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Anomalous Hall effect (AHE). Anisotropic magnetoresistance (AMR)

Spin and Charge Transport

Abstract:

The Anomalous Hall effect describes the fact that when an electrical current flows in a ferromagnetic metallic wire, an electrical current flows perpendicularly to the wire due to the magnetization of the ferromagnetic wire. The effect is originated from a magnetic interaction of conduction electrons and localized electrons (mainly spin- orbit interaction).


Note: All data of this page represent only my personal point of view, which are based on my experimental and theoretical research. The description of the common origin of the Spin Hall effect and the Inverse Spin Hall Hall, which is based on features of spin- dependent scatterings induced by the magnetic field HSO, I have developed in 2014-2019. I have realized that there are two substantial contribution to the Hall effect in a ferromagnetic metal (AHE + ISHE) in 2019.

Content

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(1). Three contributions to the Hall effect

(1a) (1st contribution): Ordinary Hall effect (OHE)
(1b) (2nd contribution): Anomalous Hall effect (AHE)
(1c) (3rd contribution): Inverse Spin Hall effect (ISHE)

(2). History of different views on origin of AHE .

(3) AHE Origin. Symmetry

(3b) AHE Origin. Uniqueness

(4) Origin of AHE: Direction- depend scattering of conduction electrons due direction dependence of their orbital moment

(4a). Possible Origin of AHE: Spin- orbit interaction
(4b). Possible Origin of AHE. sp-d exchange interaction

(5). Measurement of the OHE, AHE and ISHE Hall effects. The Hall angle αHall

(6).Hall effect in non-magnetic metal under spin-injection. Experiment

(6a) Spin injection from ferromagnetic metal to non-magnetic metal. Experiment
(6b). Hall effect induced by photo- excited spin- polarized current. Experiment

(7) spM- type Hall effect

(8). Anisotropic magnetoresistance (AMR)

(9) Why do scatterings become spin-dependent?

(10). Fitting of experimental and theoretical data

(12). Questions & Answers

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Three contributions to the Hall effect

Measurement of the Hall effect

Measurement setup

Three contribution to Hall effect
The Hall voltage or the Hall angle αHall can be measured using cross-bar structure . A bias current through the nanowire. A nanovoltmeter measures the Hall voltage αHall has 3 contributions: (contribution 1) Ordinary Hall effect αOHE, which is linearly proportional to external magnetic field H. (contribution 2) Anomalous Hall effect αAHE, which linearly proportional to magnetization M and independent from H. (contribution 3): Inverse Spin Hall l effect αISHE, which is linearly proportional to sp and non-linearly proportional to H.

click on image to enlarge it

The general Hall effect describes the fact that when an electrical current flows in a ferromagnetic metallic wire, an electrical flows perpendicularly to the wire either due to a magnetic field H or/and due to the magnetic moments of localized electrons (magnetization M) or/due to the magnetic moments of conduction electrons (which are described by the spin polarization)

(1st contribution): (Ordinary Hall effect) due to magnetic field H inside of the metallic wire

(origin of effect): The Lorentz force, which conduction electrons experience when moving in a magnetic field (See details origin of the Ordinary Hall effect here)

(proportionality to external magnetic field): The Hall angle αOHE linearly depends on external magnetic field H.

(affected group of conduction electrons): all conduction electrons (spin- polarized + spin- unpolarized)

(strength of contribution): weak . E.g. αOHE~ 0.2-0.8 mdeg/kG for FeCoB nanomagnets, which I have studied

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Three main contributions to the Hall effect in a ferromagnetic metal

(contribution 1): Ordinary Hall effect (OHE); (contribution 2): Anomalous Hall effect (AHE); (contribution 3): Inverse Spin Hall effect (ISHE);

each of 3 contributions

total (measured) αHall (sum of 3 contributions)

Hall angle αHall has 3 contributions: (1st linear contribution (blue line)) Ordinary Hall effect (OHE); (2d constant contribution (red line)) Anomalous Hall effect (AHE); (3d non- linear contribution (black line)) Inverse Spin Hall effect (ISHE);

Ordinary Hall effect (OHE)(origin) Lorentz force + Electron charge (affect) all conduction electrons: spin- unpolarized plus spin- polarized ( αHall vs magnetic field) linear dependence
Anomalous Hall effect (AHE)(origin) Total spin (total magnetic moment Md or magnetization) of localized d- electrons(affect) all conduction electrons: spin- unpolarized plus spin- polarized ( αHall vs magnetic field) a constant
Inverse Spin Hall effect (ISHE)(origin) Total spin (total magnetic moment Mcond ) of conduction electrons(affect) only spin- polarized conduction electrons; ( αHall vs magnetic field) non-linear dependence
Click on image to enlarge it

 

(2nd contribution):(Anomalous Hall effect) due to magnetic moments of localized electrons or magnetization M

(origin of effect): spin-orbit interaction (plus maybe the exchange interaction???). The origin is not fully clear yet.

(proportionality to external magnetic field): The Hall angle αAHE does not depends on external magnetic field H. It is a constant vs H.

(affected group of conduction electrons): all conduction electrons (spin- polarized + spin- unpolarized)

(strength of contribution): strong . E.g. αAHE~ 300-1500 mdeg for FeCoB nanomagnets, which I have studied

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(3d contribution): (Inverse Spin Hall effect) due to magnetic moments of conduction electrons or the spin polarization of conduction electrons

(origin of effect): Spin-orbit interaction. The Inverse Spin Hall effect (ISHE) describes the fact that a current of spin- polarized electrons creates an electrical current, which flows perpendicularly to the spin current. Conduction electrons in a ferromagnetic metal are spin- polarized. Asa result, there is a charge current perpendicular to a current of spin- polarized conduction electrons, which flows along a ferromagnetic wire. See details about Spin Hall effect and ISHE here.

(proportionality to external magnetic field): The Hall angle αISHE non- linearly depends on external magnetic field H. (See details here).

(affected group of conduction electrons): only spin-polarized electrons!!! spin- unpolarized conduction electrons do not contribute to this effect (See here)

(strength of contribution): moderate . E.g. αISHE~ 5-30 mdeg for FeCoB nanomagnets, which I have studied

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types of the Hall effect

Ordinary Hall effect (OHE)

The Charge of conduction electrons + magnetic field

Inverse Spin Hall effect (ISHE)

The Spin of conduction electrons + the Charge of localized electrons (or defects) a

Hall angle vs external magnetic field for each contribution

The Lorentz force turns the electron (green bowl) from a straight movement. The electron, which moves in a magnetic field H, experience an electrical field perpendicularly to its movement (relativistic effect). This electric field interacts with the electron charge and turns the electron movement direction. It creates an electron current (Hall current) perpendicularly to the bias current. The electrical field of a localized electron (or defect) creates a magnetic field of spin- orbit interaction HSO. The HSO depends on movent direction and position of a conduction electron. Since the electrical field (shown as red lines) is opposite at the left and right side of localized electron, the HSO is opposite at the left and right sides. When the spin of a conduction electron is along HSO , the electron energy is smaller and the probability to scattered into such state is higher. When the spin of a conduction electron is opposite to HSO , the electron energy is higher and the probability to scattered into such state is smaller. The difference in the scattering probabilities creates an electron current (Hall current) perpendicularly to the bias current. (OHE yellow line): Dependence is linear, because the Lorentz force is linearly proportional to external magnetic field Hz. (ISHE green line): Dependence is non-linear and αISHE reversed when magnetization M is reversed because the number of spin- polarized electrons (or spin polarization sp ) non-linearly increases with Hz. Polarity is following sp and therefore M. (AHE blue line): Dependence is a constant and αAHE is reversed when M is reversed, because αAHE is proportional only to M. (spM- type Hall red line): Dependence is non-linear and αspM , αspM polarity is not reversed when magnetization M is reversed because αspM is a product of M and sp. Both reverse their polarities. As a result, the polarity of does not change
Origin of ISHE: Lorentz force. See details about OHE here Origin of ISHE: Dependence of HSO on movement direction and position of a conduction electron, which originates the spin- and direction- dependent scatterings. See details about ISHE here The dependencies of αHall on Hz are substantially different for each contribution. It allows to separate each contribution from an experimental measurement of vs Hz (See here)

Anomalous Hall effect (AHE)

The Charge (or orbital moment) of conduction electrons + the Spin of localized electrons

spM- type Hall effect

The Spin of conduction electrons + the Spin of localized electrons

Total Hall angle as the sum of all contributions

A conduction electron has a orbital moment, which depends on a movement direction and position of the electron. Magnetic field, which created by the spin of localized electrons, interacts with the orbital moment of a conduction electron. The energy is higher when the spin M of localized electron is along the orbital moment of conduction electron and the energy is smaller when M is opposite to the orbital moment. The energy difference makes the scattering probability dependent on M and scattering direction, because of the directional dependence of the orbital moment. The difference in the scattering probabilities creates an electron current (Hall current) perpendicularly to the bias current. There is an exchange interaction between a conduction and localized electron, which depends on their mutual directions. Additionally, the exchange energy depends on the overlap of wavefunctions of conduction and localized electrons. Therefore, the exchange energy depends on the position and movement direction of the conduction electrons. This directional dependence of the exchange energy makes the scattering of the conduction electron spin- dependent and direction dependent. The difference in the scattering probabilities creates an electron current (Hall current) perpendicularly to the bias current. the dependence of Hall angle αHall vs Hz is no-linear, has a hysteresis loop and non-symmetric vs Hz polarity. The complex symmetry of the hysteresis loop is due to different symmetries of different contributions.
Origin of AHE: Dependence of electron scattering probability on spins of localized electron and the scattering direction. See details about AHE is here Origin of spM Hall effect: Dependence of electron scattering probability of a conduction electron on mutual spin directions of conduction and localized electrons and the scattering direction. See details about spM Hall effect is here αAHE=20 mdeg; αISHE=4 mdeg; αHall,spM=2 mdeg; αOHE=1 mdeg/KG; sp0=70% ;Hpump= 1kG;
shows a conduction electron. shows a localized d- electron. The arrow shows the electron spin, when the interaction with the spin is important.
click on image to enlarge it. Zayets 2020.03

 

 

 


History of different views on origin of AHE .

(Origin 1): Spin- orbit interaction (SO):

Origin of Anomalous Hall effect: Magnetic interaction of localized and conduction electrons

The Anomalous Hall effect (AHE) describes the current of conduction electrons, which is created perpendicularly to an electrical current in a ferromagnetic metallic wire due to the spins of the localized d- electrons. The AHE exists due to the magnetic interaction of localized and conduction electrons. Additionally to the movement along the crystal lattice, a conduction electron rotates around each atomic nuclear. It is the reason for the interaction between the steady localized d-electrons and the moving conduction electrons.

Green wavy ellipse shows the wave function of the conduction electron. Blue circles show the direction of rotation of the conduction electron around each atomic nuclear (dark spheres). Blue arrows shows the magnetic field HSO of spin-orbit interaction, which is due to rotation of conduction electron in the electrical field of the nuclear. Camera moves with the electron. 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.

Simultaneously with movement along the crystal a conduction electron rotates around each nuclear. In contrast, a localized d-electron rotates around only one nuclear.

(possible origin 1 of AHE): exchange interaction between conduction electron and localized d- electron. The exchange interaction is due to the overlap of wavefunctions of the conduction electron and the localized d- electrons

(possible origin 2 of AHE): spin- orbit interaction. The magnetic field (spin) of a localized d-electron modifies the orbital symmetry of the conduction electron and therefore enhances the magnetic field HSO of spin- orbit interaction. The HSO makes the scattering of conduction electrons the direction dependent, which results to the perpendicular electrical current (the same as in the case of the Inverse Spin Hall effect (ISHE))

click on image to enlarge it
All following origins, which are based on the SO interaction, are proposed and calculated for spin-polarized electrons. Therefore, they describe the Inverse Spin Hall effect (ISHE), but not AHE. The AHE is due to spins of localized electrons, but not spins of conduction electrons !!!

(1st proposal) of the SO as the origin of AHE: Karplus,Luttinger (1954). They used the same formalism as the formalism of the Ordinary Hall effect, but replacing the Lorentz force by the SO. It an incorrect approach since the SO cannot break the 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

See my explanation on the side-jump scatterings here.

 

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(Origin 2): sp-d exchange interaction:

This origin is proportional to the spins of the localized electrons (magnetization). Therefore, the Origin 2 is the origin of AHE, but is not the origin of ISHE (as in the case of Origin 1) !

(1st proposal) by Kondo (1962), which is based on 3 assumptions:

assumption 1: AHE is originated from the sp-d exchange interaction, but not from the spin- orbit interaction

assumption 2: the sp-d exchange interaction depends on velocity (wave vector) of a conduction electron.

assumption 3 (toughest) : the sp-d exchange interaction is different for opposite movement directions of a conduction electron

(2nd modification) by Giovannini (1973)

An additional contribution to the sp-d exchange interaction was assumed in order to explain why the sp-d exchange interaction is different for opposite movement directions of a conduction electron

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There were many science- fiction proposals on the origin of AHE. One example is that the AHE is originated by the super conductivity. Such proposals are clearly unrealistic and they will not be discussed here


Origin of Anomalous Hall effect. Symmetry

Two similar Hall effects: Anomalous Hall effect (AHE) and Inverse Spin Hall effect (ISHE)

Both the AHE and ISHE effects describes the current of conduction electrons, which is created perpendicularly to an electrical current in a ferromagnetic metallic wire (Hall effect). The spins of localized d- electrons (magnetization) is responsible for AHE effect. The spins of conduction electrons (spin polarization) is responsible for ISHE effect.

(difference 1 between AHE and ISHE effects): All conduction electrons (spin- polarized + spin- unpolarized) contribute to the AHE, but only spin- polarized conduction electrons contribute to the ISHE.

(difference 2 between AHE and ISHE effects): The AHE is proportional to perpendicular component of the total spin of all localized d- electrons. The ISHE is proportional to perpendicular component of the total spin of all spin- polarized conduction electrons.

note: in equilibrium the total spins of localized and conduction electrons are parallel
click on image to enlarge it

Asymmetry αHall with respect to reversal of magnetic field H and magnetization M:

The Hall and the Hall angle is measured by a pair of electrodes (See Fig.). Even a slight spacial misalignment of electrodes with respect to each other causes a voltage between electrodes due to the voltage gradient along the wire. This voltage is independent on H and M. In contrast, the Hall voltage and the Hall angle αHall is asymmetrical with respect to H reversal and M reversal. Therefore, it defines as a difference of αHall measured at opposite H:

From Eq. (3.1), the measured Hall angle can be only proportional to material parameters, which reverse their sign when the H is reversed. Additionally to H itself, there are two more such parameters. The first parameter is the total magnetic moment Md of localized d-electrons. The second parameter is the total magnetic moment Mcond of conduction electrons. Therefore, the Hall angle αHall can be calculated as

where the 1st term βOHE·H describes proportionality of αHall to external magnetic field ( ordinary Hall effect.); the 2nd term βAHE·M describes proportionality of αHall to magnetization of the ferromagnetic metal (Anomalous Hall effect); the 3d term βISHE· describes proportionality of αHall to the spin polarization of the conduction electrons ( Inverse Spin Hall effect)

(symmetry 1): The Hall angle αAHE due Anomalous Hall effect (AHE) is proportional to the magnetization direction:

Since the magnetization is defined as the total spin of localized electrons , the AHE should be proportional to magnetic properties of localized electrons

(symmetry 2): The Hall voltage is linearly proportional to the electrical current. The Hall voltage increases as the electrical current increases

Since the electrical current is a flow of conduction electrons, the AHE should be proportional to magnetic properties of conduction electrons

 

Symmetry- Forbidden relations for Hall angle αHall

αHall, magnetization Md (total spin of localized electrons), Mcond (total spin of conduction electrons) have the same polarity. They all change their sign when a large external magnetic field H is reversed. As a result, both the Md , Mcond are reversed and the following relations are forbidden:

αHall ≠k · (Md) 2 ->

αHall cannot be linearly proportional to the square of magnetization

αHall ≠k · (Mcond) 2

αHall cannot be linearly proportional to the square of spin polarization

αHall ≠ k ·Md ·Mcond

αHall cannot be linearly proportional to the product of magnetization and spin polarization

where k is a constant independent on H. Mcond is linearly proportional to the number of spin- polarized electrons or spin polarization sp

E.g. in the case if αHall is linearly proportional to the magnetization Md , the αHall cannot linearly depend on the magnetization Mcond of conduction electrons (and therefore the number of spin polarized electrons or the spin polarization). It means that the AHE is induced by all conduction electrons (not only spin-polarized electrons). The Hall effect, which is induced only by spin- polarized electrons is called the Inverse Spin Hall effect (ISHE). Details about ISHE are here.


Origin of Anomalous Hall effect. Uniqueness

The Uniqueness of the Anomalous Hall effect (AHE) is that is determined by the magnetic properties of two different groups of electrons: the group of conduction electrons and group of localized d- electrons.

As a result, the AHE should be originated from a magnetic interaction between localized d- electrons and conduction electrons. There are three possible interactions between localized and conduction electrons:

(interaction 1 between localized and conduction electrons): Magneto- static interaction

(interaction 2 between localized and conduction electrons): sp-d exchange interaction

(interaction 3 between localized and conduction electrons): Spin-orbit interaction

 




Origin of Anomalous Hall effect. Direction- depend scattering of conduction electrons due direction dependence of their orbital moment

Zayets 2020.03

Mechanism of Anomolous Hall effect

Dependence of electron orbital moment on its movement direction

 
 
   

click on image to enlarge it

Origin of AHE: Dependence of electron scattering probability on spins of localized electron and the scattering direction.

Explanation in short:

A conduction electron has a orbital moment, which depends on a movement direction and position of the electron. Magnetic field, which created by the spin of localized electrons, interacts with the orbital moment of a conduction electron. The energy is higher when the spin M of localized electron is along the orbital moment of conduction electron and the energy is smaller when M is opposite to the orbital moment. The energy difference makes the scattering probability dependent on M and scattering direction, because of the directional dependence of the orbital moment. The difference in the scattering probabilities creates an electron current (Hall current) perpendicularly to the bias current.

 

Origin of AHE. How the spin of localized d- electrons creates an electrical current flowing perpendicularly to a ferromagnetic wire. Step by step explanation

(step 1) Why electrical current flow across a ferromagnetic wire?

The scattering probability of a conduction electron is different to the left and to the right direction. As a result, more electrons are scattered in one direction and an electrical current flows from the left to the right.

(step 2) Why the scattering probabilities to the left and to the right are different for a conduction electron?

The scattering probability is different because the magnetic energy of a conduction electron is different when its moves to the left and the right directions..

(step 3) Why does the magnetic energy of a conduction electron depend on its movement direction?

The magnetic energy is proportional to the magnetic field, which is induced by the spins of the localized d- electrons, and the magnetic moment of a conduction electron and therefore the orbital moment of the conduction electron. The orbital moment of a conduction electron is different for its opposite movement direction

(step 4) Why does the orbital moment of a conduction electron depend on the electron movement direction?

It is an unique feature of conduction electrons. Details see here.

 

 

 

Enhancement of AHE effect due to spin-orbit interaction

 

 

 


Possible Origin of Anomalous Hall effect. Spin- orbit interaction

Spin- Orbit interaction as the origin of Inverse Spin Hall effect (ISHE)

The origin of ISHE 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 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 of 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
Even though the spin- orbit interaction and spin-dependent scattering has been considered as the possible main origin of AHE for about 70 years (See AHE history), as 2020.03 I do see the AHE mechanism, which could be related to the spin- dependent scatterings. The spin- dependent scatterings originate another type of Hall effect: Inverse Spin Hall effect (ISHE), but not AHE. The spin- orbit interaction enhances the AHE, but not originates it (See here)

(problem of Spin- orbit interaction as the origin of AHE): The Spin- Orbit interaction does induce the Hall effect, but this Hall effect is proportional to the number of the spin-polarized conduction electrons and such effect is called the Inverse Spin Hall effect. The Hall effect due to AHE should be proportional to the total number of the conduction electrons (See forbidden relations here).

(note) The Hall effect due to the spin- orbit interaction is simple and understandable (see here) , but it is proportional to the number of spin- polarized electrons and therefore called the Inverse Spin Hall effect. The Anomalous Hall effect (AHE) is proportional to the number of all conduction electrons and the magnetization (the total spin of localized electrons)

Hall effects due to the Spin- orbit interaction: (source 1): due to a non-zero orbital moment of conduction electrons; (source 2): Skew scatterings (source 3): Side-jump scatterings at defect (source 4): Side-jump scatterings across an interface. All these origin are due to spin-dependent scatterings and and they are proportional to the number of spin- polarized electrons.

Origin of the Inverse Spin Hall effect (in short)

ISHE is due to the Spin- Orbit Interaction and is proportional to the number of spin- polarized conduction electrons

(step 1) The conduction electrons experience the magnetic field HSO of spin- orbit interaction (See here) due to several possible sources: (source 1): A non-zero orbital moment of a conduction electron (Details are here); (source 2): Skew scatterings (Details are here); (source 3): Side- jump scatterings in electrical field of a defect (Details are here); (source 4): Side- jump scatterings across an interface (Details are here); . Important: Each source makes the direction, polarity and magnitude of magnetic field HSO dependent on either moving direction (kx, ky, kz) or spacial coordinates (x, y, z) of a conduction electron

(step 2) Due to the magnetic interaction of the magnetic field HSO with spin of a conduction electrons, the scatterings of conduction electrons becomes direction- and spin- dependent. The scattering probability is larger when the HSO is along the spin of a conduction electron and it is smaller when the HSO is opposite to the electron spin. E.g. the spin-up electrons are scattered more to the left and the spin-up electrons are scattered more to the right.

(step 3) The difference in scatterings probabilities creates an electrical current flowing perpendicularly to the main electrical current. Only spin- polarized conduction electrons contribute to this current. E.g. the spin direction of spin- polarized electrons is up. Since there are more spin-up electrons scattered to the left than to the right, the electron current (charge current) flows into the left direction. Therefore, only spin- polarized electrons contribute to the Hall effect.

Note: (about influence of spin- dependent scatterings on spin- unpolarized conduction electrons and the Spin Hall effect). In spins of the spin- unpolarized conduction electrons are distributed equally in all directions (See here). Due to the spin- dependent scatterings, more spin-up electrons are scattered to the left (for example) and more spin- down electrons are scattered to the right. However, in the group of spin- unpolarized electrons there are equal amounts of spin-up and spin- down electrons. As a result, the spin- dependent scatterings of spin- unpolarized electrons induce no charge current from the left to the right, but only the spin current . This effect is called the Spin Hall effect

You have divided the Hall effect in a ferromagnetic metal into two contributions: the first AHE contribution due to the total spin Md of localized electrons and the second ISHE contribution due to the total spin Mcond of conduction electrons. However, the conduction electrons in a ferromagnet metal are spin- polarized, because of their magnetic interaction with localized electrons. Therefore, Mcond is proportional to Md and both the AHE and ISHE can be combined in one Hall effect, which is only proportional to Md? Do we really need to split the Hall effect in a ferromagnetic metal into the AHE part and ISHE part?

Yes. You are correct. The conduction electrons are spin- polarized, because the spins of the localized d- electrons are aligned in one direction. The mechanism of spin alignment of conduction electrons is called the spin pumping (See here). There are two mechanisms, which aligns the spins of conduction electrons along the spins of localized electrons. (mechanism 1 (usually weaker) ) the exchange interaction between conduction and localized electrons (sp-d exchange interaction); (mechanism 2 (usually stronger) ) the scattering between localized and conduction electrons. The localized electrons, which spins are aligned in one direction, are constantly scattered into the gas of conduction electrons. Therefore, they are constantly injecting the spin into the group of the conduction electrons. Therefore, it is true that the total spin of conduction electrons is proportional to the total spin of localized electrons and the total spin of conduction electrons is aligned to the spins of localized electrons.

However, the localized and conduction electrons are two very different groups of electrons (see here) and their total spins are two independent magnetic parameters. For example, the number of spin polarized conduction electrons can be changed without affecting the localized electrons (the spin injection etc.).

In fact, the individual contributions of the AHE and ISHE are measured by changing the number of spin- polarized electrons without influence of the localized electrons. The part of Hall angle, which is changing, is the ISHE contribution and the part of Hall angle, which is staying a constant, is the AHE contribution. The spin polarization of conduction electrons can be changed by a spin injection from a neighbor region or by illumination by a circularly- polarized light. The simplest method is the applying an external magnetic field along the magnetization of a nanomagnet. The conduction electrons are aligned along the magnetic field, therefore the spin polarization increases. The spins of localized electron initially aligned along magnetic field, therefore spins of localized electron are not affected by the magnetic field (almost). More details about this measurements are here

final answer: the Hall effect in a ferromagnetic metal independently proportional to the total total spin of localized electrons and the total spin of conduction electrons. As a result, the Hall effect is the sum of the AHE contribution and the ISHE contribution. Each of contribution should be measured individually.

(conclusion) Skew scatterings; Side-jump scatterings at defect ; Side-jump scatterings across an interface. are contribute to the Hall effect, but they contribute to the ISHE effect, but not to the AHE effect. It is because these contributions are proportional to the number of spin- polarized conduction electrons, but not to the total number of conduction electrons as it should be in the case of AHE effect.


Possible Origin of Anomalous Hall effect. sp-d exchange interaction

sp-d Exchange Interactions a possible Origin of Anomalous Hall effect

Fig.11. The exchange interaction between a moving conduction electron (shown in green) and a nonmoving localized d- electrons (shown in red).

The green wavy ellipse shows the distribution if the wave function of the conduction electron. Green circles show the rotation of the conduction electron around each atomic nuclear (dark spheres). Each localized d-electron (red balls) rotates only around one nuclear (red circle)

When the wave function of conduction electron overlaps an atomic nuclear, there a stationary nonmoving part of wavefunction (green circle with arrow), which describes the rotation of the conduction electron around each nuclear

The overlap of wave functions of the localized nonmoving d- electrons (red circle) and the stationary nonmoving part of the conduction electrons determines the strength of the sp-d interaction

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.
The size of a conduction electron is substantially larger than the size of a localized electron.
click on image to enlarge it. Camera moves together with the conduction electron.
Even though the sp-d exchange interaction has been considered as the possible main origin of AHE for about 70 years (See AHE history), as 2020.03 I do see the AHE mechanism, which could be related to the sp-d exchange interaction. The sp-d exchange interaction originates another type of Hall effect: spM- type Hall effect , but no AHE. The contribution from the spM- type Hall effect is rather small. In contrast to other mechanisms of Hall effect, the spM mechanism does not have a hysteresis loop (See here)

(problem of sp-d exchange interaction as the origin of AHE): In order for the sp-d exchange interaction to be the AHE origin, the exchange interaction between the localized d- electrons and conduction electrons should have some unique properties. E.g. the strength of the exchange interaction should change significantly when the movement direction of a conduction electron is reversed. The exchange interaction should not depend on the spin of the conduction electron. The physical mechanisms, which might cause these unique properties, are still unclear.

In order for the sp-d exchange several requirements should be satisfied

(requirement 1) The overlap between wave functions of a conduction electron, which is moved along the metal, and a localized electron, which stays at the same place, should be a constant in-time.

(possible solution): Simultaneously with the movement along the metal, the conduction electron rotates around many atomic nuclears. The rotation is described by the Bloch function. The rotation is local and is stationary (does not move along metal). The wavefunction overlap of this stationary part with the nonmoving localized d- electron determines the exchange interaction. Even though the magnitude of this nonmoving stationary wavefunction part of the conduction electron is changing in time, in the average over many localized electrons the overlap does not change in time (see Fig.11) and the sp-d exchange interaction is a non-zero and does not change in time

(possible problem): Additionally to magnitude, the phase of a conduction electrons is continuously changing as the electron moves along the metal. The exchange interaction does depend on the phase. When averaged over the phase, the strength of the averaged exchange interaction might be small

(requirement 2) In order to contribute to the AHE, the sp-d exchange interaction should change significantly its strength when the moving speed of the conduction electron is changed. However, the exchange interaction depends on the stationary part, which is not moving. So why it should depend on the electron moving speed?

(possible solution): The symmetry of the stationary part depends on the electron speed. This fact is described by the kp- model (Luttinger, Kohn (1955)) and is well verified in semiconductors. The strength of the exchange interaction does depend on the symmetry of the stationary part (Bloch function) of the conduction electron and therefore on its speed.

(possible problem): The change of the symmetry vs the electron speed is very small. As a result, the change of strength of the exchange interaction is small as well. It is difficult to explain the strong AHE effect by this small change of the exchange interaction

(requirement 3) In order to contribute to the AHE, the sp-d exchange interaction should change significantly when the electron movement direction is reversed. The standard kp- model cannot explain such property.

(possible solution): Both Kondo (1962) and Giovannini (1973) have used some mathematical tricks in order to include such unexpected dependence into the sp-d exchange interaction. It is unclear how realistically these mathematical tricks correspond to the reality.

(possible problem): The exchange interaction is one of most complex effects of the nature. At present, not all its features are fully understood (See here)

(requirement 4) In order to contribute to the AHE, the sp-d exchange interaction should not depend on the spins of conduction electrons. It should depend only on spins of the localized electrons. It well-known and well- verified fact that the exchange interaction strongly depends on mutual spin directions of two interacting electrons.

(possible solution): The spins of spin- unpolarized electrons is distributed equally in all directions. The dependence of exchange interaction on the spin of the conduction electrons might be averaged out over such distribution.

(possible problem): The spin- polarized electrons contribute to ISHE (not to AHE)?

(Conclusion) At present, it is difficult to clarify whether the sp-d exchange interaction is the main origin of AHE. There are still too many unanswered questions


 

 

 

 

 

 

 

 

 


Measurement of the OHE, AHE and ISHE Hall effects. The Hall angle αHall

Measurement of Anomalous Hall effect

Measurement setup

 
 
A bias current through the nanowire. A nanovoltmeter measures the Hall voltage  

click on image to enlarge it

The OHE and AHE is measured by the Hall angle αHall. The contributions from AHE and OHE are independent: αHall=αAHE+αOHE

What is better to use the Hall resistance or Hall angle αHall

A. The Hall angle αHall. It is only correct parameter characterizing the AHE and OHE.

What are merits to use the Hall resistance for measurement of AHE and OHE?

None. It is just a number in the Ohm unit. It has no physical meaning. It depends on the device geometry. It is not a material intrinsic parameter.

What are merits to use Hall angle αHall of measurement of AHE and OHE?

merit 1: The αHall has a direct physical meaning. It the angle of deviation of electron movement from a straight line along an applied electrical field.

merit 2: The αHall is an intrinsic parameter of a material. It does not depend on the device geometry or film structure. The magnitude of the OHE and AHE in different devices and different films should be only compared by comparing their αHall.

Measurement of αHall

The Hall angle αHall is defined as

where σxx and σxy are diagonal and off-diagonal components of the conductivity tensor

nanowire with two pairs of Hall probes

measurement of Hall effect in FeB of two different thicknesses

 
 
Backside Hall probe is connected to a nanomagnet. FeB is thicker in this region and top of FeB is covered by MgO. The Front side Hall probe is connected at side of a nanomagnet. FeB is thinner in this region and at its top covered by SiO2. The distance between two Hall pairs is 11 μm.  

click on image to enlarge it

Single layer film:

In this case the Hall angle αHall is calculated as

where the Hall voltage VHall, is the Hall voltage L is wire length and w is wire width. The Hall resistance RHall can be calculated

where R is wire resistance

Double-layer metallic wire:

This case when metallic wire consists of two layers: The first layer is made of a ferromagnetic metal. The second layer is made of a non-magnetic metal. In this case the Hall angle αHall of the ferromagnetic metal is calculated as

where tferro, tisot, σferro,σisot are thicknesses and conductivities of ferromagnetic and non-magnetic metals.

or the intristic Hall angle αHall,ferro in a ferromagnetic metal is calculated from the measured Hall angle αHall,measured in a double layer nanowire as

 

to see how to obtain Eqs.(4.15),(4.24), click here to expand it

Single-layer metallic wire

Due to the Hall effect there is an electrical current current across the metallic wire, which can be calculated as

where V is the bias voltage, L is wire length, j|| and j is current density along and across the wire .

The current j makes a charge accumulation at walls of the metallic wire. This charge induces the voltage, which is called the Hall voltage VHall. The Hall voltage induces the current, which is opposite to j. The current density of this current j⊥,comp can be calculated as

where w is the width of the wire. In total, there is no current across the wire, σ=σxx is conductivity of the wire.

Since in total there is no current flow across the wire, we have

Substituting Eqs.(4.2),(4.3) into Eq.(4.4) gives

or

The Hall resistance RHall is defined as

where J|| is the current flowing throw the nanowire

where thick is wire thickness


Double-layer metallic wire

This case when metallic wire consists of two layers: The first layer is made of a ferromagnetic metal. The second layer is made of a non-magnetic metal. The Hall current is generated only inside ferromagnetic

The Hall current J, which flows across wire, can be calculated as

where j is the Hall current density , tferro is the thickness of ferromagnetic layer, w is the wire width, L is the wire length and V is the bias voltage. αHall is the Hall angle of the ferromagnetic metal. It is assumed that the Hall angle αHall=0 in the non-magnetic metal.

Due to the charge accumulation, the current flows in the opposite direction in both layers

tisot is the thickness of layer of the non-magnetic metal, σferro,σisot are conductivities of ferromagnetic and non-magnetic metals..

Since in total there is no current flow across the wire, we have

Substituting Eqs. (4.21),(4.22) into Eq.(4.22a) we have

Simplifying Eq.(4.23) gives the Hall angle in case of wire consisted of two layers as

 

 

 

 


OHE effect + ISHE effect without AHE effect. Spin injection in non-magnetic metal

(fact) In a non-magnetic metal there is only one type of Hall effect: OHE. There are no AHE effect (since there are no localized d- electrons) and no ISHE effect (since conduction electrons are not spin polarized)

Why a Hall measurement in a non-magnetic conductor under spin injection is important?

(importance 1): It is possible to separate contributions to Hall effect from localized and conduction electrons. For about 70 years it has been believed that there is only one contribution to the Hall effect in a ferromagnetic metal (the AHE contribution) and both the conduction and localized electrons jointly contribute to the AHE effect. The experimental observation of the Hall effect in a non- magnetic conductor under the spin injection clearly indicates that each localized and conduction electrons contribute individually to the Hall effect and the contributions are very different from each other ( dependence on an external magnetic field etc.)

(importance 2): Since the Hall effect (the ISHE) exists in a non-magnetic metal, which does not have a localized electrons with aligned spins, it clearly indicates that the existence of Hall effect (additional to OHE) does not require the existence of localized electrons. The conduction electrons by themselves are able to produce the Hall effect.

(importance 3 (main)): The substantial difference in contributions into the Hall effect from localized and conduction electrons clearly indicates that the spin distributions of the conduction and localized electrons are very different. The spin distribution in a ferromagnetic metal is the classical spin-up/ spin- down distribution. In contrast, the spin distribution of conduction electrons is the sum of two distributions of groups of spin- polarized and spin- unpolarized electrons. (Details see here)

 

Hall effect in non-magnetic metal under spin-injection

When spin polarized conduction electrons are injected into a non-magnetic metal, the ISHE effect exists additionally to the OHE effect.

Hall effect in non-magnetic metal under spin-injection. Spin Injection using the spin Proximity effect. (Fig.10)

non-magnetic Au nanowire under injection of spin-polarized electrons.

experimental setup

ferromagnetic FeTbB nanowire

Hall angle measured in Au(15nm)/FeBTb(9 nm) periodic structure in Au only region. Spin-polarized electrons are injected into from FeBTb into Au due to the spin proximity effect.

thin a narrow ferromagnetic stripes of FeTbB on non-magnetic Hall bar, which is made of Au. Stripes width 70 nm and gap between stripes 70 nm (See SEM image here). The conductivity of Au is substantially higher. As a result, a electrical current flow mainly in Au.

The Hall angle measured in FeBTb (20 nm)
Measured conductivity is 1.1E7 S/m2. Because of a low conductivity of FeBTb layer, contribution of this layer to Hall effect is negligible. Conduction electrons are spin- polarized in FeTbB stripes. The spin- polarized electrons diffuse into Au from FeTbB (even without an electrical current). See spin proximity effect. As a result, the ISHE effect exists in Au. The polarity of the ISHE effect is different in Au and FeTbB is different. The Hall angle is measured under stripe (left Hall pair) and in the gap between strip (right Hall pair) Measured conductivity is 0.06E7 S/m2.

Notice: Polarities of loops in Au and FeBTb are different !!!

this experiment I did in 2016. Main purpose was to study the features of the the spin proximity effect
click on image to enlarge it

Why this study is interesting?

A. The properties of the ISHE effect without disturbance of the AHE effect can be clarified and studied..

There are no localized d- electrons in a non-magnetic metal. As a result, the AHE effect does not exists in a non-magnetic metal at any conditions. (one exception is an interface with a ferromagnetic metal)

The conduction electrons are not spin- polarized in a ferromagnetic metal and there is no ISHE effect in an equilibrium. However, the spin-polarized can be injected in a non-magnetic metal. In this case, the ISHE effect starts to exist in the non- magnetic metal

(Note) The ISHE effect in a ferromagnetic metal is strong, but the similar AHE effect also exists in the ferromagnetic metal and it is difficult to separate the AHE and the ISHE effects (See details here). As a result, it is difficult to study features of the ISHE in the ferromagnetic metal, because they can be originated from ISHE.

(experiment 1) Spin injection from a ferromagnetic metal + Hall measurements in a nonmagnetic metal

this experiment I did in 2016-2018 in Au: FeTbB samples

( Main idea): To inject spin- polarized electrons from a ferromagnetic metal into a non-magnetic metal, while measuring the Hall effect in the non-magnetic metal. To use the ferromagnetic and non-magnetic metals with opposite polarity of the ISHE.

(Main challenge): The Hall effect in the ferromagnetic metal should not contribute to the measured the Hall angle

(solution 1): To use the ferromagnetic and non-magnetic metals with opposite polarity of the ISHE. As a result, the contributions from each metal can be distinguished by the polarity of the hysteresis loop.

(solution 2): the use of the ferromagnetic metal with a small conductivity and the non- magnetic metal with a high conductivity. As a result, near-all current flows in the non- magnetic metal, nearly no current flows ferromagnetic metal and therefore the main contribution to the Hall angle would be from the non- magnetic metal and only a little contribution would be from ferromagnetic metal.

 

Hall effect in non-magnetic metal under spin-injection. Conventional Spin Injection. (Fig.11)

Spin injection from top FeTbB electrode

 
 
thin a narrow ferromagnetic stripes of FeTbB on non-magnetic Hall bar, which is made of Au. Stripes width 70 nm and gap between stripes 70 nm (See SEM image here). The voltage is applied between top of FeBTb stripes and Au. One pair of Hall probe is connected under FeTbB stripe and another pair is connected between stripes. The spin polarized conduction electrons are injected from FeTbB into Au. The spin- polarized electrons induce the Hall effect in Au. (Inverse Spin Hall effect (ISHE))  

click on image to enlarge it

(possibility 2). Classical spin injection. Spin injection from top FeTbB electrode

In this case, the electrical voltage is applied between the FeTbB strips and the Au nanowire. There is a spin injection from the FeTbB

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(experiment 2) Spin injection into a semiconductor by a circular- polarized light + Hall measurements in the semiconductor

 

Hall effect induced by photo- excited spin- polarized current

Hall effect in n-GaAs/i-GaAs/ p- GaAs wire (GaAs pin- photo detector) illuminated by circular- polarized light.

ACW- circular polarized light (45 deg of λ /4 plate) -> spin- up photo- excited electrons

linear polarized light (45 deg of λ /4 plate)-> no spin polarization

CCW- circular polarized light (45 deg of λ /4 plate) -> spin- down photo- excited electrons

a positive Hall voltage There is no Hall voltage a negative Hall voltage
merit: Hall effect is induced only by spin-polarized conduction electrons (ISHE effect). There are no localized d- electrons in i- GaAs. As a result, there is no AHE effect.

Measurement of Hall angle in GaAs pin- photodetector illuminated by circular- polarized light

similar experiment is described here: Wunderlich et.al. Nat. Phys. (2009)
Output linearly- polarized light from the laser becomes circular- polarized. When the circular- polarized light illuminates the GaAs pin- photo detector, it excites spin- polarized electrons in i-GaAs, which flows from p-GaAs to n-GaAs. Their spin- polarized current is detected by a pair of Hall probe and the Hall voltage is measured.
Yellow arrow shows the polarization of light.
i- GaAs: (undoped, non-conductive); n-GaAs (donor- doped, electron- type conductivity); p-GaAs (acceptor- doped, hole- type conductivity);
click on image to enlarge it
See a similar experiment: Wunderlich et.al. Nat. Phys. (2009)

( Main idea): The circular- polarized light creates spin- polarized conduction electrons in a non- magnetic semiconductor. As a result, the the ISHE effect starts to exist under illuminations of a circular- polarized light.

 

 

 

 

 

 

 

 

 

 

 

 

 

Hall effect induced by photo- excited spin- polarized current

Controlling of spin direction of photo excited elections in pin- GaAs photodiode by rotating axis of λ /4 waveplate Hall Voltage as a function of polarization of light (schematic)

 

Rotation of λ /4 waveplate does not affect light intensity, only it changes the light polarization.
Yellow arrow shows the polarization of light. Red mark on λ /4 waveplate shows its axis direction
click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


5 types of the Hall effect:

 

(type 1 of Hall effect): Anomalous Hall effect (occurs only in a ferromagnetic metal)

The Anomalous Hall effect describes describes the fact that a change can be accumulated at the edges of a metallic wire when a magnetic field is applied perpendicularly to the electrical current. .

(type 2 of Hall effect): Inverse Spin Hall effect (may occur in both a ferromagnetic metal and a non-magnetic metal)

The Inverse Spin Hall effect describes the fact that that a change can be accumulated at the edges of a metallic wire when a magnetic field is applied perpendicularly to the electrical current.

(type 3 of Hall effect): spM Hall effect (occurs only in a ferromagnetic metal)

The spM Hall effect occurs due to the interaction between the spin of conduction electrons and spins of localized electrons.

(type 4 of Hall effect): Ordinary Hall effect (occurs in both a ferromagnetic metal and a non-magnetic metal)

The Ordinary Hall effect occurs due to the Lorentz force (change of electron movement direction in a magnetic field)

(type 5 of Hall effect ???): Spin Hall effect:

The Spin Hall effect describes the fact that spin may be accumulated at the edges of a metallic wire when spin-unpolarized drift current flows in the wire.

 

 

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Questions Answers

 

Q. The spin Hall effect (SHE), anomalous Hall effect(AHE) and inverse spin Hall (ISHE) effect are very similar. How to distinguish between these effects?

A.

Q. Since all effects: SHE, AHE and ISHE are related to scatterings, does it mean that the spin Hall effect, anomalous Hall effect and inverse spin Hall effect can occur only in "bad" metals with a large number of defects?

A. The magnitude of these effects may be large only in case of metal having a sufficient number of defects. Therefore they can be observed easily in “ bad” metals with a small conductivity. For example, FeBTb, which conductivity is 50 times smaller than the conductivity of gold and 12 times smaller than the conductivity of monocrystal iron, has a large Hall angle (~0.5 deg).

In the case when the density of defect is very large, the electron wave function overlaps several defect simultaneously, it would make all side- jump scatterings spin-independent and they would not contribute to the spin Hall effect, and inverse spin Hall.

Other important condition for existence of spin-dependent side- jump scatterings is that the electron mean-free path should be shorter than the effective radius of electrical field around a defect.

 

 

Anomalous Hall effect

Q. For the anomalous Hall effect, is it essential for electron gas to be spin-polarized?

A.

Q. Why there are no anomalous Hall effect and no Inverse Spin Hall effect in case when the magnetization and spin polarization is in-plane and along the main current (along the wire)?

A. The spin polarization of electron gas is directed along an applied magnetic field. The defect induces an electric field around itself. An electron, which moves in this field, experiences the effective magnetic field of spin-orbit interaction , which is directed either up or down, when for electrons scattered into the left or right. Because of different directions of the effective magnetic field of the spin-orbit interaction, the probability of electron scattering toward left and right may be different and it this difference is the spin-dependent. In the case when the effective magnetic field is along the spin of electron, electron energy is larger. The energy difference for electrons with spin up and spin down is the reason for the spin-dependent scatterings.

When the magnetic field is applied along the wire, the spin direction of the spin-polarized electrons (electrons of TIA assembly) is along the wire as well. The electrons of this spin direction have the same energy in magnetic field directed up and down. Therefore, the effective magnetic field of the spin-orbit interaction, which is directed up and down, does not make scattering probabilities into the left and into the right to be different and scattering to be spin-dependent. This is the reason why there is no anomalous Hall effect when a magnetic field is applied along a wire.

types of the Hall effect

Ordinary Hall effect (OHE)

The Charge of conduction electrons + magnetic field

Inverse Spin Hall effect (ISHE)

The Spin of conduction electrons + the Charge of localized electrons (or defects) a

Hall angle vs external magnetic field for each contribution

The Lorentz force turns the electron (green bowl) from a straight movement. The electron, which moves in a magnetic field H, experience an electrical field perpendicularly to its movement (relativistic effect). This electric field interacts with the electron charge and turns the electron movement direction. It creates an electron current (Hall current) perpendicularly to the bias current. The electrical field of a localized electron (or defect) creates a magnetic field of spin- orbit interaction HSO. The HSO depends on movent direction and position of a conduction electron. Since the electrical field (shown as red lines) is opposite at the left and right side of localized electron, the HSO is opposite at the left and right sides. When the spin of a conduction electron is along HSO , the electron energy is smaller and the probability to scattered into such state is higher. When the spin of a conduction electron is opposite to HSO , the electron energy is higher and the probability to scattered into such state is smaller. The difference in the scattering probabilities creates an electron current (Hall current) perpendicularly to the bias current. (OHE yellow line): Dependence is linear, because the Lorentz force is linearly proportional to external magnetic field Hz. (ISHE green line): Dependence is non-linear and αISHE reversed when magnetization M is reversed because the number of spin- polarized electrons (or spin polarization sp ) non-linearly increases with Hz. Polarity is following sp and therefore M. (AHE blue line): Dependence is a constant and αAHE is reversed when M is reversed, because αAHE is proportional only to M. (spM- type Hall red line): Dependence is non-linear and αspM , αspM polarity is not reversed when magnetization M is reversed because αspM is a product of M and sp. Both reverse their polarities. As a result, the polarity of does not change
Origin of ISHE: Lorentz force. See details about OHE here Origin of ISHE: Dependence of HSO on movement direction and position of a conduction electron, which originates the spin- and direction- dependent scatterings. See details about ISHE here The dependencies of αHall on Hz are substantially different for each contribution. It allows to separate each contribution from an experimental measurement of vs Hz (See here)

Anomalous Hall effect (AHE)

The Charge (or orbital moment) of conduction electrons + the Spin of localized electrons

spM- type Hall effect

The Spin of conduction electrons + the Spin of localized electrons

Total Hall angle as the sum of all contributions

A conduction electron has a orbital moment, which depends on a movement direction and position of the electron. Magnetic field, which created by the spin of localized electrons, interacts with the orbital moment of a conduction electron. The energy is higher when the spin M of localized electron is along the orbital moment of conduction electron and the energy is smaller when M is opposite to the orbital moment. The energy difference makes the scattering probability dependent on M and scattering direction, because of the directional dependence of the orbital moment. The difference in the scattering probabilities creates an electron current (Hall current) perpendicularly to the bias current. There is an exchange interaction between a conduction and localized electron, which depends on their mutual directions. Additionally, the exchange energy depends on the overlap of wavefunctions of conduction and localized electrons. Therefore, the exchange energy depends on the position and movement direction of the conduction electrons. This directional dependence of the exchange energy makes the scattering of the conduction electron spin- dependent and direction dependent. The difference in the scattering probabilities creates an electron current (Hall current) perpendicularly to the bias current. the dependence of Hall angle αHall vs Hz is no-linear, has a hysteresis loop and non-symmetric vs Hz polarity. The complex symmetry of the hysteresis loop is due to different symmetries of different contributions.
Origin of AHE: Dependence of electron scattering probability on spins of localized electron. See details about AHE is here Origin of spM Hall effect: Dependence of electron scattering probability of a conduction electron on mutual spin directions of conduction and localized electrons . See details about spM Hall effect is here αAHE=20 mdeg; αISHE=4 mdeg; αHall,spM=2 mdeg; αOHE=1 mdeg/KG; sp0=70% ;Hpump= 1kG;
shows a conduction electron. shows a localized d- electron. The arrow shows the electron spin, when the interaction with the spin is important.
click on image to enlarge it. Zayets 2020.03

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Q. The electrical field around defect is distributed in all directions. Why there is difference of scattering probabilities only between left and right directions as it is shown in above Fig.?

The scattering current may flow only if there is a difference in scattering probabilities into two opposite direction. The spin-orbit interaction makes this difference. The effective field of the spin-orbit interaction is directed up and down for electrons scattered into the left and into the right. Since the electrons with spin directed up and down have different energies in this field, their scattering probability is different. Similar, the electrons, which spin is directed left and right, have different scattering probabilities into up and down directions.

Q. In above Figure, spin-up electrons are scattered into the right and spin-down electrons are scattered into the left. Why not in opposite direction?

A. This direction is chosen as an example . In which direction electrons are scattered depends on a material. And specifically it depends whether the density of states in a metal increases or decreases with energy at the Fermi surface. When electron has been scattered, it experiences the electrical field at back side of the defect, which directed from front to back of the wire. This electrical field induces the effective magnetic field of the spin-orbit interaction, which is directed upward when the electron moves to the left (See here) and it is directed downward when the electron moves to the right. In the case when the effective magnetic field is along the spin of electron, electron energy is larger. When the density of states in a metal increases with increasing electron energy, the scattering probability is larger when electron spin is along the effective magnetic field. Therefore, the case shown in above Figure corresponds to a metal, in which the density of states decreases with energy.

Q. In Figure 1 hit it is only shown that only the electrons contribute into the effects. Is there any contribution of holes into the effects??

The holes (electrons of the energy lower than the Fermi energy) contributes nearly equally as the electrons (electrons of the energy higher than the Fermi energy). Importantly, the polarity of the hole and electron contributions are the same. This is the reason why the spin Hall effect, anomalous Hall effect and inverse spin Hall effect are effects with a large magnitude. It is similar to the case of the ordinary Hall effect

 


Why do scatterings become spin-dependent? The role of the spin- orbit interaction

Reason why a scattering become spin-dependent

click here to enlarge it

When a spin-up electron moves forward (central), its scattering probability is different towards left and towards right.

The Fermi-Dirac electron distributions before scattering (center), after scattering to the left (left) and to the right (right) are shown in yellow. The vertical axis is the electron energy. The horizontal axis is the occupation probability.

The electron energy depends on mutual directions of the electron spin and the effective field of the spin-orbit interaction Hso

The scattering probability is highest, when at the energy of high occupation probability before scattering (center) there is sufficient number of unoccupied states after scattering (left or right). It is the case of scattering towards left.

 

Notes:

 

 

 

When an electron is scattered on a defect, behind the defect the electrical field of the defect is directed along the electron movement. A moving electron experiences the effective magnetic field of the spin-orbit interaction only when electron moves across an electrical field.

When the electron is scattered towards left, the effective magnetic field of the spin-orbit interaction Hso is directed up and along the electron spin. Therefore, the energy the scattered electron becomes smaller.

When the electron is scattered towards right, the Hso is directed down and opposite to the electron spin. Therefore, the energy the scattered electron becomes larger.

As can be seen from Figure, for scattering towards left there are many unoccupied quantum states, therefore the scattering probability towards left is higher. In the contrast, there are almost no unoccupied quantum states for scatterings toward right, therefore the scattering probability towards right is lower.

 

 

 

 

 

 

 

 

 

 



spM- type Hall effect

spM Hall effect. Influence of hysteresis loop

Contribution of each type of Hall effect

Total Hall angle αHall

Inverse Spin Hall effect (ISHE), Anomalous Hall effect (AHE) and Ordinary Hall effect (OHE) are asymmetrical vs external magnetic field H. In contrast, the spM Hall- effect contribution is symmetrical Different symmetry of the spM Hall contribution makes the hysteresis loop non-symmetric.

Both the magnetization of the localized electrons and the magnetization of conduction electrons reverse its sign when the external magnetic field reversed. It makes the spM contribution independent on the polarity of the external magnetic field.

αHall,spM=8 mdeg; αAHE=20 mdeg; αISHE=4 mdeg; αOHE=1 mdeg/KG; sp0=70% ;Hpump= 1kG;
click on image to enlarge it

It is a join contribution of spins of localized and conduction electrons

 

The Hall angle αHall,spM from this contribution is proportional to both the magnetization Md of localized electrons and the magnetization Mcond of conduction electrons

αHall,spM ~ Md · M cond

Since the magnetization Mcond of conduction electrons is proportional to the number of spin- polarized electrons and therefore to the spin polarization, αHall,spM is calculated as

αHall,spM ~ Md · sp

where sp is the spin polarization of the conduction electrons

 

Except a few exceptions, the direction of the spin polarization (sp) of spin- polarized conduction electrons is along the spin direction of localized electrons (magnetization). As a result, αHall,spM (~ Md · sp) does not on direction of M and therefore does not have a hysteresis loop.

 

In contrast to any other types of Hall effect, the spM- type Hall effect does not have a hysteresis loop.

αHall,spM is only slightly depends on an external magnetic field H due to the dependence of spin polarization sp on H

 

 

 



 

 

Anisotropic magnetoresistance (AMR)

AMR of FeB film

click here to enlarge it

Incorrect assumption that the Lorentz force is the origin the AMR effect

The resistance of FeB wire as function the angle between applied magnetic field and the direction of electrical current in the wire. 0 and 180 deg correspond to the case when magnetic field is along the wire. -90 and 90 deg correspond to the case when magnetic field is perpendicular to the wire and in film plane.

AMR is about 0.09%.

Important! The resistance becomes larger when the magnetic field is applied along the wire.

This polarity of AMR is the same for the most of metal only with a few exceptions of some monocrystal metals.

Sample Hall 16. Measured on 2016/02/29

When an electron moves between scatterings in an magnetic field, it experiences the Lorentz force. The Lorentz force turns out the electron from a straight path. Therefore, the magnetic field slows down the movement of the electron and the electrical resistance should increase when a magnetic field is applied perpendicularly to the electrical current.

It is important: The metal resistance decreases when a magnetic field is applied perpendicularly to the wire (AMR effect) comparing to the case of no field or a field applied along the wire (See experimental measurements ).

It is clear evidence that not the Lorentz force, but spin-dependent scatterings is origin of the AMR effect.

wiki page is here and classic review paper on the AMR is T.R. Mcguire and R.I. Potter, IEEE Trans. Magn. (1975)

The effect of the anisotropy magnetoresistance describes the fact the resistance of a ferromagnetic metal becomes smaller, when the magnetization of the metal changes from parallel to perpendicular direction in the respect to the direction of the electrical current flowing in the metal. Spin- dependent scatterings originate this effect.

 

The magnitude of the AMR effect is linearly proportional to the spin-polarization of the electron gas in a ferromagnetic metal.

 

 

Q. What is the difference between cases when magnetic field is applied along electron current or perpendicular to current?

Spin polarization of electron gas is along to the applied magnetic field. When magnetic field is perpendicular to the electron current, the scattering is spin-dependent and scattering probabilities towards left and right are different. When magnetic field is along the electron current, the scattering becomes spin-independent.

 

Q. Why the resistance of a metallic wire becomes larger when a magnetic field changes from perpendicular to parallel directions with respect to the direction of electrical current in the wire?

When magnetic field is perpendicular to electron movement, the scattering probability toward left and right are different. In contrast when magnetic field is along to electron movement the scattering probability toward left and right are same.

It is important that the total scattering probability are different for perpendicular and parallel directions of the magnetic field, because of the non-linear nature the Fermi-Dirac distribution. Usually

and the metal resistivity is larger when magnetic field is along the electron current.

However, in some rare cases when there is a substantial energy dependence of the density of state at the Fermi surfers, the resistivity may be larger for a perpendicular magnetic field.

Scatterings become spin-dependent, when a electron moves perpendicular to the magnetic field

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Scatterings become spin-independent, when a electron moves along the magnetic field

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Q. The electrons in a metal move in all directions, why only electrons, which move along the electron current, contribute to the AMR effect?

A. All electrons contribute to the AMR effect. The contribution of the electrons, which move in the opposite directions, are of opposite signs. Along electron current the number of electrons, which moves along and opposite current, are different and their contributions do not compensate each other.

 

Q. Is any correlations between the AMR effect and Anomalous Hall effect, the Spin Hall effect and Inverse Spin Hall effect?

A.

 

 

 


AMR effect due to orbital moment of conduction electrons

 

 

Anisotropic magnetoresistance (AMR) due to orbital momentum of electron

case 1: orbital moment parralel to electron moving direction

Magnetization is along current

Magnetization is perpendicular to current

Magnetic moment (magnetization) of localized electrons is along y-direction. Orbital moment of electron is along its movement direction.Orbital moment of a conduction electron, which moves along y-axis, is along y-axis and therfore its magnetic energy is higher. is higher than o  

case 2: orbital moment perrpendicular to electron moving direction

Magnetization is along current

Magnetization is perpendicular to current

Magnetic  
Sphere of directional distribution of conduction electrons. The length of a vector from axis origin to sphere is proportional to the number of electrons moving in the vector direction. In absence of electrical current J, the sphere is symmetrical and an equal amount of conduction electron moving in any direction. When electrical current flows along y-axis, sphere is shifted towards possitive y- direction and more electrons flow along possitive y- direction than in opposite direction.
Red/ blue bars show amount of magnetic energy, which a conduction electron experience in magnetic field H (green arrow) of spin of localized electrons (violet ball with arrow). Rotating arrows show orbital moment of conduction electron. When spin of localized electrons is along orbital moment, the magnetic energy is largest (bar is fully red) and the scattering probability in that direction is higher.
The size of arrows in lift- top corner proportional to the scattering probability in shown direction
Orbital moment of conduction electrons is assumed to be fully quenched. (See here for details)
 
 
 
 
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Origins of Hall effects

Ordinary Hall effect (OHE)

Anomalous Hall effect (AHE)

origin: Lorentz force

origin: scattering on localized spin

When an electron moves between scatterings in a magnetic field, it experiences the Lorentz force. The Lorentz force turns out the electron from a straight path. Therefore, there is a flow of electrons towards left

When a conduction electron is scattered on a defect or an interface boundary, the probability of scattering may be larger towards the left side than towards the right side. In this case, there is a flow of electrons towards left

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

Unresolved issue

It is still unclear relation between magnitude of the AMR and magnitude of AHE & OHE. For discussion on experimental data on this topic See T.R. Mcguire and R.I. Potter, IEEE Trans. Magn. (1975).

 

 


Fitting of experimental and theoretical data

A good theoretician can always fit its own theory to experimental data. It does not matter whether the model is correct or completely wrong

A good theoretician can do impossible things

Even unfitable objects can be "perfectly" fitted to each other

Each researcher has projects, deadlines, promises etc. For some reason bureaucrats always believe that the mission is fully accomplished when an experimental data are "perfectly" fitted by a theoretical data, even if the theoretical model violates all existed Laws of Physics and the experimental data are fully unreliable. Any theoretical model has a set of free parameters. A "skillful" adjustment of free parameters can always "perfectly" fit the model to experiment, even though the model is completely wrong. The more free parameters the model, it is easier to do the fitting.

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I am very surprised that any theoretical model for the origin of the AMR, which has been proposed for the last 70 years, always has a "perfect" fit with with the experimental data. From the present view it is clear that each of the proposed model describes either only one of many contributions to the AMR and/or to the ISHE or completely incorrect.

 

I am not surprised about the present time (2000- 2020), when there is a burst of a fake and "highlight" research and when publishing of "anything" in a research paper is possible. I am surprised about the time of 1960-1970, when the research was flourished and there were many excellent ideas, models and research results. Still every researcher had to show a "perfect" fit of theory and experiment, even in the cases when it was very clear that the full understanding of the effect is still far away. For example, Smit (1955), (1958) has described correctly the skew scattering mechanism and Berger (1970) has described correctly the side-jump scattering mechanism having a very poor and primitive tools. There were no computers. The understanding of the spin- orbit interaction was poor and primitive. Only mathematical method, they have used, was the minimization of Hamiltonian (which itself was not fully correct). It is not the best tool to study the features of the spin- dependent scatterings. Even the term "Inverse Spin Hall effect" did not exists at that time. Therefore, the Hall effect was not divided into the AHE and ISHE contributions. Still they have described correctly and precisely the main essence and the main tendencies of effects. They are very amazing and talented scientists!

 


Questions & Answers

 

 

 

 

 

 

 

 


I am strongly against a fake and "highlight" research

 

 

 

I truly appreciate your comments, feedbacks and questions

I will try to answer your questions as soon as possible

 

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