends on mutual directions on mutual directions of the electron spin and the magnetic field H_{SO}. As a result, the electron scattering becomes spin dependent. Since H_{SO} depends on electron movement direction, the scattering becomes direction dependent. Due to the spin and directional dependence of electron scattering probability, the spin polarized current flows perpendicularly to a charge current in a metallic wire.
Orbital rotation of conduction electrons 



Yellow wavy ellipse shows the wave function of the conduction electron. Blue circle show the direction of rotation of the conduction electron around each atomic nucleus (dark spheres).Electrical field of each nucleus induces the magnetic field interaction H_{SO}. The electron experiences the accumulative strong H_{SO}. Even though the electrons moves along stationary nuclei,accumulative H_{SO} remains constant. 

The size of a conduction electron (size of its wave function) is relatively large. A conduction electron can cover simultaneously hundreds or thousands of nuclei.  
camera moves with the electron  
click on image to enlarge it 
Explanation in short: Due to its nonzero orbital moment, a conduction electron experiences a magnetic field H_{SO} of the spin orbit interaction (See here). The H_{SO} makes the electron scattering probability spin and direction dependent. In absence of an electrical current, the scattering is balanced in all directions and the spin and direction dependence of scattering probability does not affect the electron distribution. An electron current breaks the balance and there are more spin polarized electrons scattered in one direction perpendicularly to the current than in the opposite direction. It creates a spin polarized current flowing perpendicularly to the electrical (charge) current.
(explanation of effect):
(step 1) The conduction electron have a nonzero rotational (orbital) moment,which created magnetic field H_{SO}. There is a spin precession around H_{SO} and the spin is aligning along H_{SO} due to the damping of the spin precession.
(step 2) When there is no electrical current, there are equal numbers of electrons moving in any two opposite directions. Since the rotational (orbital) moment and H_{SO} are equal and opposite for electrons moving in opposite direction, both the total rotational (orbital) moment and total are zero for the electron gas and scattering probabilities are independent on electron movent direction and electron spin
(step 3) When there is an electrical current, the number of conduction electrons moving along current is larger than number of electrons moving in the opposite direction. As a result, the rotational (orbital) moment of electrons moving along current is not fully balanced by the opposite moment of electrons moving in the opposite direction and the total the electron gas experience a nonzero H_{SO} and the electron gas becomes spinpolarized.
(step 4) When there is an electrical current, the scattering probability of spinup electrons to the left becomes different from the scattering probability to the right. As a result, e.g. the spinup polarized current flows to the left and the spindown polarized current flows to the right.
Spin Hall effect due to a nonzero orbital moment of conduction electrons 



Fig. 3. Distribution of electrons with orbital moment and distribution of H_{SO}. 


The length of a vector from axis origin to the sphere is proportional to the number of electrons moving in the vector direction. Red arrow shows the direction of electron movement. Blue circle shows the direction of orbital moment. Violet arrow shows direction of H_{SO} induced by the orbital moment. The direction of orbital moment and H_{SO} is fixed to the electron movement direction.  
Yellow arrow shows the magnitude and direction of an electrical current. When there is a current, the distribution (the sphere) is shifted from the center. Therefore, there are more electrons moving along current than in opposite direction. The total orbital moment and total magnetic field H_{SO} of the whole electron gas are shown in top right corner of the center figure.  
H_{SO} makes different the electron scattering probability towards the left and right movement direction! 

H_{SO} makes the electron scattering probability spindependent! 

Electrical current J along ydirection creates spinpolarized currents along x and z directions 

Each election moving in a different direction experiences magnetic field H_{SO} in a different direction.  
When a conduction electron moves along crystal lattice, it simultaneously rotates around each atomic nucleus as it passes it. The electrical field of nucleus induces the magnetic field of spinorbit interaction. The spin of conduction electrons interacts with H_{SO}. There is a spin precession around H_{SO} and spin precession damping, which aligns spin to H_{SO. }  
click on image to enlarge it 
For this contribution, the magnetic field H_{SO} of the spin orbit interaction is induced of the electrical field of atomic nucleus of the host metal. As a result, it is substantial.
Spin and directional dependence of the scattering probability
(spin dependence of the scattering probability): the scattering probability is higher when the spin of the scattered electron is directed along H_{SO}, because of its higher magnetic energy. As a result, spinup electrons are scattered more to x direction and spindown electrons scattered more to +x direction (spin is referred with respect to yaxis). It makes more spinup electrons moving along x direction and more spindown electrons moving along +x direction. Therefore, a spin current flows along xdirection.
(direction dependence of the scattering probability): The direction of H_{SO} is fixed to the movement direction of a conduction electron (see Fig. 3c). As a result, the direction of H_{SO }is different for conduction electrons moving in different directions. The direction dependence of H_{SO} causes the direction dependence of the scattering probability.
(fact):Each election moving in a different direction experience H_{SO} of a different direction. It makes electron scattering spin and direction dependent, which result a flow of spinpolarized current perpendicularly to the electrical current
Additional effects of this mechanism
(additional effect 1)Inducing a magnetic field, which all conduction electrons of an electron gas experience
Each conduction electron experiences the magnetic field H_{SO} . However, each electron moving if a different direction experiences H_{SO} in different direction. E.g. electrons, which are moving in opposite directions, experience opposite H_{SO}. As a result, in absence of an electrical current the average magnetic field, which all conduction electron experience, is zero. When there is an electron current, there are more electrons moving along the current than in the opposite direction. As a result, there are more electrons, which experience +H_{SO,z} than H_{SO,z} (See Fig.3b). Therefore, in average the electron gas experience a magnetic field in the z direction
(additional effect 2)Reduction of the spin relaxation
The incoherent spin precession around H_{SO} is a substantial mechanism of the spin relaxation (See here). The spins of all spin polarized electrons are directed in one direction. In contrast, the direction of H_{SO} is different for electrons moving in different directions. As a result, the angle between electron spin and H_{SO} is different for the spin polarized electrons moving in different directions. There is a spin precession around H_{SO}. Since the directions of H_{SO} are different for conduction electrons moving in different directions, their precession directions are different as well. The precession in different directions misaligns spins of spin polarized electrons, which causes the spin relaxation.
(Similar effects): Ordinary Hall effect
In the case of the Ordinary Hall effect, the movement direction of the conduction electrons turns in an external magnetic field due to the Lorentz force. It makes the electron scattering probability to the left direction is different from the scattering probability to the right direction. It creates a Hall electrical current flowing perpendicularly to the bias electrical current.
Similarly in the case of the spin Hall effect, the magnetic field H_{SO} makes different the electron scattering probability to the left and right directions. However, the difference are not due to the Lorentz force, but due to the magnetic energy of interaction of H_{SO} with the electron spin.
No, the H_{SO} cannot induce the Lorentz force, because of the relativistic nature of H_{SO}. Therefore, H_{SO} does not cause the ordinary Hall effect.
No. All conduction electrons experience the external magnetic in one direction. Therefore, the external magnetic field cannot create the directional dependence of the scattering probability.
It is correct. A conduction electron moves along a metal and simultaneously rotates around thousands or even millions of atomic nuclei. See below for more details
Spin Hall effect. Band current. 
Origin 2. skew scatterings 

H_{SO} induced by the bias electrical field along the bias current 
Fig.11.Skew scattering as an origin of the Spin Hall effect. An electron, which moves perpendicularly to the wire and the bias current, experiences the magnetic field H_{SO} of spinorbit interaction. The directions of H_{SO} are opposite for cases when the electron moves from the left to the right and from the right to the left. It makes different the electron scattering probability to the left and right directions. As a result, there is a current of spinpolarized electrons across the bias current. 
For a spin up electron: probability to turn its movement direction after a scattering to the left is higher than the probability to turn to the right. As a result, there is a current of spinup electrons to the left. 
For a spin down electron: probability to turn its movement direction after a scattering to the right is higher than the probability to turn to the left. As a result, there is a current of spindown electrons to the right. 
Electrons, which moves perpendicularly to the applied electrical field E, experience an effective magnetic field of spinorbit interaction H_{SO}. Direction of this magnetic field is different for electrons, which moves parallel and anti parallel to the xaxis. The electrons, which move along the electric field (along the yaxis) do not experience any magnetic filed of the spinorbit interaction. When the electron is scattered, it may change its movement direction. The probability that electron movement direction is changed to the left direction is higher in the right direction. It is because The electrons may be scattered on a defect (shown as a blue ball), when wave functions of two states are overlap (shown in red). Because of the magnetic field H_{SO}, the probability for an electron to be scattered toward and opposite to the xxis direction are different. This causes a flow of spin current along the xaxis, when a drift current flows along yaxis. 
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(explanation of effect):
A conduction electron experiences a magnetic field H_{SO} of spinorbit interaction, when they move at a nonzero angle with respect to an applied bias electrical field E. The H_{SO} is in opposite directions for electrons moving to the left and to the right directions with respect to the direction of the bias electrical field. This direction dependence of H_{SO} makes the electron scattering probability both direction dependent and spin depends. Due to such dependency of scatterings, an electrical current flowing along a metallic wire (along applied bias electrical field) creates a spin polarized current, which flows perpendicularly to the wire.
In this case, Magnetic field H_{SO} is induced by a bias electrical field, along which electrical current flows
A Phonon, a magnon, an impurity and a defect can be a source of a skew scattering.
The sources of H_{SO} are different. For this origin (origin 2), the source of H_{SO} is the electrical field, along which the electrical current is flowing. In the case of origin 1, the source of H_{SO} is the electrical field of atomic nuclei, which induces due to a nonzero orbital moment of a conduction electron. Except the source of H_{SO}, the origins 1 and 2 are similar.
The scattering over 90 degrees is shown only for clarity. The effect exists for any scattering angle. It is only important that the magnetic field H_{SO} of spin orbit interaction makes different scattering probability for a change of electron movement direction to the left and to the right.
For the same electrical current, the bias voltage is large for a metallic wire of a lower conductivity.
Even with any bias electrical field the conduction electrons moves in all directions at substantial speeds, but numbers of electrons, which move in any two opposite directions, are the same. When an electrical current flows, there are more electrons, which move along the bias electrical field, than in opposite direction. For both cases with and without a bias electrical field there are conduction electrons, which move at an angle with respect to applied bias electrical field.
There are many currents for this effect, which should be distinguished somehow. The bias current is the simplest conventional charge current, which just flows along a metallic wire when a voltage applied to the wire. You took such definition from a transistor. How would you call it? The most clear and understandable name should be used.
Spin Hall effect: Scattering current 



The interaction of H_{SO} with the spin of a conduction electron makes the electron transport spindependent. 

note: The direction of H_{SO} depends at which side of the defect or the interface.  
click on image to enlarge it. Zayets 2014 
(explanation of effect):
There is an electric field around a defect in a metal. When a conduction electron passes in the vicinity of the defect, it experiences this electrical field, which induces an effective magnetic field H_{SO} of the spinorbit interaction. The H_{SO} is in opposite directions whether the electron passes from the left or the right side of defect, because of the opposite directions of the defect electrical field at opposite sides of the defect. This spacial dependence of H_{SO} makes the electron scattering probability both spacial dependent and spin dependent. When the electron is scattered from one side to another side of the defect, the scattering probability is different whether the electron scattered from the left side to the right side of the defect or from the right to the left. Due to such dependency of scattering, an electrical current flowing along a metallic wire creates a spin polarized current, which flows perpendicularly to the wire.
In this case, Magnetic field H_{SO} is induced by a electrical field of a pointlike defect or an electrical field from any a point like source of electrical field
What does it mean: the scattering probability is spacial dependent? It means that the probability that the spacial position of scattering electron to be more at the left side is higher than the probability to be at the right side with respect to the electron position before scattering.
It is because of the spacial dependence of H_{SO}. The electrical field of a point like defect is directed toward left at the left side of the defect and towards the right at the right side of the defect. Such electrical field creates H_{SO} in up direction at the left side and in down direction at the right side of the defect (See Fig.19). The scattering probability is larger when the electron spin is parallel to H_{SO} and smaller when it antiparallel to H_{SO}. As a result, the scattering probability is spin dependent.
E.g. of Fig.19, the scattering probability for a spin up electron is higher when it shifted towards the left. As a result of consequence scatterings the spin up electrons are continuously towards the left. As a result, there is a current of spin up electrons towards the left. The scattering probability for a spin down electron is higher when it shifted towards the right. As a result of consequence scatterings the spin down electrons are continuously towards the right. As a result, there is a current of spin down electrons towards the right.
The absolute position of the electron with respect to the defect does not matter for the Spin Hall effect. It only matter in which direction the electron is shifted after a scattering. E.g. it does not matter whether the electron position at the left or right side of the defect, when the scattered spin up electron is shifted to the left, it experiences a larger H_{SO} up field (or the same a smaller H_{SO} down field). when the scattered spin up electron is shifted to the right, it experiences a smaller H_{SO} up field (or the same a larger H_{SO} down field). As a result, the energy scattered spin up electron becomes smaller when it is shifted towards left and therefore the scattering probability to the left is larger.
When the average distance between defects becomes comparable with the electron meanfree path, the sidejump scatterings becomes position and spinindependent and they do not contribute to the Spin Hall effect. 
Fig 20. The wave function of each quantum state overlaps with two defects. The effective magnetic field of the opposite directions from the defects on opposite sides affects the quantum state simultaneously. Therefore, the total effective field is zero and the sidejump scatterings becomes position and spin independent. It means that at any spacial position a conduction electron experiences nearly the same H_{SO}. As a result, the scattering probability becomes independent whether the scattered electron is shifted to the left or to the right with respect to its initial position. 
It is only correct (the strength increases with for a higher defect density) when the defect density is rather small and the average distance between defects is longer than the width of a conduction electrons (its mean free path). When the distance between defects becomes shorter, the electron are affected by several defects at the same time, which results in reduction of the strength of the Spin Hall effect. Even though each defect gives the contribution of the same polarity, the average electrical field from many defect over the width of electron wave function becomes nearly constant. See Fig 20. This origin of the Spin Hall effect is proportional to the spacial gradient of electrical field, but not absolute value of the electrical field. The spacial distribution of the electrical field of one defect is sharp (~1/r), but the sum of electrical field from defects at different position + integration over the width of conduction electrons becomes nearly independent on the electron position. It means that at any spacial position a conduction electron experiences nearly the same H_{SO}. As a result, the scattering probability becomes independent whether the scattered electron is shifted to the left or to the right with respect to its initial position. Therefore, there will be no spin current and no effect.
Yes, very much. Therefore, this contribution is only substantial in a metal, in which the screening is weak. For example, in a metal with low conductivity or in vicinity of an interface or a very thin metallic wire etc.
This contribution is a largest when
1) The defects induces a significant electrical field in the crystal lattice.
2) The density of the defects is large, but the average distance between defects is still substantially larger than the electron meanfree path
Spin Hall effect: Scattering current 



The interaction of H_{SO} with the spin of a conduction electron makes the electron transport spindependent. 

note: The direction of H_{SO} depends at which side of the defect or the interface.  
click on image to enlarge it. Zayets 2014 
(explanation of effect):
This contribution is due to a difference of direction and magnitude of H_{SO} at two sides of the interface between two metals. There are two sources, which make H_{SO} different at different sides of the interface. The source 1 is due to a difference of orbitals moments of conduction electrons in different materials at two sides of interface. The source 2 is due to the electrical field, which is perpendicular to interface and which is opposite at two sides of interfaces. The electrical field is originated the difference of work functions of materials at sides of interface, which is compensated by a charge accumulation/ depletion at interface,
When a conduction electron is scattered from one to another side of interface, it experience different direction and magnitude of magnetic field H_{SO} of spin orbit interaction. This difference of H_{SO} makes the scatterings across the interface spin and direction dependent. When the electron is scattered from one side to another side of the interface, the scattering probability is different whether the electron scattered from the left side to the right side of the interface or from the right to the left. Due to such dependency of scattering, an electrical current flowing along a metallic wire creates a spin polarized current, which flows perpendicularly to the interface.
There is a charge accumulation or charge depletion at contact interface between two metals. The charge accumulation/depletion compensates the difference of the metal work functions.
In metals the region of the charge accumulation/ depletion is very thin, because of the effect of screening of electrical field and a large number of electrons in a metal. As a result, the interface electrical field in a metal is rather large, but it exists in a very thin region at interface.
In semiconductor the region of the charge accumulation/ depletion is wider, but the electrical field is weaker. It is called Schottky region (contact). For the existence of the Spin Hall effect, the Schottky contact should not be fully depleted
The electrons, which move along different sides of the contact, experience different direction of the electrical field from the accumulated (depleted) charge.
Because of the opposite electrical field, the electrons at different sides of the contact experience the opposite effective magnetic field of the spin  orbit interaction.
Therefore, the probabilities of the sidejump scatterings across interface is different whether the electron scattered from left side of the contact to the right side or in the opposite direction. Because of this difference, a spin current flows across the contact interface when a charge current flows along the contact.
The contribution due to this source is similar to the contribution 4 due to Sidejump scatterings at a defect
Spin dependent scatterings as the common origin of Spin Hall effect (SHE) and Inverse Spin Hall effect (ISHE) 

The origin of both the SHE effect and the ISHE effect is the spin dependent scatterings. There are several mechanisms, which make scatterings of conductions electrons spin and direction dependent. (mechanism 1): due to a nonzero orbital moment of conduction electrons; (mechanism 2): Skew scatterings (mechanism 3): Sidejump scatterings at defect (mechanism 4): Sidejump scatterings across an interface. Independently on the mechanism, the origin of SHE and ISHE is the same. The conduction electron experiences the magnetic field H_{SO} of the spinorbit interaction, which depends either on the electron movement direction (k_{x}, k_{y}, k_{z}) or the electron spacial position (x, y, z) . The H_{SO} makes scatterings of conductions electrons spin and direction dependent. As a result of the spin and direction dependency of scattering probability, the numbers of spinpolarized electrons, which are scattered to the left and to the right, are different and there is an electron current (charge current) flowing perpendicularly to the main current  


note:Spin dependent scatterings of only spin polarized electrons create the charge current.. Spin dependent scatterings of spin unpolarized electrons create only the spin current.  
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What is the spin dependent scattering?
A. it is a scattering, which probability is different between left and right directions with respect to the electron movement direction and the electron spin direction.
Which effect makes the scatterings of conduction electrons to be direction and spindependent?
A. The spinorbit (SO) interaction (details see here)
How the SO interaction makes a scattering to be directiondependent and spindependent?
Orbital rotation of conduction electrons 

Additionally to the movement along the crystal lattice, a conduction electron rotates around each atomic nucleus. Electrical field of the nucleus induces the magnetic field H_{SO} of the spinorbit interaction. The H_{SO} affects the electron scattering probability and therefore creates the spinpolarized current. The effect is called the spin Hall effect.  


The size of a conduction electron (size of its wave function) is relatively large. A conduction electron can cover simultaneously hundreds or thousands of nuclei.  
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Existence of a nonzero orbital moment makes scatterings of a conduction electron spin dependent. As a result, both the Spin Hall effect and the Inverse Spin Hall effect occurs in the material.
A conduction electron moves in one particular direction. The movement along material itself breaks the time inverse symmetry for the conduction electron. The movement direction of a conduction electron is reversed with the time reversal. Due to this breaking of the time inverse symmetry, a conduction electron has some magnetic properties, which do not exist for a localized electron. The localized electron stays at the same, it does not move and therefore there is no a similar breaking of the time inverse symmetry for the localized electrons
A localized electron can be represented as a sum of two electrons moving in opposite directions. Even when each of moving electron has an orbital moment with respect of its movement direction, the localized electron as a sum of these two moving electrons has no orbital moment, because the orbital moment of opposite  moving electrons is equal and opposite and their sum is zero.
A. A localized electron of spherical symmetry cannot have the orbital moment, because the time inverse symmetry is not broken for the spherical orbital (s orbital). Localized electrons of more complex symmetry have a nonzero orbital moment (e.g. p,d, f symmetries for a hydrogen atom)
A conduction electron of spherical symmetry can have a nonzero orbital moment, because its time inverse symmetry is broken along its movement direction !!!.
Why orbital moment and H_{SO} are a nonzero for a conduction electron, but zero for a standingwave conduction electron? 
A conduction electron is reflected between two objects (e.g. two defects or interfaces). The direction of orbital moment (pink ellipse) and correspondingly H_{SO} (blue arrow) is fixed to movement direction of the electron (red arrow). The orbital moment and H_{SO} are opposite for opposite movement of electron. As a result, the electron has no orbital moment and experiences no H_{SO}!! 
The standingwave electron is a conduction electron, which is constantly bounced between two imperfection. 
In contrast to an "usual" conduction electron, a standingwave electron has no orbital moment and experiences no H_{SO}!! 
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A conduction electron moves along a crystal lattice in one fixed direction. A standing wave conduction electron is bouncing back and forward between two obstacles (defects, interfaces etc.). It does not move along the lattice and it consists of opposite moving conduction electrons. The size of a conduction electron is large. E.g. it simultaneously covers of a millions of atomic nuclei. The localized electrons do not move along the crystal lattice and they are similar to standing wave electrons, but their size is much smaller and is about the size of atomic orbital.
A standing wave electron is a sum of two coupled conduction electrons, which move in the opposite directions and have the same and opposite momentum and wave vector. Therefore, the total momentum of a standingwave conduction electron is zero and it is does not move in the space.
A No.
How does the spinorbit (SO)interaction affect the scattering 



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Incorrect interpretation of the Inverse Spin Hall effect:
When an electron moves perpendicularly to an electrical field, it experiences the effective magnetic field due to the effect of the spinorbit interaction, which direction is perpendicular to the electron movement direction. When an an electron moves perpendicularly to an electrical field, it experiences the Lorentz force. It is easy to assume that the effective magnetic field of the spinorbit interaction may induce the Lorentz force or that it may induce the ordinary Hall effect. It is very incorrect.
Explanation why the spinorbit interaction can not cause the ordinary Hall effect
When an is moving in a static electric field, in the coordinate system moving together with the electron, the static electric field is relativistically transformed into the effective electric field and the effective magnetic field. The effective magnetic field is called the effective magnetic field of the spinorbitinteraction. In the coordinate system, which moves together with the electron, obviously the electron does not move. The effective magnetic field does not induce the Lorentz force on a stationary particle. Therefore, the spinorbit interaction can not cause the ordinary Hall effect.
3 types of the Hall effect 



Figure shows the side jump scatterings in the electrical field of a defect as an example. Any mechanism of spindependent scatterings contributes to these effects: mechanism 1, Skew scatterings (mechanism 2), side jump scatterings (mechanism 4, mechanism 5)  
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Detailed explanation of this Figure is here
Reason why Spin Hall effect creates the spin current and spin accumulation at wire sides , but not in the middle of wire 



Spin Hall effect creates the spin accumulation and spin current only at sides of a metallic wire. The spin directions of spin polarized electrons accumulated at left and right sides of the wire are opposite 

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The origin of the Spin Hall effect is spin depended scatterings. A spin polarized current is created from spin unpolarized electrons, because they experience the spin dependent scatterings. In the group of spin unpolarized electrons, the spins of conduction electrons are equally distributed in all directions. A scatting of a conduction electron is spin dependent when, for example, the scattering probability of spin up electrons is larger toward the left and the scattering probability of spin down electrons is larger toward the right. As a result, more electrons, which spin is directed towards up, moves to the left and more electrons, which spin is directed towards down, moves to the right. There are two spinpolarized current of the spin unpolarized electrons: (1) current flowing to the left with spin direction up and (2) current flowing to the right with spin direction down.
However, the existence of two spin currents is not sufficient to create a spin polarization (or more speaking more generally in order to break the time inverse symmetry). This fact can be understood as follows. Into each point of the wire, two spin currents are flowing. One spin current of spin down electrons flows from the neighbor left side and the second spin current of spin up electrons flows from right neighbor side. At one point of space the spin polarized conduction electrons of only one spin direction can exist. It can be either more spin up or more spindown electrons. The spin polarized conduction electrons of two different spin directions very quickly mixed up and therefore converted into the group of spin unpolarized electrons by scatterings. When the spin dependent scatterings are homogeneous in the space, the spinup and spin down currents flowing from the left and the right are equal and therefore do not create any spin polarization.
When there is a gradient of conductivity or the vicinity of an interface, the spin currents flowing from the left and the right are unbalanced and spin polarization is created.
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Spin Hall effect of type 1 
due to electron movement perpendicularly to electrical field (Schottky type) at interface 

Fig. 30 . The electron current in a GaAs stripe. The view point moves together with electrons. Because of the charge depletion at the GaAsoxide boundary, there is an electrical filed E perpendicular to the boundary (shown as the green arrows). The electrical field E induces the effective magnetic field H_{SO} of the spinorbit interaction (red arrows). The direction of the effective magnetic field is opposite at the opposite sides of the GaAs strip. Spins of conduction electrons is aligning along H_{SO}. As a result, the conduction electrons becomes spin polarized. The direction of the spin polarization is along H_{SO} 
When the conduction electrons moved near the edge of the GaAs strip, the spins precess around the magnetic field H_{SO}. Because the damping of the spin precession the spins are aligned along the effective magnetic field of the spinorbit interaction. 
The electrical field, H_{SO} and spin polarization exponentially decays from the interface deep into the wire. There are no E, H_{SO} and spin polarization at the center of the wire. 
Due to the Schottky barrier at the boundary, the electrons are depleted (accumulated) at the boundary region creating the electrical field E. 
The view point moves together with electrons. 
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The spin accumulation generated by the magnetic field of spin orbit interaction induced due electron movement perpendicularly to the electrical field at interface
The spin accumulation generated by the ordinary Hall effect is often wrongly assigned to the Spin Hall Effect
Fig. 25. The Hall effect in metal. The electron current (green balls) flows from “” to “+” . In a magnetic field the electrons turn left in respect to direction of their movement. The holes current (blue balls) flows from “+” to “” . In a magnetic field the holes turn right in respect to direction of their movement. Therefore, the charge does not accumulate. In contrast, the spin is accumulated at one edge of the sample. 
It is the case when the magnetization of a ferromagnetic metal is perpendicularly to the surface of a ferromagnetic wire.
(explanation of the effect). In a ferromagnetic metal the conduction electrons are spin polarized. When the magnetic field and the magnetization are perpendicular to the electron current, the electrons experience the ordinary Hall effect due to the Lorentz force and turns their movement direction toward one of wire edge. The electrons and holes turns in the opposite directions due to their different movement directions and charge. Since there are almost equal numbers of electrons and holes in metal (See details here) , there is only a weak charge accumulation at the edge of the wire and the ordinary Hall effect is weak in a metal. However, the polarity of the spin accumulation is the same of the electrons and holes. As a result, there is a spin accumulation at one side of wire edge and a spin depletion at another wire edge
The Hall effect is very effective to enlarge the spin accumulation at one side of the sample and to reduce another side comparing to the bulk spin polarization of a ferromagnetic metal..
How to distinguish between spin accumulations due to the Spin Hall effect and the Ordinary Hall effect??
The Spin Hall effect =====> it is independent on the magnetization direction of ferromagnetic wire
The ordinary Hall effect =========> it is largest, when magnetization is perpendicular to the wire and the electrical current.
(note) There is no influence of the spin orbit interaction on this effect
(note) This effect occurs only in a ferromagnetic metal, but not in a non magnetic metal.
(note) This effect is unable to make a non magnetic metal to become magnetic ( in contrast to the Spin Hall effect)
The spin accumulation generated by the Oersted field is often wrongly assigned to the Spin Hall Effect
Fig.6 Spin accumulation generated by an electric current. An electrical current (green arrows) flows in a conductive film (blue plate). A magnetic field (blue circles) is generated around the current (the Ampere's law). The magnetic field generates a spin accumulation (See here). There is no magnetic field at the center of the sample and the magnetic field is the largest at the the edges of the sample. Therefore, the spin accumulation is the largest at the edges of the sample. Also, the direction of the spin accumulation is opposite at opposite sides of the sample. 
Figure 6 shows a film of a nonmagnetic metallic wire, in which an electrical current flows under an applied voltage. A magnetic field (blue circles) is generated around the current (the Ampere's law). The magnetic field is small at the center of film, but it is large at the edges. The spins of conduction electrons are aligned along the magnetic field
(explanation of the effect). An electrical (charge) current generates a magnetic field around itself (Ampere's law)). The electron spins are aligned along this magnetic field. As a result, the conduction electrons become spinpolarized at the edge of the wire.
See more details how an external magnetic field makes conduction electrons spin polarized here and here.
The charge current, which flow along an AC spin current, is often wrongly assigned to the Inverse Spin Hall Effect
(similar effect): Spin detection + AC electrical current 

When conductivity of spinpolarized electrons is different from conductivity of spinunpolarized electrons, a charge is accumulated along a flow of spin polarized electrons. This effect is called the Spin detection. Details about the Spin detection is here 
A modulation of the spin current (AC spin current) modulates the amount of the charge accumulation. The modulation of charge accumulation creates an AC charge current flowing along the AC spin current. 
The detection conductivity σ_{detection} is defined as a difference between conductivities of the spin polarized and spin unpolarized electrons. It describes the spin detection effect. 
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(explanation of the effect). When conductivity of spinpolarized electrons is different from conductivity of spinunpolarized electrons, the charge is accumulated along a spin diffusion and the spin current can be detected. (This effect is called the Spin detection. Details about the Spin detection is here). When the spin current is modulated in time, the distribution of the charge accumulation becomes modulated as well. The change of the charge distribution requires a charge current. This means that along an AC spin current flows a AC charge current. The amplitude of this AC charge current may be significant and it is proportional to the amplitude of the AC spin current.
(note) This effect is similar to the Inverse Spin effect.
The Inverse Spin effect describes the generation of a charge current flowing perpendicularly to the spin current.
The effect of Spin Detection describes the generation of a charge current flowing along to the spin current (AC only).
How to distinguish between effect of the Spin Detection and the Inverse Spin effect??
(method 1) The magnitude of the Spin Hall effect and the Inverse Spin Hall Effect should be the same for DC and AC currents!!!!
(method 2) Measure the direction of charge current. If it is perpendicularly to the spin current, it is due to the Inverse Spin Effect. If it is along the spin current, it is due to the Spin Detection Effect.
The effect of the spin detection occurs when the conductivities of spin polarized and spin unpolarized electrons are different. The spin diffusion or spin current without a charge current consists of the same currents of spin polarized and spin unpolarized electrons in opposite directions. When the conductivities of spin polarized and spin unpolarized electrons are different, there is a charge accumulation along the spin diffusion. The charge accumulation can measured and therefore spin current (spin diffusion) can be detected and measured. See more about spin detection here.
The diffusion of spin current describes a flow of the spin without a flow of the charge. It is only possible when equal amounts of spin polarized and spin unpolarized electrons flow in opposite directions. When conductivity of spinpolarized electrons is different from conductivity of spinunpolarized electrons, the charge flow is balanced in opposite directions and the charge is accumulated along the flow of the spin current.
The detection conductivity σ_{detection} is defined as a difference between conductivities of the spin polarized and spin unpolarized electrons. It describes the spin detection effect.
Spin Detection using ISHE 
Fig.31. ISHE type spin detection. Nonlocal configuration. Under the applied voltage, the charge current J_{ch} (blue arrow) flow in the copper nanowire between two left electrodes. The conduction electrons in Fe are spin polarized, therefore the spinpolarized electrons are injected and accumulated in Cu. In contrast to the charge current J_{ch} (blue arrow), which can flow only along an electrical field, the charge current J_{spin} (red arrow) does require the electrical field and flows to the right. The spin current J_{spin} induces the Hall voltage due to ISHE effect, which is detected by the pair electrodes at right side 
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It is the generation of the electrical voltage proportionally to the flow of spin polarized electrons. Therefore, from the measurement of this voltage the magnitude of the spin current can be estimated.
There is another effect, which can be used for the detection of a spin diffusion current. This effect is called the spin detection or the conventional spin detection. It occurs in a material, in which the conductivities for spin polarized and spin unpolarized electrons are different.
Physical mechanism of ISHE type Spin Detection: Spin Dependent scatterings 
Fig. 30. Spin dependent scatterings as the origin of the ISHEtype spin detection. E.g. spin polarized electrons (spinup) are scattered more to the right than to the left. As a result, a charge is accumulated at the right side of the metallic wire, which can be detected by a pair of Hall electrodes. 
Spinpolarized conduction electrons flow from backside to frontside. Spinunpolarized electrons flow from frontside to backside. Spinpolarized electrons are accumulated at leftside of wire. Spinunpolarized electrons are not accumulated. 
There is a region of spin accumulation at backside of the wire. The diffusive spin current flows from the region of a higher spin accumulation to the region of a lower spin accumulation. This means that spin polarized electrons flow from back to front of the wire. In the opposite direction the spinunpolarized electrons flow. The scattering probability of spinup electrons into the right is higher. This is the reason for the charge accumulation at the left side of the wire. 
Figure shows the side jump scatterings in the electrical field of a defect as an example. Any mechanism of spindependent scatterings contributes to the ISHEtype spin detection: mechanism 1, Skew scatterings (mechanism 2), side jump scatterings (mechanism 4, mechanism 5) 
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Origin: (conventional type detection): spindependent conductivity; (ISHE type detection): spin dependent scatterings
Direction of electrical field buildup : (conventional type detection): along the spin current; (ISHE type detection): perpendicularly to the spin current
Requirement for detection electrodes: (conventional type detection): a special electrode with a strong spin dependent conductivity should be used; (ISHE type detection): no requirements. Any electrodes can be used
material properties, which determine the spin detection efficiency: (conventional type detection): spindependent conductivity if the detection electrode (MgO:Fe); (ISHE type detection): Inverse Spin Hall effect (ISHE) in nonmagnetic metal (Cu)
reliability, repeatability, controllability of the spin detection : (conventional type detection): low/moderate; (ISHE type detection): moderate/high
The ISHE exists, but the AHE does not exist for a diffusion spin current (the flow of the spin without flow of charge).
The spin diffusion current is the spin current from a region of higher to region of lower spin polarization. This current does not require any electrical field. The spin diffusion current is the sum of two opposite currents of spin polarized and spinunpolarized electrons. The flows are exactly the same, but opposite. As a result, there is a spin current, but there is no charge current
Conventional Spin Detection 
Conventional Spin Detection. Nonlocal configuration. Under the applied voltage, the charge current J_{ch} (blue arrow) flow in the copper nanowire between two left electrodes. The conduction electrons in Fe are spin polarized, therefore the spinpolarized electrons are injected and accumulated in Cu. In contrast to the charge current J_{ch} (blue arrow), which can flow only along an electrical field, the spin current J_{spin} (red arrow) does require the electrical field and flows to the right. The he spin current J_{spin} (red arrow) is detected by the right MgO:Fe electrode. 
Physical mechanism of the conventional spin detection: The conductivity Fe:MgO electrode is strongly spin dependent. The conductivity is the lowest, when spin direction of conduction electrons is along the magnetization of the Fe. It is the highest, when the spin direction is opposite to the Fe magnetization. As a result, the charge is accumulated along spin current J_{spin} (red arrow) flowing through the spin detection Fe:MgO electrode. Due to this charge accumulation a voltage is built up between the Fe:MgO electrode and right end of the Cu wire. 
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Only the spin polarized electrons experience the ISHE effect. Therefore, the Hall voltage is induced according to the flow direction of spin polarized electrons.
Both the spin polarized and spin unpolarized conduction electrons experience the AHE effect. Since flows of spinpolarized and spin unpolarized electrons are the same and opposite, in total there is no AHE effect for a spin diffusion current.
Yes. (use 1) It can be used to measure the spin polarization of conduction electrons in the ferromagnetic metal (See here). (Use 2) It can be used to measure the diffusion spin current (Similarly to a measurement of a diffusion spin current in a non magnetic metal (See fig 31.)
(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.
(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
Spin Hall effect and symmetry 
Across the wire the structure of the wire is absolutely symmetrical. There is nothing which can distinguish the left side from the right side. Still at the right side the spin direction is up and the left side of wire the spin direction is down. Why not opposite?? What does the symmetry between the left and the right? 
Fig 2 (left) 2D scan of Kerr rotation angle θ_{Kerr} in GaAs 30umwide wire. θ_{Kerr} is proportional to number of spinpolarized electrons n_{S } , The red indicates the region where the spin direction is up. The blue indicates the region where the spin direction is down. In all other region the electrons are not spin polarized. 
See above Fig 2 for more explanations 
Y. K. Kato, et. al, "Observation of the Spin Hall Effect in Semiconductors". Science 306, 19101913 (2004) 
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Figure 2 shows the measured distribution of the spin polarized electrons induced by the Spin Hall effect in a nonmagnetic material. In the direction across the wire, the wire is absolutely symmetrical. There is nothing which makes the left different from the right.
The polarity of of the Schottky barrier and the electrical field associated with it, is fixed at each side of a metallic or semiconductor wire. E.g. at the left side of the wire the electrical field is directed to the right and the direction of H_{SO} and correspondingly the direction of the spin accumulation is up. In contrast, at the right side of the wire, direction of the electrical field is right. Therefore, the directions of H_{SO} and spin accumulation is down.
For example, if all defects are negatively charged negatively, the conduction electron passing the defect from right experience the electrical field from the defect to the left direction. Correspondingly, if it passing from the right direction, the electrical field is to the left. It fixes the directions of H_{SO} and spin accumulation and spin accumulation to be up at the left side of wire and down at the right side of the wire.
In the case of positively charged defects, the spin direction of the spin accumulated electrons is opposite.
A.The spin Hall is originated from the spindependent scatterings, which make the electron current to be dependent on the spin direction.
All spin dependent scatterings are distinguished by whether the electron wave vector (skew scattering) or the position of electron (sidejump scattering) is changed after a scattering. In any material both types of the scattering coexist together. However, the properties of these two transport mechanisms are very different and, importantly, the contributions of each mechanism are very different for each individual material.
(skew scattering) The skew scattering describes all types of scattering when the movement direction of a scattered electron depends on the spin. For example, there are more spinup and less spindown scattered electrons, whose moving direction is changed to the right with respect to the electron moving direction before scattering. And there are more spindown and less spinup scattered electrons, whose moving direction is changed more to the left.
(sidejump scattering) The sidejump scattering describes all types of scattering when the position shift of a scattered electron depends on the electron spin. For example, there are more spinup and less spindown scattered electrons, whose position is shifted to the right with respect to the electron position before scattering. And there are more spindown and less spinup scattered electrons, whose position is shifted to the left.
(Origins of skew and sidejump scattering could be different and not only a defect) The scattering on a defect is only a good example, which is simple enough to understand and to visualize the feature of the effect and the transport. Both the skew scattering and sidejump scattering occurs for a scattering unrelated to a defect. An example of a stronglyspin dependent sidejump scattering is the scattering across an interface, in which one or two materials are ferromagnetic. Obviously, the scattering probability is very different whether the electron spin direction is parallel or opposite to the magnetization of the material, in which it is scattered.
(General nature of two mechanisms of transport for skew and sidejump scatterings ) In fact, it is not only the spin transport, but all types of the electron transport can be divided into two big groups depending on these two types of scatterings (often they are named differently). Nearlyall properties of these two transport mechanisms are drastically different. The bulktype transport mechanism (the common mechanism in the bulk of a metal and a semiconductor) means the probability of the scattering (like the screw scattering) is higher when the movement direction of scattered electrons is changed f towards the electrical field than toward the opposite direction (See here or here). It results that the number of electrons, whose wave vector is directed towards the electrical field, is larger than the number of electrons with opposite wave vectors. It is the same to say that the number of electrons, which moves towards the electrical field are larger than the number of electrons moving in the opposite direction, and, therefore, there is an electrical current. Another transport mechanism (hopping conductivity, the transport across a tunnel junction etc.) is based on the scatterings similar to the sidejump scattering: The change of the scattered electron position with respect to its position before scattering is matter for this mechanism. The probability of the electron scattering, after which the electron position is shifted along the direction of the electrical field, is higher than the scattering probability, after which the electron position is shifted opposite to the electrical field direction. It results in an electrical current. You can find on my Web pages these two mechanisms.
(a simple answer) Skew scattering belongs to the transport mechanism number one. It is called the band current and is due to scatterings between electron states of a different wave vector. It is a dominated transport mechanism for a metallic or semiconductor material with a relatively high conductivity. That is why the skew scatterings are dominated for a material of a higher conductivity. The sidejump scattering belongs to the second type of the conductivity, which is due to the scatterings between electron states of different special position. Examples of this type of the conductivity are tunnel current and the hoping conductivity. In a material with a moderate or high conductivity, the first transport mechanism is orders of magnitude more efficient than the second transport mechanism and, therefore, the second transport mechanism is often ignored in a calculation. The conductivity decrease usually means the 1st transport mechanism decreases. As a result, the contribution of the 2nd transport mechanism becomes important and, therefore, the contribution of the sidejump scattering becomes more substantial.
(The reason why 1st electron transport mechanism is more efficient than the 2nd transport mechanism :). It is because the probabilities of the corresponded scatterings are very different. The scattering probability is different, because of difference of overlap of electron sates with a different wave vector and electron states of a different special position. The overlap of wave functions of electrons of a different wave vector is substantial. As a result, the scattering probability between theses electrons is high (Screw scattering). The overlap of wave functions of electrons of a different spatial position is relatively small. As a result, the scattering probability between theses electrons is low (Sidejump scattering).
(underestimation of a sidejump scattering in a material of a high conductivity :) Having said that, I should note that a negligible contribution of the 2nd transport mechanism and, therefore, the sidejump scatterings in a material of a higher conductivity is an underestimation. Obviously the tunneling conductivity and the hopping conductivity are ordersofmagnitude smaller than the bulk conductivity. Therefore, historically the contribution of the 2nd transport mechanism is never even considered for the electron transport in a material of a high conductivity (e.g. in a highcrystal quality semiconductor). However, it is unfair to associate the 2nd transport mechanism only to the tunneling and hopping conductivities for the following reason: When the conductivity becomes larger, the electron mean free path becomes larger. It means that the effective length of electron wave function becomes larger (meaning the electron size increases). It results that the overlaps with electron wave function with the wave functions of the neighbor electron states of a slightly different spatial position becomes larger for a larger number of states.. Only there is a question how to verify this obvious fact and how to measure experimentally the ratio of two contributions.
(experimental estimation of the ratio of the contributions of skew and sidejump scatterings :). As an experimentator I am interesting how to measure the ratio of two contributions. I guess you know such an experimental method. I do not know it. Maybe I have missed such an experimental method. I believe it is difficult. It should be possible because the properties of the 1st and the 2nd transport mechanisms are very different. Especially, the spin transport is drastically different for these two mechanisms. For the 1st mechanism, nearly for all materials there is no difference in conductivity between spinpolarized and spinunpolarized electrons. In contrast, for the 2nd transport mechanism, such difference is substantial. As a result, there are many interesting spindependent effects (e.g. spin detection, MR) for the 2nd transport mechanism. From this difference between two transport mechanisms, it might be possible to estimate the ratio of the screw to sidejump scattering for a specific material. I don’t know. Maybe you know?
A.(magnetic or non magnetic defects): It is not necessary. The defect can be either magnetized or nonmagnetized. For a side scattering, the electrical field around the defect is important, which induces spinorbit interaction and which makes a scattering to be spindependent. For a skew scattering, the change of electron movent direction is important. Therefore, therefore the electron just should interact with another particle or object for the momentum conservation (e.g. a defect, a phonon etc.).
Scattering of spinpolarized conduction elections on magnetic defects originates the effect of Anisotropic Magnetic resistance (AMR)/ Planar Hall effect (See here) and the effect of the spindependent conductivity (See here). Also see a general discussion here.
A. It depends on type of the contribution. The effects usually are direction dependent and spin dependent. It means that for a fixed direction of current flows the direction of spin polarization is fixed.
(point 1). only contribution 1 may exists in the middle of the wire. The contributions related to the interface electrical field, interface scattering, defect electrical field and defect scattering exist only in close proximity of an interface.
(point 2 for Spin Hall effect only) . The Spin Hall effect separates conduction electrons of two opposite spin directions and therefore creates the spin polarized electrons of two opposite spin directions. However, at one spacial point the spin polarized electrons of only one spin direction can exist for a long time. The spin polarized electrons of different spin directions are mixed up together very quickly by scatterings into one group of one spin direction. This effect is called the Spin Torque effect..
It is not the defect, which is important for the Spin Hall effect, but the electrical field it produces. The electrical field and its gradient should be as large as possible. It is important that the direction of electrical is opposite at left and right sides of the defect. It is the case because the electrical field is originated at a point position of the defect.
The neighbor defects should not be too close to each other. Otherwise, their electrical fields are overlapped and there is no effect.
Additionally, the size of conduction electron should be smaller or comparable with the average distance between defects. Main essence of the scattering mechanisms is that the scattered electron experiences a different H_{SO} whether it shifted to the left or to the right after the scattering (Sidejump mechanism) or whether its movement direction is turned to the left or to the right after the scattering (Skewscattering mechanism). In average, it should be some difference in which direction the electron is shifted or in which direction the movement direction of electron is changed. Therefore, the optimum distance between defects are important for these mechanisms.
The electrical field of defect is screened by conduction electrons. Fortunately, the screening is effective mainly in a thick bulktype conductors.
Abovementioned problems do not exist for Sidejump scatterings and Skewscattering across an interface. Therefore, the Spin Hall effect due to an interface is easier to understand and optimize.
No, it is not correct. The absolute position of the electron with respect to the defect position and consequently the absolute value of H_{SO} are not as important for the scattering mechanisms of the Spin Hall effect. The change of H_{SO} is only important. It does not matter whether the H_{SO} is positive or negative before or after electron scattering. It only matters that the H_{SO} increases when the electron is scattered to the left and the H_{SO} decreases when the electron is scattered to the right (as a example). This does not depend whether the electron is at left or right side from the defect. For any electron position all contributions are complementary. However, a neighbor defect may give opposite contribution (See the question above).
I believe the use (both used calculations and used measurements) of the Hall angle and the Hall conductivity are speculative at present. Some numerical value should be used in order to characterize the strength of the Spin Hall effect and the use of the Hall angle (similar as it is done for the Ordinary Hall effect) looks fine. However, in case of the Ordinary Hall effect the use of the Hall angle is very natural. It is just the angle, at which a conduction electron turns between two consequent scatterings. For the Spin Hall effect the definition is not as clear. The spin Hall effect describes a creation of a spin current, which is proportional to the spin polarization of the electron gas. When the bias current increases, the spin polarization cannot increase infinitely. It saturates at 100 %. Therefore, there is some ambiguity. I hope some more descriptive parameter will be used in future
However, for now maybe the Spin Hall angle is fine. Surely, it should be measured and calculated correctly.
There is some co relation between the spin Hall effect and the spin relaxation.
In an electron gas the spin relaxation occurs due to an incoherent spin precession of spin polarized electrons, which move in different directions or have different spacial coordinates (see here). The spins relaxation occurs due to different directions or/and strengths of the magnetic field H_{SO} of spin orbit interaction for conduction electrons which whether move in different directions or have different spacial coordinates.
1. Yes. There is a relation between the strength of the Spin Hall effect and the rate of the spin relaxation. The different contributions to the spin Hall effect have different strengths of the contribution to the spin relaxation. Therefore, it is not "one to one" relation.
2. The simplest case to understand this relation is the relation between Origin 1 of Spin Hall effect and the the corresponded spin relaxation contribution (see here). It is the case when a conduction electron has a nonzero orbital moment. Additionally to the movement along a metal, the conduction electrons rotates simultaneously around many atomic nuclei of the host metal (See here). The electrical field of the atomic nuclei induces a magnetic field H_{SO}, which direction is different for conduction electrons moving in different directions.
2a. contribution to Spin relaxation. Spins of all spin polarized conduction electrons are directed in one direction (See details here) . There is a spin precession around H_{SO}. Since electrons, which move in different directions, experience H_{SO} in different directions, the axis of the spin precession is different for spin polarized electrons moving in different directions. It results to a misalignment of spins of spin polarized electrons and therefore to the spin relaxation.
2b. Contribution to Spin Hall effect. The dependence of H_{SO} on the electron movement direction makes the electron scattering probability spin and direction dependent. The scattering probability is higher to a state of a smaller energy, where the electron spin is aligned along H_{SO}. The scattering probability is lower to a state of a higher energy, where the electron spin is aligned opposite to H_{SO}. Without an electrical current, the difference of the scattering probabilities does not affect anything, because there is an equal amount of conduction electrons moving in any two opposite directions. When there is an electron current, there are more electrons moving along current than in the opposite direction and the difference of the scattering probabilities starts to affect the electron transport. E.g. there are more spin up electrons scattered to the left with respect to the current direction than to the right and there are more spin down electrons scattered to the right than to the left. Such difference creates a spin polarized current flowing perpendicularly to the electrical (charge) current.
2c. Both the strength of the spin relaxation and strength of the Spin Hall effect are proportional to H_{SO}. Therefore they are related!!
3. All origins of the Spin Hall effect: Origin 1, Origin 2, Origin 4 and Origin 5 have corresponded contributions to the spin relaxation. Even though these origins of the Spin Hall effect look different and have different sources, they are very similar. For each origin, the H_{SO} is different either for a different electron movement direction or for a different position of a conduction electron. 3 coordinates (x,y,z) of electron position + 3 coordinates (k_{x}, k_{y}, k_{z}) of electron speed (wave vector) are called phase space coordinates. A conduction electron is constantly scattered between its possible phase space coordinates at a high rate. In the case when the direction of H_{SO} is different for different electron phase space coordinates, there are always Spin Hall effect + spin relaxation. The spin relaxation is due the incoherent spin precession at different phase space coordinates and the Spin Hall effect is due to the spin and phase space coordinate dependence of the electron scattering probability. Both the strength of the spin relaxation and strength of the Spin Hall effect are proportional to the difference of H_{SO} for possible phase space coordinates. Therefore they are related!!
4. The contribution due to the Ordinary Hall effect is very similar to the Spin Hall effect and it is rather strong in a ferromagnetic metal. However, it is absolutely not related to the spin relaxation.
There are many currents for this effect, which should be distinguished somehow. The bias current is the simplest conventional charge current, which just flows along a metallic wire when a voltage applied to the wire. I took such definition from current definitions in a transistor. How would you call it? The most clear and understandable name should be used.
(about defect density) See here
(about the strength of different contributions):
Fig,3 shows the contribution to the Hall effect, which is called the bulk contribution, which practically does not exist in reality, but it is very popular among theoreticians.
The purpose of Fig.3 is mainly educational to simplify the understanding of the origin of the Spin Hall effect, which is the spindependent scatterings and how the spin dependent scattering leads to the spin accumulation in a nonmagnetic material.. I hope Fig.3 is not confusing. The mechanism of the skew scattering, which is one of main mechanisms of the Spin Hall effect, is very similar to the bulk contribution, but it is slightly more complex to explain. Therefore, Fig.3 might be a good starting point to understand the spindependent scatterings and how they create the Spin Hall effect. The problem of the bulk contribution is that it is based on many "ifs":
(1st "if":) It occurs only if the electron distribution of the orbital moments depends on the electron wavector.
(2nd "if":) It occurs only if the electron distribution of the electron spins depends on the electron wavector, which means that the spin distribution follows the distribution of the orbital moments. The electron scatterings make these two conditions very improbable In contrast, the existence of the skew scattering only requires the depence of the scattering probability on the electron spin and the wavector, which is very often the case. (under the same symmetry requirements as for the bulk contribution). Therefore, I believe that the bulk contribution is always negligible in comparison to the contribution from the skew scattering.
(1st "if":) Why is it possible:. It is one of the predictions of the kpmodel. The kp model fixes the orbital moment of a conduction electron to its orbital moment. For example, in a direct band semiconductor the conduction electrons have zero orbital moment at k=0, but the orbital moment becomes nonzero and increases, when k increases (or electron energy increases). Additionally, it could be some dependence on the polarity of k in a system with broken geometric symmetry. For example, along a direction perpendicular to an interface or along a direction of a consequence of 3 different atoms like: atom1atom2atom3atom1atom2. In the reversal direction, the atomic consequence is different: atom1atom3atom2atom1atom3. The reason why the orbital moment is fixed to the wave vector k can be understood by a comparison of the electron wave function to an electromagnetic wave. A conduction electron of a + or  orbital moment is similar to a + or  circular polarized wave. A circular polarized wave can be an eigen wave in a nonmagnetic metal with a birefringence.
(1st "if":) The problem: The energy difference for the opposite k is usually smaller than kT. It makes the dependence of the electron distribution (the orbital moment distribution) on the polarity of k to be very weak.. (2nd "if":) Why is it possible:. There is a magnetic field of the spinorbit interaction Hso along the orbital moment. The electron spin is aligned along Hso. The non zero orbital moment means that the timeinverse symmetry of the Bloch function for a conduction electron is broken. In this electric field of nucleus creates Hso.
(2nd "if":) The problem: It takes time for the spin to align along Hso. This time is called the precession relaxation time. This time is not short, because precession relaxation requires an involvement of an external nonzerospin particle. However, the electron scatterings misalign the spins of the conduction electrons with a very fast speed. There is just not enough time for the electron spin to align along Hso before occurring of a scattering event, which misaligns spin with a very high probability.
The electron gas, in which the spin distribution depends on the electron wavevector k and, more specifically, on the polarity of k, is called the Rashbatype electron gas. To my present knowledge, the Rashbatype electron gas exists only below a very low temperature. At a higher temperature, it is destroyed by the electron scatterings.
To conclude, the bulk contribution to the Spin Hall effect may exist theoretically, but it is very small and often negligible in comparison to the contribution due to the skew scatterings. The spindependent and kdependent skew scattering exist almost at the same conditions for the broken geometry symmetry as required for the bulk contribution, the skew scatterings are independent of abovedescribed "ifs" and occur with a high probability.
I will try to answer your questions as soon as possible