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Spin Hall effect

Spin and Charge Transport

The Spin Hall effect describes the fact that a spin current may be generated across a flow of charge current due to the spin-orbit interaction. The origin of the Spin Hall effect is spin-dependent scatterings.

 



The Spin Hall effect is the effect describing accumulation of the spins at a surface of a metallic wire, when an electrical current flows through the wire due to the Spin-Orbit interaction.

 

for a Wikipedia explanation of the Spin Hall effect, click here (note: I do not agree with "intuitive" explanation given there)
Explanation of the Spin Hall effect from the model of the spin-up/spin-down bands is here (my view)

 


 

Origin of the Spin Hall effect

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.

The Origin of the Spin Hall effect is the scattering current. See more about the scattering current as origin of the Spin Hall effect here

 

Origin of the Spin Hall effect in short:

The origin of the Spin Hall effect is the spin-dependent scattering current

In order for the scattering current to flow, the electron scattering probability in one direction should be different from the scattering probability in the opposite direction (to be non-reciprocal). The spin-orbit interaction makes the scatterings spin-dependent and non-reciprocal.

 

Material Parameters required for observation of the Spin Hall effect

1) The scattering probability should be spin-dependent.

That means that the scattering probability of a delocalized electron from a quantum state, in which the effective magnetic field of the spin orbit-interaction is parallel to the electron spin, to a quantum state, in which the effective magnetic field of the spin orbit-interaction is anti-parallel to the electron spin, is different from probability of scattering in the opposite direction.

It is not always the case!!!

 

 


In a solid there two kinds of the spin-orbit interaction:

1) intrinsic. It is induced by the electric field of a nuclear. It is proportional to the orbital moment of the delocalized electrons.

2) extrinsic. It is proportional to the applied electrical field or structural electrical field (for example, the electrical field in the vicinity of the contact or QW due to a charge accumulation) . It is proportional to the speed and the movement direction of the electron.

Generally the strength of the intrinsic. spin-orbit interaction is larger than the extrinsic.

Both types of the spin-orbit interaction may contribute to the Spin Hall effect.

For more details See here.


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 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.


 

How the Spin Hall effect makes an electron gas to be spin-polarized?

The spin-orbit interaction induces a magnetic field. This magnetic field induces a spin polarization in the electron gas. (Why? See here). The electrons, which moves in opposite directions, experience the magnetic field in opposite directions. Therefore, there are two spin accumulation with opposite spin directions.

Since in an electron gas, two spin accumulations of different spin directions can not coexist at same point, the spin accumulation of opposite spin directions annihilates with each other. However, because the difference of the scattering probability for electrons scattered towards the x-axis and into the opposite direction, there are more electron in one direction and even after annihilation some spin accumulation remains. This is the reason why the scattered current is the spin-polarized.


 


Physical mechanisms, which cause The Spin Hall Effects

 

1. Skew scatterings

2. Side-jump scatterings

3. Intrinsic. Inhomogeneous spin distribution.


 

 

Skew scatterings

Origin of the Spin Hall effect

skew scatterings

to enlarge picture click here or on the picture

Fig. 11. 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. 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.

 

 

the Spin Hall effect due to skew scatterings.

When an electron changes its movement direction after a scattering, it may experience different the effective magnetic field of the spin -orbit interaction whether it changes direction to the left or to the right. This makes different the scattering probability for an electron scattered to the left or to the right and a spin scattered current flows perpendicularly to the a charge drift current.

It is a largest when

(1) an electron changes its movement direction over 90 degrees.

(2) the spin-orbit interaction in metal is large.

(3) delocalized electrons have a large orbital moment in the direction perpendicular to the direction of the drift current

or

(4) An electrical field induces a larger orbital moment for delocalized electrons.

note: The electrical field, along which the drift current flows, itself induces Hso. It is negligibly small in a metal with high conductivity and it is larger in a metal with lower conductivity.

It occurs only when

(1) an electron changes its movement direction after a scattering.

(2) For two opposite directions of the electron movement, the sign of the spin-orbit interaction is opposite (or at least the magnitude of the spin-orbit interaction is different). It is the case when the orbital moment of delocalized electrons is different in the opposite directions

 

 

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

 

 

 


 

 

Bulk-type side-jump scatterings

Origin of the Spin Hall effect

side-jump scatterings

To enlarge picture click here

To see it it slower rate, click here

or even slower click here

Fig. Scattered of electrons between delocalized states. The electrical field around a defect induces the effective field due to the spin-orbit interaction. The effective magnetic field is in opposite directions when a delocalized electron moves from the left or from the right side from the defect. Because of the effective magnetic field of the spin-orbit interaction, the scattering probability is different whether the electron is scattered from left to the right side or from the right to the left side of the defect.

The view point moves together with electrons.

Distribution of the wavefunction of delocalized electrons is shown as the dark-yellow ellipses. The delocalized electron is shown as the green ball. The blue balls are defects. The green halo shows the distribution of the electrical field around defect. The green arrows shows the direction of the electrical field. The red arrows shows the direction and the magnitude of the effective magnetic field of the spin-orbit interaction.

the Spin Hall effect due to side-jump scatterings.

 

There is an electric field around a defect in a crustal. When a delocalized electron passes in the vicinity of the defect, it experiences an effective magnetic field of the spin-orbit interaction, which is originated from this electrical field. The effective magnetic field of the spin-orbit interaction opposite whether the electron passes from the left or the right side of defect. 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. This cause a flow of a spin current across the charge current.

 

 

It 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

 

Only a side-jump scattering on crystal defect may be spin-dependent and contribute to the Spin Hall effect. Side-jump scatterings on phonons and magnons are spin-independent and they do not contribute to the Spin Hall effect.

 

 

 

 

 

 

 

 

 

 

 

 

 

When the average distance between defects becomes comparable with the electron mean-free path, the side-jump scatterings becomes spin-independent and they do not contribute to the Spin Hall effect.

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 spin- independent.

 

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

 

 

 

 

 

 

 

 

 

 

 

 


 

Origin of the Spin Hall effect

Side-jump scatterings across an interface

To enlarge picture click here

Fig. 20 The electron current flows along a contact between two metals (the blue wall) . There is a charge accumulation at the contact (the violet balls). The charge may be accumulated because of the different work functions of the metals. The electrical field of the accumulated charge induces the effective magnetic field of the spin-orbit interaction for electrons flowing along the interface. The effective magnetic field is of opposite sign for electrons flowing at each side of the contact. The electron scattering probability across contact interface is different whether an electron is scattered from the left to the right metal 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 view point moves together with electrons.

Distribution of the wavefunction of delocalized electrons is shown as the dark-yellow ellipses. The delocalized electron is shown as the green ball. The violet balls are charge accumulated at the interface. The yellow halo shows the distribution of the electrical field around the charge. The yellow arrows shows the direction of the electrical field. The red arrows shows the direction and the magnitude of the effective magnetic field of the spin-orbit interaction.

Side-jump scatterings across an interface

The side-jump scatterings across an interface are one of strongest contributors to the Spin Hall effect.

 

The side-jump scatterings across an interface can be:

1) extrinsic. It occurs because of a charge accumulation at the interface between two metals.

1) intrinsic. It occurs because of the deformation of the orbital of electrons moving in the vicinity of the interface (See here).

Extrinsic contribution to the spin-dependent side-jump scatterings

The contact interface between two metals are charged. The charge may be accumulated because of the difference of the metal work functions. The charge accumulation at Schottky contact and the charge accumulation at sides of a pn-junction are the examples.

The electrons, which move along different sides of the contact, experience different direction of the electrical field from the accumulated 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.

In the case when there is a tunnel contact, the charge accumulation can be changed significantly by applying a voltage to the contact. By this method the magnitude of the spin-orbit interaction can be modulated and the magnitude of the Spin Hall effect can be modulated as well.

Since the electrical field at the contact interface may be significant, this contribution to the Spin Hall effect may be large.

 

Intrinsic contribution to the spin-dependent side-jump scatterings.

In the vicinity of an interface the electron orbital deforms. It causes a substantial magnetic field of the spin-orbit interaction. Mainly this effective magnetic field acts on the localized d-electrons. However, the delocalized electrons, which moves along the contact interface, also may experience a significant effective magnetic field. On each side of the contact, the effective magnetic field is in opposite directions. It makes the scattering across the contact interface to become spin-dependent (See Fig.10)

The effective magnetic field of the spin-orbit interaction for localized d-electrons in the vicinity of an interface may be very large. It may reach 1-30 kOe and larger. The effective magnetic field for the delocalized electrons is smaller, but still it may be very large. Therefore, in the vicinity of the interface the Spin Hall effect can be substantial.

 


 

 

 

 

Intrinsic contribution to the Spin Hall effect.

Origin of the Spin Hall effect

Intrinsic contribution

To enlarge picture click here

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 perpendicular to the boundary (shown as the green arrows). The electrical field induces the effective magnetic field of the spin-orbit interaction. The direction of the effective magnetic field is opposite at the opposite sides of the GaAs strip.

When the electrons the center of the strip, their spin direction is distributed randomly, because spin rotations after frequent spin-independent scatterings.

When the electrons moved to edge of the GaAs strip, the spins precess around the effective magnetic field. Because the damping of the spin precession the spins are aligned along the effective magnetic field of the spin-orbit interaction.

Note: The Oersted magnetic field, induced by the electron current, also created the spin polarization by the similar mechanism (See here)

This contribution is defined as a contribution, which electron gains not during a scattering, but during the movement between scatterings.

Because of its relativistic nature, the spin-orbit interaction can not change a trajectory of an electron.

 

Incorrect interpretation of the intrinsic. contribution to the 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.

 

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.

 

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.

 

 

Since the spin-orbit interaction can not change an electron trajectory, what is the intrinsic contribution to the Spin Hall effect???

 

Two mechanisms for the intrinsic contribution to the Spin Hall effect

(1) The spin polarization induced by the effective magnetic field of the spin-orbit interaction (See Fig. 30)

(2) Inhomogeneous spin distribution


The effects which are originated by spin-dependent scatterings

The effective electrical field, which is induced by the defect, is shown by the green arrows. The electrical field induces the spin-orbit interaction. The direction of the effective magnetic field of the spin-orbit interaction is different for electrons scattered into the left and into the right. This makes different the probabilities of scattering into the left and into the right.

Anomalous Hall effect

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

click here or on picture to enlarge it

Spin Hall effect

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

click here or on picture to enlarge it or another version

Inverse Spin Hall effect

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

click here or on picture to enlarge it

Under an applied voltage the 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 (spin directed up (TIA assembly)) flow from back to front of the wire. In the opposite direction the spin-unpolarized electrons (spin directed in all directions (TIS assembly)) 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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

 

 

 

Detailed explanation of this Figure is here

 

 



 

 

Effects often confused with the Spin Hall effect.


 

The spin Hall effect is always small!!!

(1) The scattering current is less efficient than the conventional running wave electron current

(2) The spin-orbit interaction is small.

(3) Only small part of induced spin accumulation remaining, because of two spin accumulation of opposite spin direction is induced

 

 

effect 1: Ordinary Hall effect

 

effect 2: Spin polarization induced by the Oersted magnetic field of electrical current (the Ampere's law)

 

effect 3: Spin detection

 

 

 

 

 

 

 

 

 

 

 

effect 1: Ordinary Hall effect

The spin accumulation generated by the ordinary Hall effect is often wrongly assigned to the Spin Hall Effect

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.

 

The Hall effect is very effective to enlarge the spin accumulation at one side of the sample.

How to distinguish between effects??

The Spin Hall effect =====> the spin accumulation at two opposite edges of the sample

The ordinary Hall effect =========> the spin accumulation at one edge of the sample

 

 

 

 

 

 

 

 

 

effect 2: 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

 

Figure 6 shows a film of a non-magnetic material, 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.

 

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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

effect 3: Spin Detection

The charge current, which flow along an AC spin current, is often wrongly assigned to the Inverse Spin Hall Effect

 

 

When is non-zero, the charge is accumulated along a spin diffusion and the spin current can be detected.

The AC spin current is accompanied by a charge current.

When is non-zero in a material, the charge is accumulated along a spin diffusion and the spin current can be detected. 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.

 

This AC charge current usually surpasses significantly the current induced by the Inverse Spin Hall Effect

 

For this reason, AC charge current, which is induced due to the spin detection effect, is often wrongly assigned to the Inverse Spin Hall Effect.

 

How to distinguish between effects??

The magnitude of the Spin Hall effect and the Inverse Spin Hall Effect should be the same for DC and AC currents!!!!

 

 

 

 

 

 

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