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Spin and Charge Transport

Spin-Torque Current

The spin-torque current is a current flowing between regions, which have different spin directions of the spin-polarized electrons. The spin-torque current induces a spin torque on the spin-polarized electrons. The torque rotates spin directions in neighbor regions toward each other. The spin-torque current aligns the spins of all spin-polarized electrons in one direction over the whole sample. In contrast to the spin current, which is the diffusion of the spin accumulation, the spin-torque current is the diffusion of the direction of the spin accumulation.


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.
The same content can be found in V. Zayets JMMM 356 (2014)52–67 (click here to download pdf) or (http://arxiv.org/abs/1304.2150 or this site) .Chapter 9 (pp.25-27).
An explanation can be found in Slide 12 of this Audio presentation or here

Spins of all spin-polarized electrons all over whole sample are trying to be in one direction !!!!

What will happen if in some region the spin direction of spin-polarized electrons turns out of the common spin direction?

A. The spin-torque current will flow between different regions of the sample until all spin-polarized electrons realigned along one common spin direction.

If there is an external force, which keeps spin direction

Does spin-torque current affect the localized -d and -f electrons?

No. The spin torque current is the cureent of the conduction electrons only. Of course, the sp-d exchange interaction the spins of localized and delocalized (conduction) electrons. (See Spin-Transfer torque current)


 

Fig.1 Animated picture. Spin accumulation current. Conventional spin current.

 

 

Spin accumulation current is the current of spins. The spins diffuse from a region of higher spin accumulation into a region of smaller spin accumulation.

 

 

 

 

 

Fig.2 Animated picture. Spin-torque current.

 

Spin-torque current is the current of the spin direction. The spin direction diffuses between regions of different directions of spin accumulation. The spin-torque current aligns spins in neighboring region in one directions.

 

 

 

 

 

 

 

click here to see the spin torque current explained from the classical model of spin-up/spin-down bands

Spin accumulation current

Fig.3 Animated picture. Spin accumulation current. Conventional spin current.

The spin accumulation current flows from a region of higher spin accumulation to a region of lower spin accumulation.

 

The spin accumulation current requires a spin source, but it does not require a spin drain.

 

 

 

 

 

 

 

 

Spin torque current

Fig.4Animated picture. Spin torque current.

 

The spin torque current flows between regions in which spin directions of spin-accumulated electrons are different.

 

The spin torque is trying to make spins aligned either parallel or antiparallel over whole sample.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

Properties of spin torque current

1) The spin-torque current is trying to align spins of all spin-polarized electrons in one direction over the whole sample.

2) The spin-torque current is greater in a metal with a shorter spin diffusion length

3) The spin-torque current is always accompanied by an additional spin relaxation. This means that the spin life time becomes shorter in regions where the spin-torque current flows.


 

The results in short

In the case when there is a spacial gradient of spin direction of spin-polarized electrons, the spin-polarized electrons experience the spin torque, which can be calculated as

and there is a spin relaxation (conversion of electrons from the group of spin-polarized electrons into the group of spin-polarized electrons) associated with spin-torque current with rate:

where is a unit vector along the spin direction of the spin-polarized electrons, is the spacial density of spin-polarized electrons and D is the diffusion coefficient.


 

Origin of spin torque current

If the spin directions of spin-polarized electrons are different in two close regions in a metal, there is a mutual diffusion of electrons between these two regions. Since the spin direction of the diffused electrons is different, they cause a spin torque (For details about the spin torque, click here), which rotates the spin direction of the spin-polarized electrons. Because of this spin torque, the spin directions of each region are turning toward each other.

 


 

Comparison

 

Charge current -> flow of charge

Spin current -> flow of spin

Spin-torque current -> flow of spin direction


 

Spin-torque current in the 1D geometry calculated by the random walk model

Note: The following calculation should be considered as approximate. The purpose of the following calculations is to clarify the basic properties of the spin-torque current.

Values to be calculated:

1) The spin torque due an electron diffusion from the surroundings.

2) Additional rate of the spin relaxation, because of the spin-torque current.

This is because the spin torque is always accompanied by an additional spin relaxation

 

Method:

The spin-torque and the spin relaxation are linearly proportional to the diffusion rate of the electrons (Details, see here).

This injection rate was calculated from the random-walk model. The method is similar to the method described here (see derivation of Derivation of Fick's 1st law in 1 dimension)

Assumptions:

We will consider a spin current as a diffusion of uncharged particles, which have a defined spin direction. In this case, the random walk approximation may be used.

We assume that the diffusion rate of the spin-polarized electrons does not depend on the amount of spin-polarized electrons.

 

Let us consider the spin diffusion between 3 points: , where . Each point has a different spin direction and a different amount of spin-polarized electrons. As a result of the random walk, the number of particles, which diffuse from a point in some direction (for example, from point x towards point ) is proportional to the number of particles at that point

where D is the diffusion constant and n(x) is the density of particles at point x.

In the case when spin relaxation is weak and the conversion of electrons from the spin relaxation is slow, it is possible to assume that the diffusion of spin-polarized and spin-unpolarized electrons are independent. Than, the number of spin-polarized electrons, which diffuse from point to point x, is calculated as

The number and spin direction of spin-polarized electrons at point can be approximated as

Since the spin direction at points x and are different, electrons, that diffuse from point cause a spin torque on the electrons at point x. The spin torque can be calculated by substitution Eqs (11),(12),(13) into Eq(20) here

The spin torque is accompanied by spin relaxation (conversion of electrons from the group of spin-polarized electrons to the group of spin-unpolarized electrons). The conversion rate is calculated by substitution Eqs (11),(12),(13) into Eq(21) here

Electrons also diffuse from point x-delta_x towards point x. Similarly, the spin torque due to diffusion in this direction is calculated as

and the spin relaxation can be calculated as

Summing up Eqs. (14) and (16), the spin torque, which the spin-polarized electrons experience due to the diffusion from the surroundings, can be calculated as

The spin relaxation due to the spin-torque current can be calculated as


 

The case when the spin accumulation exponentially decays along the diffusion distance

Note: it is not always that the decay of spin accumulation is exponential. One example is the spin drain effect.

In this case of the exponential decay along the x -coordinate, the number of spin-polarized electrons are

where is the spin diffusion length.

Substituting Eq.(2) into Eqs. (18) and (19), the spin torque and the spin relaxation rate are obtained as

To see how to obtain the spin-diffusion equation from the random walk model in 1D geometry by the similar method, click here

For the similar method check Derivation of Fick's 1st law in 1 dimension

Here it is assumed that the diffusion of spin-polarized and spin-unpolarized electrons are independent.

Firstly, we calculate an amount of spin-polarized electrons, which diffuse from point x out to surrounding.

According to the random-walk model, the electrons diffuse from point x to the left at the rate:

The electrons diffuse from the point x to the right at the rate

From Eqs. (p0a) and (p0b) the number of spin-polarized electrons, which diffuse out of point x per unit time, is

Secondly, we calculate an amount of spin-polarized electrons, which diffuse from surrounding into point x.

Electrons diffuse into point x from points x+delta_x and x-delta_x. The number of spin-polarized electrons, which diffuse into point x per unit time, is

There is an additional spin relaxation due to the spin-torque current. The number of electrons, which are converted at point x from the group of spin polarized electron into the group of the spin-unpolarized electrons, is

where is the spin life time.

The continuity equation for the spin-polarized electrons at point x will be

Substituting (p1)-(p3) into (p4) gives

Defining the spin diffusion length as

the spin diffusion equation is obtained from Eq. (p5) as

 

 

 

 

 

 

 

 

 

 

 

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