The scattering current occurs because of changing of possition of an electron after a scattering. In the electron gas the electrons are constantly scattered from between quantum states. Because the quantum states have different coordinates in the phase space, each scattering causes an electron movement in the phase space. The scattering current can only exists when there is a difference in an electron scattering probability between two opposite directions. For example, the scattering conduction is significant in a vicinity of interface, because a substantial difference of an electron scattering probability toward and outward of the interface. The hopping conductivity, the Spin Hall effect and anomalous Hall effect occur because of a flow of the scattering current.

 

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The same content can be found in this paper (http://arxiv.org/abs/1410.7511 or this site for a more upgraded version) .Chapter 9, pp. 22-25

Possible confusion!!: from 2014 to 2017 I have used names TIA and TIS for groups of spin-polarized and spin-unpolarized electrons, respectively. The reasons are explained here.

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Why do we need the Boltzmann Transport Equations?????

 

From the Boltzmann Transport Equations, the charge conductivity , spin conductivity , detection conductivity and injection conductivity can be found

 

 

Results in short

I. The Boltzmann transport equations were solved for the scattering current

 

2. It was found the main required condition for a flow of scattering current:

The scattering probability in one direction should be different from the scattering probability in the opposite direction!!!

 

a) current in the vicinity of the interface ======> scattering probability towards and outwards the interface is different

b) the Spin Hall effect ======> scattering probability to the left and to the right with respect of the flow direction of the drift current is different

c) hopping conductivity ======>scattering probability along the electrical field and in the opposite to the electrical field is different

 

3. The origin of the Spin Hall effect is explained

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Scattering current

Note: The scattering current is less effective for the transport of the charge and the spin than the current of the running-wave electron .

Note: Both the running-wave electrons and the standing-wave electrons contribute to the scattering conductivity.

The running-wave-electron conduction occurs because of the electron movements between scatterings. The scattering conduction occurs because of the electron movements during scatterings. In an electron gas the electrons are constantly scattered between quantum states. Because the quantum states have different coordinates in space, each scattering causes an electron spatial movement. In the case when the electron scattering probability is the same in all directions, there is no scattering current. The scattering current can only exists when there is a difference in the electron scattering probability between two opposite directions. For example, the scattering conduction is significant in the vicinity of an interface, because of a substantial difference in the electron scattering probability toward and away from the interface. The hopping conductivity and the Spin Hall effect are examples of the scattering conductivity.

 

The scattering current is significantly less efficient for the spin and charge transport than the running-wave-electron current. Its contribution to the total current becomes substantial only when the current of running-wave electrons becomes small (for example, the hopping conductivity or the conductivity through a high-resistance contact (a tunnel barrier or a semiconductor-metal contact)) or when the scattering current flows perpendicularly to the current of the running-wave electrons (the Spin Hall effect).

 

Where the scattering current occurs?

It occurs in case when there is a difference in an electron scattering probability between two opposite directions.

 

Examples of scattering conduction:

(1) the Spin Hall effect;

(2) the hopping conduction;

(3) conduction near an interface;

(4) the tunneling conduction

The scattering conductivity is nearly zero in bulk of semiconductors and metals with a low density of defects.

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3 types of scattering currents

The current of scattering current occurs, because of the scattering of an electron from one state to another state.

Depending between each states the scatterings occur, 3 types currents can be distinguished:

(1) Electron scattering current. half-filled + "empty" states

(2) Hole scattering current. half-filled + full-filled states

(3) full-filled / "empty"scattering current. "empty" + full-filled states & half-filled + half-filled states

 

Electron scattering current. half-filled + "empty" states	Hole scattering current. half-filled + "empty" states	full-filled / "empty"scattering current. half-filled + full-filled states & half-filled + half-filled states
Consequence scattering of an electron from a half-filled state into "empty" state corresponds to movement of half-filled states from right to left and to movement of "empty" from left to right . The spin and "-e" charge are transported in the same direction from the right to the left.	Consequence scattering of an electron from a full-filled state into a half-filled state corresponds to movement of half-filled states from right to left and to movement of full-filled from left to right. The spin and "-e" charge are transported in opposite directions. The spin is transported from the right to the left. The "-e" charge is transported from the right to the left.	Consequence scattering of an electron from a full-filled state into a "empty" state and next from a half-filled state to a half-filled state. This current does not transports the spin. It transports only the charge.

Source & destination

For each current there are two contributions of different Source & destination

(1) Electron scattering current.

First contribution: source: half-filled-----> destination: "empty"; Second contribution: source: "empty"-----> destination: half-filled

(2) Hole scattering current. half-filled + "empty" states

First contribution: source: half-filled-----> destination: full-filled; Second contribution: source: full-filled-----> destination: half-filled

(3) full-filled / "empty"scattering current.

First contribution: source: full-filled-----> destination: "empty"; Second contribution: source: "empty"-----> destination: full-filled

 

 

 

Solution of the Boltzmann equation

for electron scattering current

The electron current occur because of the electron scattering between half-filled states and full-filled states. (See left figure above). Contribution: source: half-filled-----> destination: "empty";

Scattering probability is proportional to

(1) The number of states from which the electron is scattered

(2) The number of states to which the electron is scattered

(3) Overlap integral of wave functions between which the scattering occurs.

(4) Probability that the states interacts with a defect or phonon or other scattering source.

 

This probability may be different for a electron scattered in forward direction p_forward and electron in scattered in backward direction p_backw

where are wave functions of half-filled and "empty" states, l_scat is the average distance through which an electron moves after one scattering.

 

For running-wave electrons the Probability that the states interacts with a defect or phonon or other scattering source can be calculated as follows. During time dt an electron moves a distance . The mean-free path is an average distance an electron moves between scattering. The probability for electron to be scattered during time dt is

Change of distribution function due to scatterings

 

For electron current 4 scattering events should be considered

Event 1: scattering of an electron from “spin” state at position x forward to “empty” state. Because of his event, the number of spin states at point x decreases

Event 2 : scattering of an electron from “spin” state at position x backward forward to “empty” state. Because of his event, the number of spin states at point x decreases

Event 3: scattering of an electron to “empty” state at position x from forward “spin” state. Because of his event, the number of spin states at point x increases

Event 4: scattering of an electron to “empty” state at position x from backward “spin” state. Because of his event, the number of spin states at point x increases

 

Summing up all probabilities Eqs. (30.4-30.5) under the condition that l_scat is small, we

In Eq.(30.8) it is assumed that the during a scattering an electron moves along only the x-direction. In a general case the electron may move also in the y- and z-directions. It can be assumed that during a scattering, an electron moves along its propagation speed. Then, the average distance l_scat,x through which an electron moves along the x direction after one scattering is calculated as

Substituting Eq. (30.8a) into Eq. (30.8) gives

Eq. (30.9) describes the scattering when an electron is scattered from a half-filled state into an "empty". Therefore, the half-filled state is the source of the scattering and the "empty" state is the destination. Another type of the scattering can occur between a half-filled state and "empty" state. It is when "empty" state is the source and the half-filled state is the destination. It could be understood as a hole is scattered from the "empty" state to half-filled state. The second contribution can be calculating similar to Eq. (53) but replacing :

Summing up Eqs. (30.9) and (30.10) the total contribution of the scattering current can be calculated as

The Boltzmann equation Eq.(2) only with the scattering term Eq.(30.11) and relaxations term Eq(12) in a static case for half-filled states is given as

Using the approximation of a small external perturbation and substituting Eq(12) and Eq (14.4) into Eq (30.12) gives the Scattering term of the Boltzmann equation for the electron scattering current as

 

 

The electron scattering current can be calculated from Eq. (11.8) here as

Similarly, the hole scattering current and empty/full scattering current can be calculated as

 

 

 

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The electron scattering probability between two half-filled states

 

Since in the TIA assembly all spin are parallel, an electron can not be scattered between two spin states of the TIA assembly.

The probability for scattering of an electron between a half-filled state of the TIS assembly a half-filled state of the TIA assembly is 1/2.

 

The probability for scattering of an electron between a half-filled state of the TIS assembly a half-filled state of the TIS assembly is 1/2.

 

 

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Spin Hall effect

 

to note: The scattering current induced due to the Spin Hall effect flows perpendicular to the drift current. This reason why the scattering current is not oversreened by the larger current of the running-wave electrons

 

Origin of the Spin Hall effect in short:

In order for the scattering current to flow, scattering probability in one direction should be different from the scattering probability in the opposite direction. The spin-orbit interaction makes the scattering probability different between scatterings to the the left and to the right with respect to the flow direction of the drift current.

The origin of the Spin Hall effect:

1. The effective magnetic field of spin-orbit interaction is opposite for the electrons scattered to the left and to the right with the respect to the flow of the drift current.

2. The effective magnetic field of spin-orbit interaction is opposite for the electrons of opposite spin directions.

3. The effective magnetic field of spin-orbit interaction makes the probability

 

 

The Spin Hall effect is an example of a scattering current. The electrons, which are scattered into the left side and into the right side with respect to the flow direction of the drift current, may experience opposite direction of the effective magnetic field of the spin-orbit interaction. This may cause a different electron scattering probability to the left and right directions. The difference depends on the spin direction. This difference may cause a flow of the spin-polarized scattering current perpendicularly to the drift current. This effect is called the Spin Hall effect

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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