My Research and Inventions ###### click here to see all content or button bellow for specific topic ### Reticle 11 ### Reticle 11 ### Reticle 11 ### Reticle 11

Scattering Current.

### Spin/Charge 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

## Scattering hole current  The scattering electron current in n-type semiconductor. The yellow cells indicates quantum states. In a n-type semiconductor the most states are "empty". There are a few half-filled states. Electrons are drifted through the sample by the scattering from a half-filled state into "empty" state. At left side of the sample the directions of movement of half-filled and "empty" states are shown. At right side of the sample the directions of movement of a positive charge, a negative charge and spin are shown. The spin moves from "-" to "+". It is opposite direction to the case of p-type semiconductor The scattering hole current in p-type semiconductor. Spin-up current. The yellow cells indicates quantum states. In a p-type semiconductor the most states are full-filled. There are a few half-filled states. Electrons are drifted through the sample by the scattering from a full-filled state into half-filled state. At left side of the sample the directions of movement of half-filled and full-filled states are shown. At right side of the sample the directions of movement of a positive charge, a negative charge and spin are shown. The spin moves from "+" to "-". It is opposite direction to the case of n-type semiconductor

# 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.

### (4) the tunneling conduction

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

### 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

### 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 ### Defect and interfaces are good for spin transport Click on image to enlarge it.
Improvement due to the defect and interfaces:
1. Enlargement of spin transport by an electrical current (spin injection)
2. Enhancement of the spin detection
3. Enhancement of the exchange interaction between localized d- electrons and conduction electrons (sp-d interaction)

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 ### Two mechanisms of the Spin&Charge transport by Running-Wave Electrons.

##### (2) Scattering current. Origin: electron scattering from one state to another Blue balls represent defects. Brown ellipse shows distribution of wave function of the running-wave electrons.

A scattering occurs when wave functions of two states are overlapped (shown in red color)

Note: An electron current is shown. A hole current of the running wave electrons is shown here.

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   ## 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.

proof (click to expand)

The probability of scattering of an electron between two half-filled state, when an angle between their spin directions is theta, can be calculated as (See here) The angle distribution of the electrons in the TIS assembly is Noticing that Substituting Eq.(c3) into Eq.(c1), the probability for scattering of an electron between a half-filled state of the TIS assembly a half-filled state of the TIA assembly can be calculated as 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.

# Spin Hall effect

##### Example of scattering current

Spin Hall effect

###### difference of electron scattering probabilities along and opposite to the x-axis is due to different direction of Hso 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.

### 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