more Chapters on this topic:IntroductionScatteringsSpinpolarized/ unpolarized electronsSpin statisticselectron gas in Magnetic FieldFerromagnetic metalsSpin TorqueSpinTorque CurrentSpinTransfer TorqueQuantum Nature of SpinQuestions & Answersmore Chapters on this topic:IntroductionScatteringsSpinpolarized/ unpolarized electronsSpin statisticselectron gas in Magnetic FieldFerromagnetic metalsSpin TorqueSpinTorque CurrentSpinTransfer TorqueQuantum Nature of SpinQuestions & Answersmore Chapters on this topic:IntroductionScatteringsSpinpolarized/ unpolarized electronsSpin statisticselectron gas in Magnetic FieldFerromagnetic metalsSpin TorqueSpinTorque CurrentSpinTransfer TorqueQuantum Nature of SpinQuestions & Answersmore Chapters on this topic:IntroductionScatteringsSpinpolarized/ unpolarized electronsSpin statisticselectron gas in Magnetic FieldFerromagnetic metalsSpin TorqueSpinTorque CurrentSpinTransfer TorqueQuantum Nature of SpinQuestions & Answersmore Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11

Spin and Charge Transport
SpinTorque CurrentThe spintorque current is a current flowing between regions, which have different spin directions of the spinpolarized electrons. The spintorque current induces a spin torque on the spinpolarized electrons. The torque rotates spin directions in neighbor regions toward each other. The spintorque current aligns the spins of all spinpolarized electrons in one direction over the whole sample. In contrast to the spin current, which is the diffusion of the spin accumulation, the spintorque 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 spinpolarized and spinunpolarized 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.2527).An explanation can be found in Slide 12 of this Audio presentation or hereSpins of all spinpolarized electrons all over whole sample are trying to be in one direction !!!!What happen if in some region the spin direction of spinpolarized electrons turns out of the common spin direction?A. The spintorque current flows between different regions of the sample until all spinpolarized electrons realigned along one common spin direction. If there is an external force, which keeps spin direction of conduction electrons unchanged?Due to the spdexchange interaction and spd electron scatterings the spindirection of spinpolarized conduction electrons aligns along the spin direction of the localized delectrons (See the spin polarization). When there is some additional torque (e.g. an external magnetic field, illumination by light, etc) , the spin direction of spinpolarized conduction electrons may rotate away from the spindirection of the localized delectrons (see Spin torque). When this additional is removed, the spin direction of the spinpolarized conduction electrons returns back to be along the spin direction of the localized delectrons. Does spintorque current affect the localized d and f electrons?The spin torque current is the current of the conduction electrons. It aligns the spins of spinpolarized conduction electrons in one direction. In the case when the spin direction of the conduction electrons is rotated away from the spin direction of the localized delectrons, the delectrons experience a torque due to the spd exchange interaction and their spins are rotated following the spin rotation of the conduction electrons. (See SpinTransfer torque )
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.
Spintorque current is the current of the spin direction. The spin direction diffuses between regions of different directions of spin accumulation. The spintorque current aligns spins in neighboring region in one directions.
click here to see the spin torque current explained from the classical model of spinup/spindown bands
Spin accumulation 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
The spin torque current flows between regions in which spin directions of spinaccumulated electrons are different.
The spin torque is trying to make spins aligned either parallel or antiparallel over whole sample.
Properties of spin torque current1) The spintorque current is trying to align spins of all spinpolarized electrons in one direction over the whole sample. 2) The spintorque current is greater in a metal with a shorter spin diffusion length 3) The spintorque current is always accompanied by an additional spin relaxation. This means that the spin life time becomes shorter in regions where the spintorque current flows.
The results in short In the case when there is a spacial gradient of spin direction of spinpolarized electrons, the spinpolarized electrons experience the spin torque, which can be calculated as and there is a spin relaxation (conversion of electrons from the group of spinpolarized electrons into the group of spinpolarized electrons) associated with spintorque current with rate: where is a unit vector along the spin direction of the spinpolarized electrons, is the spacial density of spinpolarized electrons and D is the diffusion coefficient.
Origin of spin torque currentIf the spin directions of spinpolarized 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 spinpolarized 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 Spintorque current > flow of spin direction
Spintorque current in the 1D geometry calculated by the random walk modelNote: The following calculation should be considered as approximate. The purpose of the following calculations is to clarify the basic properties of the spintorque 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 spintorque current. This is because the spin torque is always accompanied by an additional spin relaxation
Method:The spintorque and the spin relaxation are linearly proportional to the diffusion rate of the electrons (Details, see here). This injection rate was calculated from the randomwalk 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 spinpolarized electrons does not depend on the amount of spinpolarized electrons.
Let us consider the spin diffusion between 3 points: , where . Each point has a different spin direction and a different amount of spinpolarized 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 spinpolarized and spinunpolarized electrons are independent. Than, the number of spinpolarized electrons, which diffuse from point to point x, is calculated as The number and spin direction of spinpolarized 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 spinpolarized electrons to the group of spinunpolarized electrons). The conversion rate is calculated by substitution Eqs (11),(12),(13) into Eq(21) here Electrons also diffuse from point xdelta_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 spinpolarized electrons experience due to the diffusion from the surroundings, can be calculated as The spin relaxation due to the spintorque 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 spinpolarized 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 spindiffusion 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 spinpolarized and spinunpolarized electrons are independent. Firstly, we calculate an amount of spinpolarized electrons, which diffuse from point x out to surrounding. According to the randomwalk 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 spinpolarized electrons, which diffuse out of point x per unit time, is Secondly, we calculate an amount of spinpolarized electrons, which diffuse from surrounding into point x. Electrons diffuse into point x from points x+delta_x and xdelta_x. The number of spinpolarized electrons, which diffuse into point x per unit time, is There is an additional spin relaxation due to the spintorque current. The number of electrons, which are converted at point x from the group of spin polarized electron into the group of the spinunpolarized electrons, is where is the spin life time. The continuity equation for the spinpolarized 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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