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Spin Hall effect. Inverse spin Hall effect

Spin and Charge Transport

The 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 spin-polarized electrical current. Both effects have the same origin, which is spin- dependent scatterings due to the spin-orbit interaction



Uniqueness of the Spin Hall effect: It causes a completely non-magnetic 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 Spin-Orbit torque). It is not only a feature of a non-magnetic 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 HSO (See details here). Some descriptions might be different from an official view. My use of terms "Skew scatterings" and "Side-jump 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 2014-2019. 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.


Content

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(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 spin-orbit interaction create the spin current?

(3). Mechanism of Spin Hall effect: (Origin 1) due to a non-zero orbital moment of conduction electrons

(4). Mechanism of Spin Hall effect: (origin 2) Skew scatterings

(5). Mechanism of Spin Hall effect: (origin 4) Side-jump scatterings at defect

(6). Mechanism of Spin Hall effect: (origin 5) Side-jump 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 effect

similar effect 1: Electron movement in electrical field (Schottky- type) at interface
similar effect 2: Ordinary Hall effect
similar 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 side-jump or skew scatterings. Scattering on a defect or a general scattering?
skew scattering
side-jump scattering
Origins of skew and side-jump scattering could be different and not only a defect
General nature of two mechanisms of transport for skew and side-jump scatterings
(q1a) Why is Skew scattering dominant in higher mobility samples whereas, side jump is dominant in low mobility samples?
a simple answer
The reason why 1st electron transport mechanism is more efficient than the 2nd transport mechanism
underestimation of a side-jump scattering in a material of a high conductivity
experimental estimation of the ratio of the contributions of skew and side-jump scatterings
(q2) The defect should be magnetized or not?
(q3) About direction of the spin current
(q4) About Spin Hall effect in the middle of the bulk of the wire.
(q5) Is the Spin Hall effect proportional to the number of defects?
(q6) Influence of the Position of conduction electron with respect to the defect position. Averaging over all positions.
(q7) about Spin Hall angle
(q8) spin relaxation time (the spin life time) of conduction electrons and the strength of the spin Hall effect.
(q9) about orbital-dependent scatterings & bulk contribution & Fig.3

 

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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 spin-orbit interaction, the scattering probability for spin-up electrons is higher for a scattering to the right than to the left and in contrast the scattering probability for spin-down electrons is lower for a scattering to the right than to the left. As a result, the spin-up electrons is accumulated at the right side of wire and the spin-down 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 spin-up or spin-down 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)
click on image to enlarge it

(What is the Spin Hall effect?) The Spin Hall effect describes the generation of spin current, which flows perpendicularly to the electrical charge current

(Origin of the Spin Hall effect. All contributions) The Spin Hall effect occurs due to the dependence of the magnetic field HSO of spin- orbit interaction on the phase coordinates of a conduction electron. 6 phase coordinates= wave vector ( kx, ky, kz) + spacial coordinates (x,y,z)

(Different contributions (origins) to the Spin Hall effect) Different contributions corresponds to different sources, which make the HSO dependent on the electron phase coordinates.

(origin 1)Source: orbital moment of a conduction electron It makes HSO dependent on electron movement direction.

(origin 2)Source: skew scattering on defects It makes HSO dependent on electron movement direction.

(origin 4)Source: side- jump scattering on defects It makes HSO dependent on electron spacial position.

(origin 4)Source: side- jump scattering across an interface It makes HSO dependent on electron spacial position with respect to the interface.

(source of spin-polarized current) Spin- dependence of scattering probability of conduction electron. The scattering probability is higher when the spin of the scattered electron is parallel to HSO and the scattering probability is lower when the spin of the scattered electron is anti parallel to HSO

(magnetic field HSO of spin- orbit interaction) The HSO is the magnetic field, which an electron experiences when it moves perpendicularly to an electrical field. See about the spin- orbit interaction here.


 

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 spin-up electrons is higher to the left than to the right and correspondingly the scattering probability of spin-down electrons is higher to the right than to the light. As a result, more spin- up electrons flow to the left and more spin-down 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 HSO of the spin-orbit interaction, which the electron experience whether it is scattered to the left or to the right. E.g. if the direction of HSO is up in the left direction, the scattering probability of spin-up electrons is higher than of spin- down electrons for scatterings to the left. Correspondingly, the direction of HSO is down in the right direction and the scattering probability of spin-down 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 HSO depends on the electron movement direction. For the scattering current, the HSO depends on the electron position.


 

 


Spin Hall effect. Band current.

Origin 1. Spin Hall effect due to a non-zero orbital moment of conduction electrons

Distribution of speeds, movement directions, orbitals moments and magnetic field HSO for conduction electrons (with and without electrical current)

rotational moment of a conduction electron

Fig. 3. Distribution of electrons with orbital moment and distribution of HSO. The length of a vector from axis origin to sphere is proportional to the number of electrons moving in the vector direction. The direction of orbital moment and HSO 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 HSO of the whole electron gas are shown in top- right corner. Red arrow shows the direction of electron movement. Blue circle shows the direction of orbital moment. Violet arrow shows direction of HSO induced by the orbital moment.
The scattering probability is proportional to the electron energy. The energy is different for electron spin- parallel and antiparallel to HSO. It makes the scattering probability spin- dependent. Directional dependence of HSO makes the electron scattering probability direction- dependent. E.g. a spin-left conduction electron, which is moving along y- direction and scattered in xy plane, experiences a larger HSO,x along its spin for scatterings to the left than to the right. As a results, it scattering probability to the left is larger than to the right. When a conduction electron moves along crystal lattice, it simultaneously rotates around each atomic nuclear as it passes it. The electrical field of nuclear induces the magnetic field of spin-orbit interaction. The spin of conduction electrons interacts with HSO. There is a spin precession around HSO and spin precession damping, which aligns spin to HSO
(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 y-axis) 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 x-direction. 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 HSO, because of a higher energy.. As a result, spin-up electrons are scattered more to -x direction and spin-down electrons scattered more to +x direction (spin is referred with respect to y-axis). It makes more spin-up electrons moving along -x direction and more spin-down electrons moving along +x direction. Therefore, a spin current flows along x-direction.
note: This contribution exists only when a conduction electron has a non-zero orbital (rotational) moment
(fact):Each election moving in a different direction experience HSO of a different direction. It makes electron scattering spin- and direction- dependent, which result a flow of spin-polarized current perpendicularly to the electrical current
(additional fact) In total, average HSO, which experience by all conduction electrons (shown in left-top corner), is a non-zero. It is because, there are more along y-direction than in opposite direction. As a result, the rotational moment of conduction electrons is not compensated and the whole electron gas experience a non-zero HSO 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

click on image to enlarge it

A single common origin and physical mechanisms, which originate the SHE and ISHE.

 

(step 1): The electrical current in a non-magnetic metal creates the magnetic field HSO of the spin-orbit interaction.

(step 2): The interaction of HSO with electron spin makes the electron transport spin-dependent. The electron scattering probability becomes dependent on HSO and electron spin. As a result, the electron transport becomes both direction- dependent and spin- dependent. E.g. more spin-up electrons move, for example, to the left and more spin-down 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.

Two types of perpendicular current, which are created by SHE and ISHE

(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.

When a conduction electron is scattered, it may change slightly its moving direction. E.g. the electron current is in x-direction, before scattering a conduction electron moves along x-direction ( its wave vector is kx) and after the scattering its movement direction slightly turns to the right ( its wave vector becomes kx+ Δky) or to the left ( its wave vector becomes kx- Δky). The spin-orbit interaction makes the probability to change the moving direction to the left different from probability to change direction to the right. As a result, the amounts of electrons moving to the left and to the right becomes different and therefore a current flows perpendicularly to the x-direction.

(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.

When a conduction electron is scattered, it may change slightly its position. E.g. the electron current is in x-direction, before scattering the position of electron is (x,y) and after the scattering its position is slightly shifted to the right (x+Δx,y+Δy) or to the left (x+Δx,y-Δy). The spin-orbit interaction makes the probability, that the scattered electron is shifted to +y- direction (Δy>0), different from probability that the electron is shifted to -y- direction (Δy<0). As a result, in average the amount of electrons shifted along +y direction and -y direction are different. As a result, there is a drift of electrons along y- direction and therefore a spin-polarized current along the y-direction

(Just an example) The result of mechanism 1 +mechanism 2 + origin 1(example): More spin-up scattered electrons are moving to the left than to the right with respect to the electrical current direction. In opposite, more spin-down scattered electrons are moving to the right than to the left. Therefore, more spin-up electrons are moving to the left and more spin-down electron are moving to the right.

band current:

HSO depends on wave vector or movement direction of a conduction electron

(origin 1): due to a non-zero orbital moment of a conduction electron

(origin 2): Skew scatterings due to electrical field of defects

 

scattering current:

HSO depends on spacial position of a conduction electron

(origin 4):side-jump scatterings induced by defects

(origin 5):side-jump scatterings across an interface

 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Complementariness of the Spin Hall effect (SHE) and the Inverse Spin Hall effect (ISHE)

SHE and ISHE are fully complementary effects. They both are originated from the same origin (the spin-dependent scatterings), are induced by the same mechanisms and co-exits at the same conditions.

(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) Spin-depending scatterings. The dependence of magnetic field HSO of spin-orbit interaction on the movement direction (kx, ky, kz) and spacial position (x, y, z) of a conduction electrons.

(only difference between effect origins): Which electrons create the effect. Spin-polarized conduction electrons create ISHE effect. Spin-unpolarized 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, spin-polarized 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 spin-up electron scattered to the right and more spin-down electrons are scattered to the left. Since the group of spin- unpolarized conduction electrons has equal amounts of spin-up 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 spin-up electrons flow to the right and more spin-down electrons flow to the left. It means there is a spin current flowing from the left to right.

All mechanisms of SHE and ISHE (Skew scatterings; Side-jump scatterings at defect ; Side-jump scatterings across an interface etc.) have the same and the same origin!

Note: The spin and the charge are not complementary and not similar features of the nature. The spin describes the time-inverse symmetry breaking. The charge describes the gauge- symmetry breaking. (see details here). These symmetries are very different. E.g. the properties of charge current and spin spin current are drastically different. The striking full symmetry between the charge and the spin in SHE and ISHE effects should be considered as a coincidence.

Spin Hall effect: Scattering current

origin 4: side-jump scatterings due to defects origin 5: side-jump scatterings across interface

HSO induced by electrical field of a defect HSO induced by electrical field at interface

Fig.1a .Side- jump scattering mechanism at defects as the mechanism of the Spin Hall effect. Electrical field of a defect (shown in bright green) induces the magnet field HSO of spin-orbit interaction (red arrow). The electrical field is in opposite directions at the left and right sides of the defect. Correspondingly, the direction of HSO is opposite at the left and right sides of defects. The distribution of electron field is shown as green ellipse.

Fig.1b .Side- jump scattering mechanism across the interface as the mechanism of the Spin Hall effect. Electrical field at interface (shown in yellow) induces the magnet field HSO of spin-orbit interaction in the vicinity of the interface (blue plane). The origin of electrical field is the difference of work functions of materials at sides of interface and therefore a charge accumulation at interface. The case of electron accumulation is shown. The electrical field is in opposite directions at the left and right sides of the defect. Correspondingly, the direction of HSO is opposite at the left and right sides of defects. The distribution of electron field is shown as green ellipse.
For a spin- up electron: scattering probability to shift position to the left is higher than the probability to the right. As a result, there is a current of spin-up electrons to the left. For a spin- up electron: scattering probability to shift position to the left is higher than the probability to the right. As a result, there is a current of spin-up electrons to the left.
For a spin- down electron: scattering probability to shift position to the right is higher than the probability to the left. As a result, there is a current of spin-down electrons to the right. For a spin- down electron: scattering probability to shift position to the right is higher than the probability to the left. As a result, there is a current of spin-down electrons to the right.

Direction of HSO is up from left side of defect and down from right side. Therefore, the magnetic energy of a spin-up electron become larger when after a scattering its position is shifted towards left side, where HSO is along electron spin. As a result, the scattering probability of the spin-up electron is larger when the position of electron is shifted to the left with respect to the defect position. In contrary, for spin-down electron, the energy is larger at right side of defect and therefore scattering probability is larger when the position of electron is shifted to the right.

Direction of HSO is up in a material at left side of interface and HSO is down in a material at right side. Therefore, the magnetic energy of a spin-up electron become larger when it is scattered from the right side to the left side of the interface, where HSO is along electron spin. As a result, the scattering probability of the spin-up electron is larger when it scattered across interface from left to right than when it is scattered from right to left. The case of the spin down electrons is opposite. The scattering probability across the interface is larger when electron is scattered from the right to left material.

See below more details about Origin 4 of the Spin Hall effect

See below more details about Origin 5 of the Spin Hall effect

The interaction of HSO with the spin of a conduction electron makes the electron transport spin-dependent.

note: The direction of HSO depends at which side of the defect or the interface.
click on image to enlarge it. Zayets 2014

What is difference between the Spin Hall effect and the Inverse Spin hall effect? Do they have the similar physical origin?

Yes, they are complementary effects and they have the same physical origins.

(Spin Hall effect) acts on spin- unpolarized conduction electrons:

More spin-up electrons flows to the left and more spin-down electrons flows. As a result, there is a spin current flowing perpendicularly to the charge current. E.g. more spin-up electrons flows to the left and the same amount of the spin-down electrons flows to the right.

(Inverse Spin Hall effect) acts on spin- polarized conduction electrons:

E.g. If a spin-polarized electrical current has more spin-up electrons than spin-down 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 spin-polarized current as there is a flow of electrons to the left.

There are only one group of spin- unpolarized electrons and one group spin- polarized electrons of one specific direction (see details here). The electron scatterings quickly mix up the spin-polarized electrons of different spin directions. This fact makes the above analysis slightly more complicated, but the result remains the same.

Symmetry & Dependence on parameters:

Spin Hall effect is linearly proportional to

(1) charge current j;

(2) The number of spin-unpolarized electrons or (1-sp), where sp is the spin polarization. The effect decreases, when the spin polarization sp increases.

symmetry: spin direction of accumulated spin-polarized 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 spin-polarized 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

HSO 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 HSO of spin-orbit interaction. The directions of HSO 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 spin-polarized 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 spin-up 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 spin-down electrons to the right.
Electrons, which moves perpendicularly to the applied electrical field E, experience an effective magnetic field of spin-orbit interaction HSO. Direction of this magnetic field is different for electrons, which moves parallel and anti parallel to the x-axis. The electrons, which move along the electric field (along the y-axis) do not experience any magnetic filed of the spin-orbit 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 HSO, the probability for an electron to be scattered toward and opposite to the x-xis direction are different. This causes a flow of spin current along the x-axis, when a drift current flows along y-axis.

See below more details about Origin 2 of the Spin Hall effect

click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

How does the spin-orbit interaction make a non-magnetic material become magnetic. How does it create the spin accumulation and the spin current?

The electrical current induces the magnetic field HSO of the spin-orbit interaction (see details here). The interacts differently with conduction electrons, which spin is parallel and opposite to HSO. It causes spin-dependent current (spin current)

The spin-orbit interaction cannot break the time-inverse symmetry. It cannot create a magnetic field by itself. Therefore, how can it make a material to become magnetic and a create the spin accumulation and the spin current?

The electrical current breaks the time-inverse symmetry. The spin-orbit interaction (SO) only enhances that breaking.

 

 

 

 

 

 

 

 

 

 

 

 

 

two groups of origins of Spin Hall effect

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

group 1: Band current

group 2: Scattering current

1st Eq. shows that scattering probability p() to turn toward the left is different from probability to turn to the right. 2d Eq. shows that scattering probability with a turn of movement direction is spin-dependent.

1st Eq. shows that scattering probability p for a conduction electron to shift its position to the left is different from probability to shift to the right. 2d Eq. shows that scattering probability with a position shift is spin-dependent.

it is also called the intrinsic contribution to the spin Hall effect

it is also called the extrinsic contribution to the spin Hall effect

(origin 1) Spin- dependence and directional- dependence of electron scattering direction.

(origin 2) Spin- dependence and directional- dependence of electron spacial shift after a scattering

(directional dependence):When a conduction electron is scattered, it may change slightly its moving direction. E.g. There is an electron current in x-direction. Before scattering a conduction electron moves along x-direction ( its wave vector is kx) and after the scattering in its movement slightly bent ( its wave vector becomes kx+ Δky) . The spin-orbit interaction makes the probability to change the direction to the left different from probability to change direction to the right. As a result, the amounts of electrons moving to the left and to the right becomes different and therefore a current flows perpendicularly to the x-direction. (directional dependence) When a conduction electron is scattered, it may change slightly its position. E.g. There is an electron current in x-direction. Before scattering the position of electron is (x,y) and after the scattering its position is slightly shifted and becomes (x+Δ x,y+Δy). The spin-orbit interaction makes the probability, that the scattered electron is shifted to +y- direction (Δy>0), different from probability that the electron is shifted to -y- direction (Δy<0). As a result, in average the amount of electrons shifted along +y direction and -y direction are different. It creates an electron current along the y-direction
(spin dependence): E.g. there are more spin-up scattered electrons, which movement direction is changed to the left than to the right. In contrast, there are more spin-down scattered electrons, which movement direction is changed to the right than to the left (spin dependence): E.g. there are more spin-up scattered electrons, which are shifted to the left than to the right. In contrast, there are more spin-down scattered electrons, which are shifted to the right than to the left

group 1 contains: (1) origin 1: due to a non-zero orbital moment of conduction electrons; (2) origin 2: Skew scatterings;

group 2 contains: (1) origin 4: Side-jump scatterings at a defect; (2) origin 5: Side-jump scatterings across an interface

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


How does the spin-orbit interaction create the spin current flowing across an electrical (charge) current?

Spin-dependent scatterings:

Origin of the Spin Hall effect

from: spin ↑↓ HSO;

into: spin ↑↑ HSO

from: spin ↑↑ HSO

into: spin ↑↓ HSO

The Spin Hall effect occurs in a material, in which there are spin-dependent scatterings. A scattering is spin-dependent 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 spin-orbit 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.

How does the spin-orbit interaction makes electron scatterings spin- dependent and direction- dependent?

Spin-dependent scatterings as the Origin of the Spin Hall effect

 

The spin- orbit interaction is a relativistic effect (it is not quantum- mechanical effect). It describes the fact that an electron experience a magnetic field HSO, when it moves perpendicularly to an electrical field. Except the existence of the magnetic field HSO, the spin- orbit interaction does not produce anything!!!!

 

Spin-dependence of scattering probability .That means that the scattering probability of a conduction electron from a quantum state, in which the effective magnetic field HSO of the spin orbit-interaction is parallel to the electron spin, to a quantum state, in which the HSO is anti-parallel 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.

 

 

Scatterings.

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 side-jump scatterings. The

The scatterings, after which the electron changes only its movement direction, are defined as skew scatterings

Both the side-jump scatterings and the skew scatterings may be spin-dependent and they both may contribute to the Spin Hall effect.

See more about the scattering current here

Direction- dependence and spin-dependence of electron scatterings as the origin of the Spin Hall effect

Origin 1 of Spin Hall effect: Band current

Origin 2 of Spin Hall effect: Scattered current

Spin-up electrons flow to the left

Spin-up electrons flow to the left

HSO depends on movement direction of electrons (black arrow). HSO is up, when electron moves to the left, and it is down, when electron moves to the right. Scattering probability (yellow arrow) is higher when electron movent direction turns to the left, because its spin is along HSO for movement to the left HSO depends on position of a conduction electron. HSO is up, when electron position is at the left side (e.g. of a defect or interface), and it is down, when electron position at the right side. Scattering probability (yellow arrow) is higher when electron is shifted after scattering to the left, because its spin is along HSO for the left shift
Spin-up electrons flow to the right Spin-up electrons flow to the right
In contrast, for the spin-down electron the scattering probability is higher when electron movent direction turns to the right, because in the right movement direction the spin and HSO are in the same direction (down) In contrast, for the spin-down electron Scattering probability for the right shift is higher., because down- spin is along HSO for the right shift
   

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

click on image to enlarge it

Why do you discuss the influence of the spin- orbit interaction only on scatterings. Does the spin- orbit interaction influence the electron movement or some properties of conduction electrons (e.g. energy, wave function, etc)?

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 (kx, ky, kz). 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 x-direction makes scattering probability to a state (kx+ Δkx, ky, kz) larger than to the state (kx- Δkx, ky, kz) . 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 x-direction makes scattering probability to a state (x+Δx, y, z) larger than to the state (x-Δx, y, z) As a result, the electrons are constantly drifted (moving) along x-direction 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 low-conductivity metal (See here)

There are only two possible origins of Spin Hall effects: band current and scattering current

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!

Of course, there are some exotic currents likes a current through a defect or a quantum dot or a single-electron transistor, which is not discussed here

 


Origin of the Spin Hall effect. Step by step.

Both effects have the same origin: spin-dependent scattering (or spin-dependent 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 HSO due to the flow of the bias charge current and the spin-orbit 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 HSO.

(effect origin. Step 1): Generation of HSO

note: the spin-orbit interaction cannot break the time-inverse symmetry by itself. It can be broken externally and SO just enhances the breaking. In the case of the Spin Hall and Inverse Spin Hall effects, the electrical current j breaks the time-inverse symmetry
The spin-orbit interaction (HSO) exists when an electron moves in an electrical field. Which electrical field induces HSO, which consequently originates the spin Hall effect?

(contribution 1) : the electrical field of the nuclear 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)

 

(effect origin. Step 2): Creation of spin- polarized current and spin polarization

creates current flowing perpendicularly to the bias current and this current is spin- dependent

(contribution 1) (main) : the spin dependent scatterings

(contribution 2) : spin precession around HSO 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 non-magnetic GaAs wire created by an electrical current

Fig.2 (left) 2D scan of Kerr rotation angle θKerr in GaAs 30-um-wide wire. θKerr is proportional to number of spin-polarized electrons nS , 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 spin-up and spin-down 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 HSO , 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, 1910-1913 (2004)
click on image to enlarge it

 

(case of non-zero orbital moment of conduction electrons) Even without an electrical current the conduction electrons move in the electrical field of a defect. It should produce HSO and a spin accumulation and the material should become magnetic. Why this is not the case?

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 HSO and all spin-dependent 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 +HSO than -HSO It causes more spin-up electrons moving, for example, to the left and more spin-down electrons moving to the right.

Note: Even without an electrical current, the spin-orbit interaction induced by the movement of the conduction electrons may influence the electron properties. One example is the energy splitting between heavy and light holes in a semiconductor (See here and here)

Figure 2 shows the experimental observation of spin accumulation in non-magnetic GaAs wire, when an electrical current flows throw it. There are spin-polarized electrons at sides of the wire. The spin-polarized 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 spin-accumulation.

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 cm-3), 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 HSO , 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.

Often two contributions to the Spin Hall effect are distinguished. They are called by different names (e.g. intrinsic and extrinsic). What it is?

Without

The electrical field if a defect in a conductor are screened by the conduction electrons. How does it influence the Spin Hall effect?

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 nuclear of the host atoms contributes similarly to the Spin Hall effect.

 
 

 

 


 

 

 

 

 


 


 


 

How the Spin Hall effect creates the spin-polarized current?

(fact 1): The electrical current breaks the time-inverse symmetry. The breaking of the time-inverse symmetry is the required condition for the electron gas to be spin-polarized.

(fact 2): The electrical current + the spin-orbit interaction create a magnetic field HSO, which interacts with the electron spin. The electrons, which moves in opposite directions, experience the magnetic field HSO in opposite directions.

(fact 3): The electron scattering probability depends on mutual directions on mutual directions of the electron spin and the magnetic field HSO. As a result, the electron scattering becomes spin- dependent. Since HSO 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.


 


Physical mechanisms, which cause The Spin Hall Effects

Origin 1, Band current:

(Mechanism 1): due to a non-zero orbital moment of conduction electrons

(Mechanism 2): Skew scatterings

Origin 2, Scattering current:

(Mechanism 3). Side-jump scatterings at a defect

(Mechanism 4). Side-jump scatterings across an interface


Mechanism of Spin Hall effect: (origin 1) due to a non-zero orbital moment of conduction electrons

Zayets 2019.12

It is major contribution to the Spin Hall effect in the case of a metal with a high conductivity

Orbital rotation of conduction electrons

Additionally to the movement along the crystal lattice, a conduction electron rotates around each atomic nuclear. Electrical field of the nuclear induces the magnetic field HSO of the spin-orbit interaction. The HSO affects the electron scattering probability and therefore creates the spin-polarized current. The effect is called the spin Hall effect.

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 nuclear (dark spheres).Electrical field of each nuclear induces the magnetic field interaction HSO. The electron experiences the accumulative strong HSO. Even though the electrons moves along stationary nuclears,accumulative HSO 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 nuclears.
camera moves with the electron
click on image to enlarge it

 

Explanation in short: Due to its non-zero orbital moment, a conduction electron experiences a magnetic field HSO of the spin- orbit interaction (See here). The HSO 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 non-zero rotational (orbital) moment,which created magnetic field HSO. There is a spin precession around HSO and the spin is aligning along HSO 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 HSO 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 non-zero HSO and the electron gas becomes spin-polarized.

(step 4) When there is an electrical current, the scattering probability of spin-up electrons to the left becomes different from the scattering probability to the right. As a result, e.g. the spin-up polarized current flows to the left and the spin-down polarized current flows to the right.

 

 

 

Spin Hall effect due to a non-zero orbital moment of conduction electrons

No electrical current. Distribution of speeds, movement directions, orbitals moments and magnetic field HSO for conduction electrons

There is an Electrical current J (yellow arrow).Distribution of speeds, movement directions, orbitals moments and magnetic field HSO for conduction electrons

rotational moment of a conduction electron

(Fig.3a) (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 (Fig.3b) (there is an electrical current along y-axis) 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 x-direction. This is origin 1 of the Spin Hall effect. (Fig. 3c) (Moving conduction electron experiences SO magnetic field HSO ). Red arrow shows the direction of electron movement. Blue circle shows the direction of orbital moment. Violet arrow shows direction of HSO induced by the orbital moment.

Fig. 3. Distribution of electrons with orbital moment and distribution of HSO.

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 HSO induced by the orbital moment. The direction of orbital moment and HSO 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 HSO of the whole electron gas are shown in top- right corner of the center figure.

HSO makes different the electron scattering probability towards the left and right movement direction!

HSO makes the electron scattering probability spin-dependent!

Electrical current J along y-direction creates spin-polarized currents along x- and z- directions

Each election moving in a different direction experiences magnetic field HSO in a different direction.
When a conduction electron moves along crystal lattice, it simultaneously rotates around each atomic nuclear as it passes it. The electrical field of nuclear induces the magnetic field of spin-orbit interaction. The spin of conduction electrons interacts with HSO. There is a spin precession around HSO and spin precession damping, which aligns spin to HSO.
click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

For this contribution, the magnetic field HSO of the spin- orbit interaction is induced of the electrical field of atomic nuclear of the host metal. As a result, it is substantial.

Spin- and directional- dependence of the scattering probability

In this case, Magnetic field HSO is induced by the electrical field of atomic nuclears of the host metal due to a non-zero orbital moment of a conduction electron
The magnetic field HSO makes the scattering probability of a conduction electron spin- and direction- dependent. This dependence is origin of this contribution of the Spin Hall effect

 

(spin- dependence of the scattering probability): the scattering probability is higher when the spin of the scattered electron is directed along HSO, because of its higher magnetic energy. As a result, spin-up electrons are scattered more to -x direction and spin-down electrons scattered more to +x direction (spin is referred with respect to y-axis). It makes more spin-up electrons moving along -x direction and more spin-down electrons moving along +x direction. Therefore, a spin current flows along x-direction.

 

(direction- dependence of the scattering probability): The direction of HSO is fixed to the movement direction of a conduction electron (see Fig. 3c). As a result, the direction of HSO is different for conduction electrons moving in different directions. The direction dependence of HSO causes the direction dependence of the scattering probability.

(fact):Each election moving in a different direction experience HSO of a different direction. It makes electron scattering spin- and direction- dependent, which result a flow of spin-polarized current perpendicularly to the electrical current

 

Additional effects of this mechanism

except of the creation of the spin- current

(additional effect 1)Inducing a magnetic field, which all conduction electrons of an electron gas experience

Each conduction electron experiences the magnetic field HSO . However, each electron moving if a different direction experiences HSO in different direction. E.g. electrons, which are moving in opposite directions, experience opposite HSO. 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 +HSO,z than -HSO,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 HSO 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 HSO is different for electrons moving in different directions. As a result, the angle between electron spin and HSO is different for the spin- polarized electrons moving in different directions. There is a spin precession around HSO. Since the directions of HSO 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 HSO 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 HSO with the electron spin.

May HSO induce the Lorentz force and the ordinary Hall effect?

No, the HSO cannot induce the Lorentz force, because of the relativistic nature of HSO. Therefore, HSO does not cause the ordinary Hall effect.

It is important: The effective magnetic field of the spin-orbit interaction does not induce the Lorentz force and it can not be a cause the ordinary Hall effect. The effective magnetic field of the spin-orbit interaction only affects the electron spin and it can not change the electron trajectory.

Can an external magnetic field make the electron scattering probability spin- and directional- dependent and therefore create the spin- polarized current flowing perpendicularly to the electric (charge) current.

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.

 

A conduction electron moves along the metal. How it is possible that the conduction electron simultaneously rotates around an atomic nuclear?

It is correct. A conduction electron moves along a metal and simultaneously rotates around thousands or even millions of atomic nuclears. See below for more details

 

 

 



Mechanism of Spin Hall effect: (Origin 2) Skew scatterings

Spin Hall effect. Band current.

Origin 2. skew scatterings

HSO 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 HSO of spin-orbit interaction. The directions of HSO 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 spin-polarized 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 spin-up 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 spin-down electrons to the right.
Electrons, which moves perpendicularly to the applied electrical field E, experience an effective magnetic field of spin-orbit interaction HSO. Direction of this magnetic field is different for electrons, which moves parallel and anti parallel to the x-axis. The electrons, which move along the electric field (along the y-axis) do not experience any magnetic filed of the spin-orbit 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 HSO, the probability for an electron to be scattered toward and opposite to the x-xis direction are different. This causes a flow of spin current along the x-axis, when a drift current flows along y-axis.
click on image to enlarge it

 

This contribution is substantial only in a metal with a low conductivity.

(explanation of effect):

A conduction electron experiences a magnetic field HSO of spin-orbit interaction, when they move at a non-zero angle with respect to an applied bias electrical field E. The HSO 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 HSO 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 HSO is induced by a bias electrical field, along which electrical current flows

The magnetic field HSO makes the scattering probability of a conduction electron spin- and direction- dependent. This dependence is origin of this contribution of the Spin Hall effect

 

A Phonon, a magnon, an impurity and a defect can be a source of a skew scattering.

 

What is difference between this origin (origin 2) and origin 1 of the spin Hall effect?

The sources of HSO are different. For this origin (origin 2), the source of HSO is the electrical field, along which the electrical current is flowing. In the case of origin 1, the source of HSO is the electrical field of atomic nuclears, which induces due to a non-zero orbital moment of a conduction electron. Except the source of HSO, the origins 1 and 2 are similar.

Fig. 11 shows that after the scattering the electron movement direction changes over 90 degrees. Is such extreme scattering rare?

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 HSO of spin- orbit interaction makes different scattering probability for a change of electron movement direction to the left and to the right.

Why is this contribution substantial only in a metal with a low conductivity?

For the same electrical current, the bias voltage is large for a metallic wire of a lower conductivity.

Why does the conduction electrons move perpendicularly to the bias electrical field? Are the electron flowing along the electrical field?

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.

(from Jan) what you meant with "bias current"?

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.

 

 


 

Mechanism of Spin Hall effect: (Origin 4) Side-jump scatterings at a defect

Spin Hall effect: Scattering current

origin 4: side-jump scatterings due to defects

HSO induced by electrical field of a defect

Fig.19 .Side- jump scattering mechanism at defects as the mechanism of the Spin Hall effect. Electrical field of a defect (shown in bright green) induces the magnet field HSO of spin-orbit interaction (red arrow). The electrical field is in opposite directions at the left and right sides of the defect. Correspondingly, the direction of HSO is opposite at the left and right sides of defects. The distribution of electron field is shown as green ellipse.

For a spin- up electron: scattering probability to shift position to the left is higher than the probability to the right. As a result, there is a current of spin-up electrons to the left.
For a spin- down electron: scattering probability to shift position to the right is higher than the probability to the left. As a result, there is a current of spin-down electrons to the right.

Direction of HSO is up from left side of defect and down from right side. Therefore, the magnetic energy of a spin-up electron become larger when after a scattering its position is shifted towards left side, where HSO is along electron spin. As a result, the scattering probability of the spin-up electron is larger when the position of electron is shifted to the left with respect to the defect position. In contrary, for spin-down electron, the energy is larger at right side of defect and therefore scattering probability is larger when the position of electron is shifted to the right.

The interaction of HSO with the spin of a conduction electron makes the electron transport spin-dependent.

note: The direction of HSO depends at which side of the defect or the interface.
click on image to enlarge it. Zayets 2014

 

This contribution is substantial only in a metal with a low conductivity.

(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 HSO of the spin-orbit interaction. The HSO 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 HSO 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 HSO is induced by a electrical field of a point-like defect or an electrical field from any a point- like source of electrical field

The magnetic field HSO makes the scattering probability of a conduction electron spin- and direction- dependent. This dependence is origin of this contribution of the Spin Hall effect

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.

Why the electron scattering probability is spin- dependent?

It is because of the spacial- dependence of HSO. 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 HSO 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 HSO and smaller when it antiparallel to HSO. As a result, the scattering probability is spin- dependent.

 

note 1. Scattering can be spin-dependent on both the magnetized and non-magnetized defect.
note 2. It is the electrical field around a defect, in which a conduction electron experiences the spin-orbit interaction, which makes the electron scattering on the defect spin-dependent.

 

How the spacial dependence of the scattering probability create a spin current?

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 direction and magnitude of HSO depend on mutual positions of the conduction electrons and the defect. Since a conduction electron can have any position with respect to the defect and may experience both up- and down- HSO, does still this contribution exist when it is averaged over all possible electron positions?

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 HSO -up field (or the same a smaller HSO -down field). when the scattered spin- up electron is shifted to the right, it experiences a smaller HSO -up field (or the same a larger HSO -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 mean-free path, the side-jump scatterings becomes position- and spin-independent 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 side-jump scatterings becomes position- and spin- independent. It means that at any spacial position a conduction electron experiences nearly the same HSO. 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.

How does the defect density influence this contribution? Since the contribution is proportional to the electrical field of a defect, the effect increases, when the defect density increases. Is it correct?

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 HSO. 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.

Details about the size of conduction electron and the mean- free path are here.

 

When the density of defects increases, the side-jump scatterings becomes position- and spin- independent and they do not contribute to the Spin Hall effect.

 

 

The electrical field around a defect is screened by conduction electrons. Does it influence the strength of this contribution?

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 mean-free path

 

 

 


 

Mechanism of Spin Hall effect: (origin 5) Side-jump scatterings across an interface

Spin Hall effect: Scattering current

origin 5: side-jump scatterings across interface

HSO induced by electrical field at interface (source 2)

Fig.1b .Side- jump scattering mechanism across the interface as the mechanism of the Spin Hall effect. Electrical field at interface (shown in yellow) induces the magnet field HSO of spin-orbit interaction in the vicinity of the interface (blue plane). The origin of electrical field is the difference of work functions of materials at sides of interface and therefore a charge accumulation at interface. The case of electron accumulation is shown. The electrical field is in opposite directions at the left and right sides of the defect. Correspondingly, the direction of HSO is opposite at the left and right sides of defects. The distribution of electron field is shown as green ellipse.

For a spin- up electron: scattering probability to shift position to the left is higher than the probability to the right. As a result, there is a current of spin-up electrons to the left.
For a spin- down electron: scattering probability to shift position to the right is higher than the probability to the left. As a result, there is a current of spin-down electrons to the right.

Direction of HSO is up in a material at left side of interface and HSO is down in a material at right side. Therefore, the magnetic energy of a spin-up electron become larger when it is scattered from the right side to the left side of the interface, where HSO is along electron spin. As a result, the scattering probability of the spin-up electron is larger when it scattered across interface from left to right than when it is scattered from right to left. The case of the spin down electrons is opposite. The scattering probability across the interface is larger when electron is scattered from the right to left material.

The view point moves together with electrons.

The interaction of HSO with the spin of a conduction electron makes the electron transport spin-dependent.

note: The direction of HSO 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 HSO at two sides of the interface between two metals. There are two sources, which make HSO 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 HSO of spin- orbit interaction. This difference of HSO 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.

The magnetic field HSO makes the scattering probability of a conduction electron spin- and direction- dependent. This dependence is origin of this contribution of the Spin Hall effect

 

 

 

 

(source 1 of HSO) Difference of orbitals moments in materials at sides of interface

In this case, Magnetic field HSO is induced by a electrical field of atomic nuclear of the host metal

 

 

(source 2 of HSO) Electrical field in the vicinity of interface

In this case, Magnetic field HSO is induced by a electrical field induced by a charge accumulation/ depletion at 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 side-jump 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 Side-jump scatterings at a defect


Spin- dependent scatterings as the common origin of the Spin Hall effect and the Inverse Spin Hall effect

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 non-zero orbital moment of conduction electrons; (mechanism 2): Skew scatterings (mechanism 3): Side-jump scatterings at defect (mechanism 4): Side-jump scatterings across an interface. Independently on the mechanism, the origin of SHE and ISHE is the same. The conduction electron experiences the magnetic field HSO of the spin-orbit interaction, which depends either on the electron movement direction (kx, ky, kz) or the electron spacial position (x, y, z) . The HSO makes scatterings of conductions electrons spin- and direction- dependent. As a result of the spin- and direction- dependency of scattering probability, the numbers of spin-polarized 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

Skew scattering is one several mechanisms of creation of HSO due to spin- orbit interaction

How does the spin-orbit (SO) interaction affect the scattering

Flow of conduction electrons in a metallic wire. The y - coordinate is along the wire length.. The electrical field E (blue arrow) applied along wire (along y- axis) and the main electron current flows along the y- axis. The spacial distribution of the conduction electrons are shown as the green ellipses.
When the conduction electron is scattered (e.g. on a defect (blue ball)), it may change its movement direction. E.g. to have movement component along the x-direction. The movement perpendicularly to E induces magnetic field HSO of spin-orbit interaction (red arrow)
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 HSO of spin-orbit interaction due to bias electrical field E. Only when there is perpendicular component of movement with respect to E, a conduction electron experiences HSO. The directions of HSO 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. The explanation of spin- and direction- dependency of scattering probability of a conduction electron. In bottom part, the energy distributions are shown for the different electron movement direction. The case before scattering is shown at center when the electron moving in the forward direction (along y- axis) and does not experience any HSO. When the electron is scattered to the left , it experiences HSO, which is along its spin direction. Due to the additional magnetic energy, the distribution is shifted to the lower energies (left distribution). When the electron is scattered to the right , it experiences HSO, which is opposite to its spin direction and the distribution is shifted to the higher energies (right distribution). When the conduction electron is scattered to the left, there are many available unoccupied states at the same energy. As a result, the scattering probability to the left is high. When the conduction electron is scattered to the right, the most of available states are occupied. As a result, the scattering probability to the right is low.
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.
click on image to enlarge it

 

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 spin-dependent?

A. The spin-orbit (SO) interaction (details see here)

How the SO interaction makes a scattering to be direction-dependent and spin-dependent?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Orbital moment of conduction electron as origin 1 of Spin Hall effect

Orbital rotation of conduction electrons

Additionally to the movement along the crystal lattice, a conduction electron rotates around each atomic nuclear. Electrical field of the nuclear induces the magnetic field HSO of the spin-orbit interaction. The HSO affects the electron scattering probability and therefore creates the spin-polarized current. The effect is called the spin Hall effect.

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 nuclear (dark spheres).Electrical field of each nuclear induces the magnetic field interaction HSO. The electron experiences the accumulative strong HSO. Even though the electrons moves along stationary nuclears,accumulative HSO remains constant. Camera moves with the electron

Simultaneously with movement along the crystal a conduction electron rotates around each nuclear.
The size of a conduction electron (size of its wave function) is relatively large. A conduction electron can cover simultaneously hundreds or thousands of nuclears.
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Existence of a non-zero 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.

 

Additional breaking of time- inverse symmetry for a conduction electron (in comparison to a localized electron)

Zayets 2019.12

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.

Can an electron of spherical symmetry (either a localized electron or a conduction electron) have an orbital moment?

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 non-zero orbital moment (e.g. p-,d-, f- symmetries for a hydrogen atom)

A conduction electron of spherical symmetry can have a non-zero orbital moment, because its time- inverse symmetry is broken along its movement direction !!!.

 

 

Why orbital moment and HSO are a non-zero for a conduction electron, but zero for a standing-wave 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 HSO (blue arrow) is fixed to movement direction of the electron (red arrow). The orbital moment and HSO are opposite for opposite movement of electron. As a result, the electron has no orbital moment and experiences no HSO!!

The standing-wave electron is a conduction electron, which is constantly bounced between two imperfection.
In contrast to an "usual" conduction electron, a standing-wave electron has no orbital moment and experiences no HSO!!
click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

What is difference between a localized electron, a conduction electron and a standing- wave electron?

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 nuclears. 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.

Every conduction electron has a non-zero momentum  and a non-zero wave vector. How it is possible that the standing- wave conduction electrons does not move?

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 standing-wave conduction electron is zero and it is does not move in the space.

Is the orbital symmetry for conduction electrons only s- and p-type and for localized electrons only p-, d- and f- type?

A No.

 


Physical mechanisms of spin depend scatterings

How does the spin-orbit (SO)interaction affect the scattering

An electron at different sides from a defect experiences the opposite magnetic field of SO. As a result, it has a different energy and symmetry. It causes a different scattering probability towards left and right. Since only the electron spin interacts with the magnetic field of SO, the difference of scattering probabilities is spin-dependent. The first origin of direction- dependency and spin- dependency of the electron scattering is the direction dependency of electron energy, which is induced by SO. More details see here. The second origin is the direction dependency of the electron symmetry.
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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 spin-orbit 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 spin-orbit interaction may induce the Lorentz force or that it may induce the ordinary Hall effect. It is very incorrect.

 

Explanation why the spin-orbit 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 spin-orbit-interaction. 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 spin-orbit interaction can not cause the ordinary Hall effect.

 

 


click on image to enlarge it

3 types of the Hall effect

Inverse Spin Hall effect (ferromagnetic metal)

The charge is accumulated, when a spin-polarized drift current flows

Spin Hall effect

The spin is accumulated, when a spin-unpolarized drift current flows

Inverse Spin Hall effect (non-magnetic metal)

The charge (and spin) is accumulated, when a spin diffusion current flows.

Under an applied voltage the electrical current (drift current) flows in the metal wire. When the metal is ferromagnetic, the drift current is spin-polarized. Therefore, there are more electrons with spin directed up. It causes more electrons be scattered into the left than into into the right. This is the reason for the charge accumulation at the right side of the wire. Under an applied voltage the drift current flows in the non-magnetic metal wire. The drift current is spin-unpolarized and the electrons have spin in any direction with an equal probability. Since the probability to be scattered to the left is higher for electrons with spin directed up and the probability to be scattered to the right is higher for 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 spin-unpolarized electrons flow. The scattering probability of spin-up electrons into the right is higher. This is the reason for the charge accumulation at the left side of the wire.
This effect is used to measure the intrinsic spin polarization in a ferromagnetic metal. See details here   This effect is called the ISHE- type spin detection. See details here
Figure shows the side- jump scatterings in the electrical field of a defect as an example. Any mechanism of spin-dependent scatterings contributes to these effects: mechanism 1, Skew scatterings (mechanism 2), side- jump scatterings (mechanism 4, mechanism 5)
click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

 

 

 

Detailed explanation of this Figure is here

 


Condition for generation of a spin current by the Spin Hall effect

Reason why Spin Hall effect creates the spin current and spin accumulation at wire sides , but not in the middle of wire

Spin current and spin accumulation created by spin- dependent scatterings (Spin hall effect)

experimental verification

Schematic diagram illustrating creation of spin current and spin accumulation due to spin- dependent scattering (Spin Hall effect). Brown lines show sides of a metallic wire. Yellow balls are charged defects. Green balls are conduction electrons. Red arrows shows the scattering path of spin- up electrons and blue arrows shows the scattering path of spin- down electrons Spin accumulation at sides of non-magnetic GaAs wire created by an electrical current.2D scan of Kerr rotation angle θKerr in GaAs 30-um-wide wire. θKerr is proportional to number of spin-polarized electrons.
Before scattering,the electrons are spin- unpolarized and there is equal number of spin-up and spin- down electrons. The spin-up are more scattered to the left and spin-down are more scattered to the right. At center of wire, there are equal number of spin- up electrons moving to the left and spin- down electrons moving to the right right. The electrons of opposite spin polarization are quickly mixed into group of spin- unpolarized electrons (see here for details). As a result, there are no current and no spin accumulation at the center of the wire. At the left side of wire, there are more spin- up electrons. As a result, there are spin and spin accumulation of spin-up electrons at the left side. At the right side of wire, there are more spin- down electrons. As a result, there are spin and spin accumulation of spin-down electrons at the left side. The red and blue regions at wire side are regions of spin accumulation. Red and blue colors corresponds to spin-up and spin-down directions of spin accumulation. White color corresponds to regions without any spin accumulation. (Y. K. Kato, et. al),

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

click on image to enlarge it

 

(fact 1) A spin current and therefore spin accumulation can be created by the Spin Hall effect only in a wire having a gradient of conductivity (e.g. near wire side or an interface)

(fact 2) The spin Hall effect creates two spin currents, which flows in opposite directions and have opposite spin directions of the spin polarized electrons. When these two currents interacts, their spin- polarized electrons are converted back into the group of the spin unpolarized electrons. Only in case when these spin current flowing from left and right neighbor regions are not equal, the spin polarized electrons can be accumulated.

(fact 3) Homogeneously distributed spin- dependent scatterings cannot break the time- inverse- symmetry and therefore convert the spin-unpolarized electrons to the spin- polarized (create or increase the spin polarization). Additionally, the breaking of the spacial symmetry is required in order to break the time- inverse symmetry and create the spin- polarized electrons.

(fact 4) The spin- dependent scatterings, which originate the Spin Hall effect, do not rotate the spin or align it along one direction (e.g. as a magnetic field does). The spin- dependent scatterings only re- distribute the conduction electrons of different spins into different direction. Only with specific conditions, the conduction electrons of one spin direction can be accumulated locally.

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 spin-polarized 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 spin-down 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 spin-up 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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Effects similar to the Spin Hall effect

The following effects do not create a spin-polarized current across a charge current. Therefore, they cannot be assigned as a contribution to the Spin Hall effect. However, similar to the Spin Hall effect they create a spin-polarization at the edge of metallic wire. Therefore, they can easily be confused with the Spin Hall effect.

similar effect 1: Electron movement in electrical field (Schottky- type) at interface

similar effect 2: Ordinary Hall effect

similar effect 3: Spin polarization induced by the Oersted magnetic field of electrical current (the Ampere's law)

similar effect 4: Spin detection


(similar effect 1): Spin polarization created by electrical field at interface. Spin-orbit effect of type 1

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 GaAs-oxide 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 HSO of the spin-orbit 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 HSO. As a result, the conduction electrons becomes spin- polarized. The direction of the spin- polarization is along HSO

When the conduction electrons moved near the edge of the GaAs strip, the spins precess around the magnetic field HSO. Because the damping of the spin precession the spins are aligned along the effective magnetic field of the spin-orbit interaction.
The electrical field, HSO and spin polarization exponentially decays from the interface deep into the wire. There are no E, HSO 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.
click on image to enlarge it

The spin accumulation generated by the magnetic field of spin- orbit interaction induced due electron movement perpendicularly to the electrical field at interface

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

(similar effect 2): Ordinary Hall effect

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.

 

more about the Ordinary Hall effect is here

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) This effect is very similar to the spin Hall effect. Similarly to the Spin Hall effect, a spin- polarized current is created flowing perpendicularly to a charge current due to this effect. I am constantly hesitating whether to include this effect as a contribution to the Spin Hall effect or not.

(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)

 

 

 

 

 

 

 

 


(similar effect 3): Spin polarization induced by the Oersted magnetic field of electrical current (the Ampere's law)

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 non-magnetic 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 spin-polarized at the edge of the wire.

See more details how an external magnetic field makes conduction electrons spin- polarized here and here.

 

 

 

 

 

 

 

 

 

 

 

(similar effect 4) Spin Detection

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 spin-polarized electrons is different from conductivity of spin-unpolarized 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 σdetection is defined as a difference between conductivities of the spin- polarized and spin- unpolarized electrons. It describes the spin detection effect.
click on image to enlarge it

(explanation of the effect). When conductivity of spin-polarized electrons is different from conductivity of spin-unpolarized 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.

What is the spin detection? At which condition does it occur? Why a charge accumulated along a diffusion spin current?

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 spin-polarized electrons is different from conductivity of spin-unpolarized electrons, the charge flow is balanced in opposite directions and the charge is accumulated along the flow of the spin- current.

The detection conductivity σdetection is defined as a difference between conductivities of the spin- polarized and spin- unpolarized electrons. It describes the spin detection effect.

 


Spin detection using Inverse Spin Hall effect (ISHE)

Spin Detection using ISHE

Fig.31. ISHE- type spin detection. Non-local configuration. Under the applied voltage, the charge current Jch (blue arrow) flow in the copper nanowire between two left electrodes. The conduction electrons in Fe are spin- polarized, therefore the spin-polarized electrons are injected and accumulated in Cu. In contrast to the charge current Jch (blue arrow), which can flow only along an electrical field, the charge current Jspin (red arrow) does require the electrical field and flows to the right. The spin current Jspin induces the Hall voltage due to ISHE effect, which is detected by the pair electrodes at right side

click on image to enlarge it

 

What is the spin detection?

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.

 

Origin of the ISHE- type spin detection: Spin- dependent scatterings

E.g. spin- polarized electrons (spin-up) 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.

 

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.

Origin of the conventional- type spin detection: Spin- dependent conductivity

The spin diffusion current is the sum of two opposite currents of spin- polarized and spin-unpolarized electrons. The flows are exactly the same, but opposite. When the conductivity is spin-independent, there is a balance between opposite electron flows and the spin diffusion current does not transform any charge. In contrast, when the conductivity is different for the spin-polarized and spin- unpolarized electrons, there is no balance and a different number of electrons flowing in opposite directions. As a result, a charge is accumulated along the flow of spin diffusion current and therefore an electrical voltage is built-up along the flow of the spin-current

 

 

Physical mechanism of ISHE- type Spin Detection: Spin- Dependent scatterings

Fig. 30. Spin- dependent scatterings as the origin of the ISHE-type spin detection. E.g. spin- polarized electrons (spin-up) 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.

Spin-polarized conduction electrons flow from back-side to front-side. Spin-unpolarized electrons flow from front-side to back-side. Spin-polarized electrons are accumulated at left-side of wire. Spin-unpolarized 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 spin-unpolarized electrons flow. The scattering probability of spin-up 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 spin-dependent scatterings contributes to the ISHE-type spin detection: mechanism 1, Skew scatterings (mechanism 2), side- jump scatterings (mechanism 4, mechanism 5)
click on image to enlarge it

Differences between ISHE- type and conventional- type of the spin detection:

Origin: (conventional- type detection): spin-dependent conductivity; (ISHE- type detection): spin- dependent scatterings

Direction of electrical field build-up : (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): spin-dependent conductivity if the detection electrode (MgO:Fe); (ISHE- type detection): Inverse Spin Hall effect (ISHE) in non-magnetic metal (Cu)

reliability, repeatability, controllability of the spin detection : (conventional- type detection): low/moderate; (ISHE- type detection): moderate/high

 

 

Do the Inverse Spin Hall effect and Anomalous Hall effect exist for a diffusion spin current (the flow of the spin without flow of charge)? Can these effects be used for the detection of existence of the diffusion spin current (the spin detection)?.

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 spin-unpolarized 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. Non-local configuration. Under the applied voltage, the charge current Jch (blue arrow) flow in the copper nanowire between two left electrodes. The conduction electrons in Fe are spin- polarized, therefore the spin-polarized electrons are injected and accumulated in Cu. In contrast to the charge current Jch (blue arrow), which can flow only along an electrical field, the spin current Jspin (red arrow) does require the electrical field and flows to the right. The he spin current Jspin (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 Jspin (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.

click on image to enlarge it

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 spin-polarized and spin- unpolarized electrons are the same and opposite, in total there is no AHE effect for a spin diffusion current.

Can the ISHE-type spin detection be used in a ferromagnetic metal?

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.)

 

 

 


 

History

(note): Until recently the Anomalous Hall effect (AHE) and the Inverse Spin Hall effect have not been distinguished as two separated effects and considered as one AHE effect

(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 time-inverse symmetry and cannot contribute in the similar way. The used approximation was too rough. The error was correctly noticed by Smit (1955), but the error was not accepted by Luttinger (1958).

(2d proposal): skew scatterings: Smit (1955), Smit(1958)

It correctly describes the fact that the SO makes the scattering probability dependent on the electron movement direction and electron spin.

See my explanation on the skew scatterings here.

(3d proposal): side-jump 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 the symmetry puzzle?

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 θKerr in GaAs 30-um-wide wire. θKerr is proportional to number of spin-polarized electrons nS , 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, 1910-1913 (2004)
click on image to enlarge it

Figure 2 shows the measured distribution of the spin polarized electrons induced by the Spin Hall effect in a non-magnetic material. In the direction across the wire, the wire is absolutely symmetrical. There is nothing which makes the left different from the right.

Why the direction of spins accumulated is up at the left and down at the right? Why not opposite? What does it make the left to be different from the right?

The distribution of spin shown in Fig.2 means that the scattering probability for the spin up electrons is higher to the left direction than to the right and therefore spin- up electrons are accumulated at the left. In contrast, the scattering probability for the spin down electrons is higher to the right direction than to the left and therefore spin- down electrons are accumulated at the right. Why does such directional difference for scattering probability exist in fully symmetrical structure?

 

(reason 1 for breaking symmetry:) Schottky barrier

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 HSO 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 HSO and spin accumulation is down.

(reason 2 for breaking symmetry) : specific charge of defects

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 HSO 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.

 


Questions & Answers

 

. (from Pramod) What factor decides whether the electron will undergo a side-jump or skew scattering? Also, since they both originate due to impurity scattering in presence of spin orbit interaction, why are they different??

A.The spin Hall is originated from the spin-dependent 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 (side-jump scattering) is changed after a scattering. In any material both types of the scattering co-exist 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 spin-up and less spin-down scattered electrons, whose moving direction is changed to the right with respect to the electron moving direction before scattering. And there are more spin-down and less spin-up scattered electrons, whose moving direction is changed more to the left.  

(side-jump scattering)  The side-jump scattering describes all types of scattering when the position shift of a scattered electron depends on the electron spin. For example, there are more spin-up and less spin-down scattered electrons, whose position is shifted to the right with respect to the electron position before scattering. And there are more spin-down and less spin-up scattered electrons, whose position is shifted to the left.

(Origins of skew and side-jump 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 side-jump scattering occurs for a scattering unrelated to a defect. An example of a strongly-spin dependent side-jump 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 side-jump 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). Nearly-all properties of these two transport mechanisms are drastically different. The bulk-type 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 side-jump 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.

Why is Skew scattering dominant in higher mobility samples whereas, side jump is dominant in low mobility samples?

(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 side-jump 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 side-jump 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 (Side-jump scattering).


(underestimation of a side-jump 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 side-jump scatterings in a material of a higher conductivity is an underestimation. Obviously the tunneling conductivity and the hopping conductivity are orders-of-magnitude 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 high-crystal- 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 side-jump  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 spin-polarized and spin-unpolarized electrons. In contrast, for the 2nd transport mechanism, such difference is substantial. As a result, there are many interesting spin-dependent 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 side-jump scattering for a specific material. I don’t know. Maybe you know?

 

. Do the Spin Hall effect, and Inverse Spin Hall effect occur because of the electron scattering on magnetized defects (defects with non-zero magnetic moment). Are the defect should be magnetized ?

A.(magnetic or non- magnetic defects): It is not necessary. The defect can be either magnetized or non-magnetized. For a side- scattering, the electrical field around the defect is important, which induces spin-orbit interaction and which makes a scattering to be spin-dependent. 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 spin-polarized conduction elections on magnetic defects originates the effect of Anisotropic Magnetic resistance (AMR)/ Planar Hall effect (See here) and the effect of the spin-dependent conductivity (See here). Also see a general discussion here.

 

 

What is spin direction of the spin current induced by the Spin Hall effect? Is the spin direction of generated spin current described by the left- or right- hand rule? Is the spin direction different for different contributions to the Spin Hall effect?

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.

Does the the Spin Hall and the Inverse Spin Hall effects occur at sides of a wire or they can occur in the middle of the sample?

(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..

 

 

(from Jan) i just read that you can see this contribution only in the clean Limit. So that s really strange for me cause the skew scattering should be stronger if you have more impurities, so why it is stronger if you have a cleaner material? That whole Argumentation is based on the Thing that the skew scattering is proportional to the so called Momentum Relaxation time. That s the Point which doesn't make sense to me because the time tells you Long it will take when an electron will get scatter. For skew scattering it should be inverse proportional to it cause if your electrons Momentum Relaxation time is Shorter, you get more scattering. And therefore more contribution to a spin current.

Both the Skew-scattering and Side-jump scattering mechanisms are due to electron scatterings on defects. Why contributions of these mechanisms are not linearly proportional to the number of defects in a conductor?

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 HSO whether it shifted to the left or to the right after the scattering (Side-jump mechanism) or whether its movement direction is turned to the left or to the right after the scattering (Skew-scattering 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 bulk-type conductors.

Above-mentioned problems do not exist for Side-jump scatterings and Skew-scattering across an interface. Therefore, the Spin Hall effect due to an interface is easier to understand and optimize.

A scattering of a conduction electron depends on mutual positions of the electron and the defect. E.g. when a conduction electron is at left side of a defect, it experiences a positive HSO , but the electron is at right side of the defect, it experiences a negative HSO. Since the electron may have any position with respect to the defect, there are both negative and positive contributions, which in average cancel each other. Is it correct?

No, it is not correct. The absolute position of the electron with respect to the defect position and consequently the absolute value of HSO are not as important for the scattering mechanisms of the Spin Hall effect. The change of HSO is only important. It does not matter whether the HSO is positive or negative before or after electron scattering. It only matters that the HSO increases when the electron is scattered to the left and the HSO 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).

(from Jan) i don't know if you know about the spin hall angle and spin hall conductivity cause i cant find it on your Website but i give it a try and try to ask you?

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.

What is relation between the spin relaxation time (the spin life time) of conduction electrons and the strength of the spin Hall effect.

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 HSO of spin- orbit interaction for conduction electrons which whether move in different directions or have different spacial coordinates.

(from Jan) also read your answer about the spin relaxation time and his relation to the strength of spin hall effect. I think you wanted to answer this question above with that answer. But i don't really get what you want to say with that...

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 non-zero orbital moment. Additionally to the movement along a metal, the conduction electrons rotates simultaneously around many atomic nuclears of the host metal (See here). The electrical field of the atomic nuclears induces a magnetic field HSO, 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 HSO. Since electrons, which move in different directions, experience HSO 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 HSO 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 HSO. The scattering probability is lower to a state of a higher energy, where the electron spin is aligned opposite to HSO. 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 HSO. 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 HSO 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 (kx, ky, kz) 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 HSO 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 HSO 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.


 

(from Jan) what you meant with "bias current"?

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.

(from Jan about the strength of different contribution to the Spin Hall effect and about their dependence on impurity density) If you know a paper from Sagasta et al. about Pt or from Hoffmann about the spin hall effect in metals, there is a description about the relation of the spin life time (or relaxation time) and the spin hall conductivity. Well and they will tell you that the spin hall conductivity is constant in the "moderately dirty" regime (which i also find strange but that's not the main question). I find it totally strange, that it is the INTRINSIC mechanisms which is dominating here. i would naively thinking if you have more impurities you will have a stronger skew scattering part. After these papers its not the case but i don't really see why...?

(about defect density) See here

(about the strength of different contributions):

 


(from Hetian Chen about orbital-dependent scatterings & bulk contribution & Fig.3) Thank you for your detailed explanation. But I do not fully understand why the direction of the orbital moment is fixed to the electron movement direction mentioned in fig. 3. Looking forward to your explanation. ?

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 spin-dependent scatterings and how the spin- dependent scattering leads to the spin accumulation in a non-magnetic 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 spin-dependent 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 kp-model. 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 non-zero 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: atom1-atom2-atom3-atom1-atom2. In the reversal direction, the atomic consequence is different: atom1-atom3-atom2-atom1-atom3. 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 non-magnetic 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 spin-orbit interaction Hso along the orbital moment. The electron spin is aligned along Hso. The non- zero orbital moment means that the time-inverse symmetry of the Bloch function for a conduction electron is broken. In this electric field of nuclear 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 non-zero-spin 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 Rashba-type electron gas. To my present knowledge, the Rashba-type 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 spin-dependent and k-dependent 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 above-described "ifs" and occur with a high probability.

 


 

 

 

 

 

 


 

for a Wikipedia explanation of the Spin Hall effect, click here
A simple explanation of the Spin Hall effect based the model of the spin-up/spin-down bands is here

I am strongly against a fake and "highlight" research

 

I truly appreciate your comments, feedbacks and questions

I will try to answer your questions as soon as possible

 

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