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: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
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Spin Torque
Magnetism of electron gasThe spin torque occurs when the electron gas is already spinpolarized and a small amount of spinpolarized electrons of different spin direction are injected. As result, the spin direction of all spinpolarized electrons rotates toward the spin direction of the injected electrons.The spin torque is always accompanied by an additional spin relaxation.Contentclick on the chapter for the shortcut(1). Explanation in short(2). Cases of two spinpumping sources(3).The interaction of two groups of spinpolarized electrons with different spin directions(Case 1). electron gas in a magnetic field(Case 2). electron gas is illuminated by light(Case 2a). AllOptical Magnetization Reversal(4). Calculations of spin torque(5). Interaction of two groups of spinpolarized electrons of different spin directions(case 1). two groups of same number of electrons(case 2). two groups of different numbers of electrons(6). Numerical simulation(7). Spin Torque(8). Property of spin torque.........
Can two groups of spinpolarized electrons with different spin directions coexist together. For example, is it possible that the spin direction of one group of spinpolarized electrons is along the xaxis and another group of spinpolarized electrons is along the yaxis?A. No. At any point of space only one group of spinpolarized electrons and one group of spinunpolarized electrons can coexist. At one point of space, the spins of all spinpolarized electrons are in one direction. At different points of space, the spinpolarized electrons may have different spin directions (See here and here). What happens when there are two spinpumping sources and the first source creates spinpolarized electrons with spin direction along xaxis and the second source creates spinpolarized electrons with spin direction along yaxis?A. The conduction electrons will be spinpolarized. The spin direction of the spinpolarized electrons will be in the xyplane.
This page describes the interactions of two (or more) groups of spinpolarized electrons of different spin directions 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 8 (pp.2225).When the number of spinpolarized electrons in electron gas there is n_{TIA1} and spin direction of spinpolarized electrons is along normal vector , and spinpolarized electrons of different spin direction are injected with the injection rate of , the spinpolarized electrons of the electron gas experiences the spin torque, which is calculated as where A(φ) is the spin torque coefficient, which depends on the angle φ between the spin directions of injected and existed electrons. (See calculations of A(φ) below)
The spin torque causes an additional spin relaxation or conversion from group of spinpolarized electrons into group of spinunpolarized electrons at the rate where n_{TIS} is the number of spinunpolarized electrons, R(φ) is the spinrelaxation coefficient, which depends on the angle φ between the spin directions of injected and existed electrons. (See calculations of R(φ) below)
Properties of spin torque:
Two or more SpinPumpsCases when there are two or more spinpumping sources of different spin directionscase 1. magnetic field applied to a ferromagnetic metalspinpumping source (1): magnetic field (see here). Spin Direction: along magnetic filed spinpumping source (2): spd exchange interaction and spd scattering (see here and here); Spin Direction: along magnetization case 2. a ferromagnetic metal is illuminated by circularpolarized lightspinpumping source (1): spin of a photon (see here).Spin Direction: along light propagation direction spinpumping source (2): spd exchange interaction and spd scattering (see here and here); Spin Direction: along magnetization case 3. Spinorbit torque (SOT) in a ferromagnetic metal (see here)
spinpumping source (1): spin Hall effect (see here) due to scatterings of spinunpolarized electrons (see here). Spin Direction: along current spinpumping source (2): spin Hall effect (see here) due to scatterings of spinpolarized electrons (see here). Spin Direction: perpendicularly to the magnetization spinpumping source (3): spd exchange interaction and spd scattering (see here and here); Spin Direction: along magnetization case 4. spintorque current in a ferromagnetic metal (see here)it is the case when the spin direction of spinpolarized electrons is different at different points of the sample. It is the case ,for example, magnetization direction is different in different magnetic domains.spinpumping source (1): spin diffusion from a neighbor region spinpumping source (2): spd exchange interaction and spd scattering (see here and here); case 5. antiferromagnetic , compensated ferromagnetic, ferrimagnetic metals (see here)it is the cases of antiferromagnetic , compensated ferromagnetic, ferrimagnetic metals, the magnetization of localized electrons are different from a atom to a atom. For example in FeTb the coupling between Fe atoms and Tb atoms is antiferromagnetic. Therefore if magnetization of Fe atom is up, the magnetization of Tb atom is down.spinpumping source (1): spd exchange interaction (scatterings) with spinup localized electrons Direction: along magnetization spinpumping source (2): spd exchange interaction (scatterings) with spindown localized electrons; Direction: opposite to magnetization
The spin torque occurs as a result of the interaction of two groups of spinpolarized electrons. The features of this interaction define the properties of the spin torque. The interaction of two groups of spinpolarized electrons with different spin directionsIt is possible that at some moment in time in a metal there are two or more groups of spinpolarized electrons. However, within a very short time all spinpolarized electrons of two groups combine into one group of spinpolarized electrons and some electrons are converted into the group of the spinunpolarized electron (additional spin relaxation). How long does it take for two groups of spinpolarized electrons with different spin directions to combine into one group with the same spin direction? A. Between very quickly and almost immediately. The time of interaction of two groups of spinpolarized electrons significantly depends on the relative number of electrons in each group. (almost immediately): In the case of the interaction of two groups with the same number of spinpolarized electrons, only (!!!) one scattering event 5 between each electron of two groups is sufficient to combine spinpolarized electron into one group with the same direction of spin. (very quickly): In the case of the interaction of two groups with a different number of spinpolarized electrons, one spinpolarized electron should experience several scattering event 5 until the two groups fully combine into one group with the same direction of spin. (See calculation below)
Spin torque. Case 1. Spinpolarized electron gas in a magnetic field. Magnetic field is applied at some angle with respect to the spin direction of spin accumulated electrons.
Case: External magnetic field applied to a ferromagnetic metal at an angle to its magnetization direction. There are two spinpumping sources to create the spinpolarized conduction electrons: (1) magnetization (localized delectrons); (2) external magnetic fieldThe spin torque affects the spinpolarized conduction electrons both in equilibrium and during the precession damping. spin torque in equilibrium: (a) magnitude of the spin polarization slightly changes (b) spin direction of spinpolarized electron slightly changes spin torque during precession damping: Pressing damping of spinpolarized conduction electrons increases during the precession damping Why it is important to understand and to calculate the spinpolarization of conduction electron in the case when a magnetic field is applied at an angle with respect to the magnetization direction?It is important for measurement of anisotropy field and the strength of perpendicular magnetic anisotropy (PMA) (See here) Does the magnetization always align along direction of magnetic field?A. Not always. When an external magnetic field is applied, the magnetization aligns itself very quickly (within 10100 ns) along an effective magnetic field due to the precession damping. However, the direction of the effective magnetic field may be different from the direction of external and internal magnetic fields due to influence of the spinorbit interaction and the exchange interaction: (influence 1) spinorbit interaction (SO) Due to some relativistic effect (see here), an electron experience an additional magnetic field. The magnitude and direction of this magnetic field depends on orbital deformation. As a result, the conduction electrons and localized delectrons may experience the SO magnetic field of different directions and magnitude(influence 2) exchange interaction Due to the quantum nature of electron, there is spindependent field between two or more electrons. Since the exchange interaction depends on the electron spin, it may align the electron spin along a direction different from the direction of the applied magnetic field. Note, in contrast to the SO interaction, the change interaction does not induce any magnetic field.Origin of the spin torque in a magnetic fieldA magnetic field converts the electrons from the group of spinunpolarized to the group of spinpolarized electrons. This effect is called the spinpumping (See here and here). The spin direction of the converted electrons is along the magnetic field. In the case when the spin direction of the existed spinpolarized electrons in electron gas is different than the spin direction of the converted electrons, the existed spinpolarized electrons experience a spin torque, which rotates their spin towards the direction of the magnetic field. Example: a magnetic field is applied inplane to a ferromagnetic metal with a perpendicular magnetic anisotropy (PMA). This configuration is used to measure the anisotropy field (See here) In the this case, inpolarization linearly depends on the magnetic field. The linear dependence makes easier to measure the strength of the PMA with a high precision (See here). The influence of the spin torque on this measurement should be taking into calculations. Fact: Decrease or increase of the spin polarization in an external magnetic field, depending on angle between the magnetic field and spin of existed spinpolarized electrons.A magnetic field constantly creates the spinpolarized electrons (spin pumping). The spin direction of these created spinpolarized electrons is along the magnetic field. When direction of the magnetic field is along the spin direction of existed spinpolarized electrons, the spin polarization of the electron gas in the magnetic field increases (experiment is here). When direction of the magnetic field is opposite to the spin direction of existed spinpolarized electrons, the spin polarization of the electron gas in the magnetic field decreases. For other angles of the magnetic field, the spin polarization either increases or decreases. 67 degrees: the margin angle between applied magnetic field and spin direction of existed spinpolarized conduction electron smaller than 67^{0} Spin polarization increases smaller than 67^{0} Spin polarization decreases The spin torque is always accompanied by an additional spin relaxation. When existed spinpolarized electrons combine with spinpolarized electron of a different spin angle created by the magnetic field, the total number of spinpolarized electrons decreases. This decrease can be large so resulting number spinpolarized electrons becomes smaller than priorexisted number and the spin polarization decrease in the magnetic field. Or the decrease can be small so resulting number spinpolarized electrons becomes larger than priorexisted number due to newly created spinpolarized electrons and the spin polarization increase in the magnetic field. The margin angle between these two cases is 67^{0}(see calculations below)
Influence of spin torque on precession damping of spinpolarized electronsIn a magnetic field there is a precession of spins of spinpolarized electron around the magnetic field. Also, the spin direction of the spinpolarized electrons slowly aligns itself along the direction of the magnetic field. The alignment of spin direction of spinpolarized electrons along the magnetic field is joint work of precession damping and the spin torque. Why the spin torque affects the precession damping of spinpolarized conduction electrons?In a magnetic field the spinpolarized electrons are constantly created (spin pumping). The spin direction of these created is along the magnetic field. The spin of existed spinpolarized conduction electrons precess along the magnetic field. Both groups of the spinpolarized electrons combine into one group, which spin direction becomes closer and closer to the direction of the magnetic field. Can the spin precession exist in an equilibrium (for conduction electrons or localized delectrons)?A. No. Any spin precession means that the magnetization changes its direction in time. According to the Maxwell's equations, any change of magnetization generates an electromagnetic wave (See here). The emission of photons removes both the energy and spin from the electron gas. Therefore, the electron gas is not in an equilibrium. The spin precession may exist in an equilibrium only if there is an additional source of the energy and the spin, which compensates the energy and spin loss due to the magnetization precession. Note, the emission of photons is one of origins of precession damping (See here).
Spin torque. Case 2. Spinpolarized electron gas is illuminated by light. Illumination by circularly polarized light, which is spinpolarized. The transfer of the photon spin to an electron gas
When absorbed, a circularly polarized photon may convert a conduction electron from the group of spinunpolarized electrons or from group of spininactive electrons to the group of spinpolarized electrons (See here about spin distributions in different groups of electrons). The spin of circularly polarized photon is one (See here) and it transformed to the electron gas when a photon is absorbed by a conduction electron. The spin direction of the opticallycreated spinpolarized electrons is along the incident direction of light. In the case when the electron gas is already spinpolarized (e.g. in a ferromagnetic metal), the spindirection of photoexcited spinpolarized electrons may not coincide with the spin direction of alreadyexisted spinpolarized electrons. In this case, light induces the spin torque, which turns the spin direction of the existed spinpolarized electrons toward the incident direction of light. Circularly polarized light may change the magnetization direction in a ferromagnetic metal. For example, when circular polarized light incidents normally on a thin film of a soft ferromagnetic metal (See Fig.2), which magnetization is in plane, it induces a spin torque. The spin torque rotates the spin direction of the spinpolarized conduction electrons from the inplane direction to the normaltoplane direction. Since 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. Note: The interaction of light with electron spin is very complex. It depends on the electron spin, orbital moment, orbital quenching, etc. See here for detailsAllOptical Magnetization ReversalThere are several possible physical origins of the alloptical magnetization reversal. The mechanism of reversal depends on a material where it occurs: (case 1 of AllOptical Magnetization Reversal): a ferromagnetic metalOrigin: spin torqueLight interacts with delocalized conduction electrons. Light transfer its spin and creates the spinpolarized conduction electrons. These electrons creates the spin torque and the spin direction of the spinpolarized electrons turns away from its equilibrium direction. Due to the spd exchange interaction and scattering the spin direction of the localized delectrons follows the turn of the spin of conduction electrons Two steps of spin rotation by light in a ferromagnetic metal: (step 1) Turn of spin direction of conduction electrons Circularlypolarized light excites spinpolarized electrons in the electron gas. Spin direction of the lightexcited spinpolarized electrons is along light propagation direction and this spin direction is not necessarily the same as the spin direction of already existed spinpolarized electrons. Since the spinpolarized electrons of two different spin directions cannot coexist at the same place, all spinpolarized electrons quickly (nearly immediately) combine into one group of the spinpolarized electrons. The spin direction of this combined group of the spinpolarized electrons is between the spin directions of existed and lightexcited spinpolarized electrons. As a result, the spin polarization of the spinpolarized turns away from its equilibrium direction toward the spindirection of photoexcited electrons. (step 2) Turn of spin direction of localized delectrons The spd exchange interaction and scatterings between delocalized conduction electrons and localized delectrons aligns spins of both types electrons in one same direction. When spin of conduction electrons turn out from the its equilibrium direction and the direction the localized d electrons, the spin of localized delectrons follows the rotation and turns to be along the spin of the conduction electrons. With which electrons light interacts more effectively? With conduction electrons or with localized delectrons?A. Light interacts more effectively with conduction electrons, because their larger size. Interaction between two particles are most effective when they have the same size. The size of a photon is about 0.1 mm 1 mm (from a light bulb) and 1 mm 1 m (from a laser). The size an electron is substantially smaller. The length of a conduction electrons is called the meanfree path λ _{mean}. The size of a conduction electrons ( λ _{mean}) is about 30200 nm in a semiconductor and 0.5~20 nm in a metal. The size of a localized delectron is about the size of atomic orbital ~0.1 nm. As a result, the interaction of a photon with a conduction electron is more effective than with a localized electron. (case 2 of AllOptical Magnetization Reversal): an oxide and transparent dielectric (for example, YIG)Origin: excitement of bonding electrons. As a result> distortion of the bonding and spin alignmentLight excites the electrons, which are responsible for the superexchange interaction or the exchange interaction, to a higher energy level. The properties (spin, orbit and so forth) of excited electrons are different from the properties they had at unexcited state. It breaks the equilibrium spin arrangement. mechanism: Light excites an electron responsible for the superexchange (exchange) interaction on a higher energy level. This locally reduces or turns off the superexchange (exchange) interaction between them and the magnetization start to precess or it may even be reversed. Note: in this case the lightinduced magnetization reversal mechanism is not related to the spin torqueCalculations of spin torqueWhat is calculated below?The case is calculated when there are two groups of spinpolarized electrons with different spin directions. It is calculated how these two groups are mixed up by scatterings to combine into a single group of the spinpolarized electrons of a single spin direction. How are calculations done?(calculation 1) The final number of spinpolarized electrons and their direction is calculated from the conservation law of the timereversal symmetry and the total spin of the electron gas. The spin is a quantummechanical object. The spins cannot be simply integrated or sum up. The quantummechanical rules have to be used. It is very difficult to do in the case of many electrons with different directions of spins. Additional difficulties is that the spin direction of each electron may be changed after a scattering (See here). The conservation law of the timereversal symmetry greatly helps such calculations. In an equilibrium, there is only one group of spinpolarized electrons of the same spin direction. Additionally there are spinunpolarized and spininactive electrons (See here). In the case when the electron gas does interact with any external "spin" objects (close system), the electron scatterings do not change the total spin of the electron gas and degree of its time inverse symmetry. Such scatterings are called spinindependent scatterings(See here). Since the timeinverse symmetry does not change during mixing of two groups of spinpolarized electrons by scatterings, the final spin direction and spin polarization is evaluated from comparison of initial degree of time inverse symmetry of two groups of spinpolarized electrons with final degree time inverse symmetry of one group of spinpolarized electrons. Note: The calculation of the degree of time inverse symmetry is very similar to the calculation of the total spin of the electron gas, but simpler (see here).(calculation 2) Tracing of the evaluation of spin distribution in time In this case, the change of spin distribution is calculated after each scattering event. The calculation traces step by step evolution of spin distribution from initial state, which is the sum of two distributions with different spin angles, to the final distribution of a single spin angle. The spin properties of all possible scatterings between spinpolarized, spin
Calculation Results: Interaction of two groups of spinpolarized electrons of different spin directions
(simple case 1): Two groups of spinpolarized electrons with the same number of electrons resulting spin angle: exactly between initial two angles +=resulting number of spinpolarized and spinunpolarized electrons: where n_{TIA1} is the number of spinpolarized electrons in each initial group of before interaction and phi is the initial angle between spin directions of two groups of spinpolarized electrons.
+=200%All initial spinpolarized electrons remain spinpolarized. None of them become spinunpolarized+=0 %All initial spinpolarized electrons become spinunpolarized. None of them remain spinpolarized+=100 %A half of initial spinpolarized electrons remain spinpolarized. Another half become spinunpolarized.(simple case 2): Number of electrons in one group spinpolarized electrons much smaller than in other group resulting spin angle: spin angle of the larger group slightly rotates (general case): Two groups of spinpolarized electrons with different numbers of electrons In the case of the interaction of two groups of different numbers of spinpolarized electrons, it takes several scattering events 5 until they relax into one group of one spin direction. For example, let us consider the groups named TIA1 and TIA2 when the number of spinpolarized electrons in group TIA1 is greater than the number of spinpolarized electrons in group TIA2. The spin direction of TIA1 is along the zaxis and the spin direction of TIA2 has angle φ with respect to the zaxis. After the first scattering event 5 the spin direction of TIA2 group will have the angle φ/2 with respect to the zaxis, the spin direction of TIA1 will not change. The number of spinpolarized electrons n_{TIA1} , n_{TIA2} in TIA1 and TIA2 groups and number of spinunpolarized electrons n_{TIS} after the first scatterings are calculated as (See scattering event 5): After the second scattering event 5, the angle between the spin directions of two groups of spinpolarized electrons will be φ/4 and the numbers of spinpolarized and spinunpolarized electrons can be calculated as where the number of spinpolarized electrons in the group TIA2 is assumed to be larger than in group TIA1: n_{TIA2}> n_{TIA1} Therefore, after each scattering event 5, the spin angle between the two groups of spinpolarized electrons is reduced by a factor of 2. The spin direction of the group, in which there were fewer spinpolarized electrons, rotates and the number of electrons in this group increases. The spin direction of the group, in which there were more spinpolarized electrons, does not rotate and the number of electrons in this assembly decreases. How long time does it take to convert two spinpolarized electrons into one equilibrium group?The most of the spinpolarized electrons make one group of the same spin direction very quickly (See the numerical simulation below). Do the spin and energy electron distributions changed during interactions of two groups of spinpolarized electrons?
It should be noted that the probability of scattering event 5 is not constant, but it is proportional to the number of spinpolarized electrons in the TIA1 and TIA2 groups. The probability p_{scat5 }of one scattering event 5 in the electron gas can be calculated as (See scattering event 5 ) where n_{TIA1} , n_{TIA2} are numbers of spinpolarized electrons in TIA1 and TIA2 groups, n_{TIS} is the number of spinunpolarized electrons and p_{scat5,0}is the probability of a single scattering event 5. You calculate the time evaluation in unit of number of scattering event 5. How does it relate to the real time?The most of the spin
Why TIS and TIA abbreviations are used for the spin unpolarized and spinpolarized electrons?TIS means "timeinverse symmetrical". TIA means "timeinverse asymmetrical". TIA1 and TIA2 describe two group of spinpolarized electrons with different directions of spin..In fact, the group of the spinunpolarized contains the electrons with a defined direction of spin, but their spins are distributed equally in all directions. There are third group of electrons, which is called spininactive or deeplevel electrons. These electrons do not a defined direction of spin. (See details here). The distinguish property of the spinunpolarized electrons is that it is timeinverse symmetrical. The group of spinpolarized electrons is timeinverse asymmetrical. This important property is used to distinguish in the calculation between the spinpolarized and spinunpolarized electrons.
Numerical simulation of interaction of two groups of spinpolarized electrons having different spin anglesInitial conditions: Figure 1 shows the calculated interaction of two groups TIA1 and TIA2 of spinpolarized electron when initially all electrons are spinpolarized (sp=100 %), 80 % of the spinpolarized electrons belongs to group TIA1 (80 % of electrons have the same spin direction), 20 % of the spinpolarized electrons belongs to group TIA1 (20 % of electrons have a different spin direction from electrons of TIA1 group), and there are no spinunpolarized electrons. The initial angle between the spin directions of two groups is 145 deg. Figure 2 shows a similar case of the interaction of the TIA1 and TIA2 groups of the spinpolarized electrons . However, initially 98 % of the spinpolarized electrons are in the TIA1 group and only 2 % of the spinpolarized electrons are in the TIA2 group. Similarly, the initial angle between the spin directions of two groups is 145 deg. Number of spinpolarized electrons in each group: After each scattering event 5 the angle between the spin directions of two groups of spinpolarized electrons decreases and some electrons become spinunpolarized. During the first 34 scattering events 5 , a significant number of spinpolarized electrons becomes spinunpolarized (there is a significant spin relaxation). Additionally, the number of spinpolarized electrons in the TIA1 group decreases and the number of spinpolarized electrons in the TIA2 group increases until the numbers of spinpolarized electrons of both spin angles become nearly equal. After that, the numbers oscillate approaching the same number. Spin direction: During a few first scatterings, the spin direction of the spinpolarized electrons of TIA2 group (a smaller number of electrons) changes substantially. In contrast, the change of spin direction of the spinpolarized electrons of TIA1 group (a larger number of electrons) is very small (nearly no change). The change becomes larger and substantial after 57 scatterings, when the numbers of spinpolarized electrons in both groups of different spin angles become comparable.
Interaction time: After 1015 scatterings, the angle between the spin directions of spin polarized electrons of two groups becomes very small and the two groups may be considered as one group of spinpolarized electrons of one spin direction..
When the initial spin directions of two group of spin polarized electrons are opposite, the resulting remaining number of spinpolarized electrons is difference between spinpolarized electrons of two groups. The resulting spinangle is the along the spin direction of the group of the larger amount of electrons. Figure P11 shows the case when the angle between spin directions of two is 179 degrees. It is near opposite. In this case, the spinpolarized electrons of group TIA2 become spinunpolarized very quickly. Even the most of the electrons of the TIA2 group are converted into spinunpolarized already after 1st scattering, a tiny amount of electrons still remains in TIA2 group. Afterwards, after each scattering the electron amount in this group sharply increases. After 12 scatterings, the amount of electrons in the TIA2 group becomes comparable with the amount of electrons in the TIA1 group.
Figure 3 shows the rotation angle and the amount of electrons converted from both groups of spinpolarized electrons to the group of spinunpolarized electrons as a function of related amounts of spin polarized electrons in each group. The probability of 0.5 means that there is an equal amount of electrons in each group. To note: The spin rotation angle and the amount of spinpolarized electrons, which become spinunpolarized (spin relaxation), are linearly proportional to the amount of spinpolarized electrons in group, which contains a smaller amount of electrons.It is the case, when the number of spinpolarized electrons of one group is significantly smaller than amount in another group (less than 10 %),
Figure 4 shows the rotation angle and the amount of electrons converted from both groups of spinpolarized electrons to the group of spinunpolarized electrons as a function of the angle between spin directions of electrons of two groups. no spin rotation: There is no spin rotation in the cases when the spin direction of spinpolarized electrons of group TIA1 and TIA2 are parallel or antiparallel of spin direction of electrons of the group TIA2.
The largest spin rotation: The rotation is largest in the case when the angle between the spin directions is around 67 degrees.Does the final amount of spinpolarized electrons increases or decreases after interaction of two groups of spinpolarized electrons?It always decreases comparing to the total amount of initial spinpolarized electrons (TIA1+TIA2). The final amount of spinpolarized electrons may either decrease or increase comparing to the initial amount in the larger group (TIA1). It depends on the spin directions in two groups. The number of spinpolarized electrons, which become spinpolarized, increases when the angle between the spin directions of electrons of two groups increases. When the angle is 120 degrees and larger, the number of converted spinunpolarized electrons is about 2%. This is twice the initial amount of spin polarized electrons in the group TIA2. For angles smaller than 67 degrees, the amount of converted spinunpolarized electrons is smaller than the initial amount of spinpolarized electrons in the group TIA2. For angles smaller than 67 degrees, the final number of spinpolarized electrons is larger than the initial number in the group TIA1 (the group of the larger initial number of electrons). Therefore, we may say that the number of spinpolarized electrons in the TIA1 increases as a result of interaction with group of TIA2. In contrast, in the case of angles greater than 67 degrees, the number of spinpolarized electrons in the group TIA1 decreases due to the interaction.
Spin torqueThe spin torque describes the case when a very small amount of spinpolarized electrons is continuously injected into the existed spinpolarized electrons of a significantly larger amount. The spin direction of the injected electrons is different from the spin direction of the existed electrons. As a result of the injection, the spin direction of the existed spinpolarized electrons rotates towards the spin direction of the injected electrons. The effective torque of the rotation is called the Spin Torque.Calculations of the Spin TorqueHow the calculations are done?The spin torque is calculated from the condition of an equilibrium of injected rate of spinpolarized electrons (TIA2) and the conversion rate of spinpolarized electrons from group TIA2 into group TIA1 of the existed spinpolarized electrons and the group of spinunpolarized electrons (See Eq.(25)). Even though the calculations are done numerically using the method of calculations of Fig.1 and Fig.2 , the results are expressed analytically with calculated parameters A(φ) and R(φ), which are defined as the spin torque coefficient and the spinrelaxation coefficient. The dependence of the spin torque on different material and injection parameters can be calculated from calculated dependencies of A(φ) and R(φ) on those parameters. Results:In the case when a small amount of spinpolarized electrons of the group TIA2 is continuously injected at a rate into the region where there are spinpolarized electrons of the TIA1 group, the spinpolarized electrons of TIA1 group experience a spin torque, which can be calculated as as where n_{TIA1} is the number of spinpolarized electrons in the group TIA1, are unit vectors directed along spin directions of electrons of the TIA1 and TIA2 groups,, respectively, and A(φ) is the spin torque coefficient, which depends on the angle φ between the spin directions of injected and existed electrons. The conversion rate (spin relaxation rate) of spinpolarized electrons into spinunpolarized can be calculated as where R(φ) is the spinrelaxation coefficient, which depends on the angle φ between the spin directions of injected and existed electrons. old form:
Figure 5 shows the spin torque coefficient A(φ) and the spinrelaxation coefficient B(φ) (TIS conversion coefficient) as a function of angle φ between the spin directions of injected and existed electrons (angle between spins of TIA1 and TIA2 groups). Note: For small angles φ (φ<30 deg) , A(φ) ~57 deg and R(φ) ~0. Figure 6 shows the spin torque coefficient A(φ) and the spinrelaxation coefficient B(φ) (TIS conversion coefficient) as a function of the amount of injected spinpolarized electrons (TIA2) with respect to the number of existed spinpolarized electrons (TIA1) for different angles φ . The number is scanned from 0 to 40%. Note: the value of the coefficient B(φ) is between 1 and 2 (the variation is small). In the calculations shown in Fig.5 and Fig.6, a slow injection rate was assumed. In the case of a faster injection rate, the spin torque coefficient A(φ) and the spinrelaxation coefficient B(φ) become dependent on the injection rate.Property of spin torque:
the spin torque is linearly proportional to the injection rate . Note: It is only the case of a slow injection rate, when the spin torque coefficient A(φ) and the spinrelaxation coefficient B(φ) do not depend on the injection rate (the case of a slow injection rate).
How to determine whether the injection rate is slow or fast?As was calculated in Fig.1 and Fig.2, the interaction of two groups of spinpolarized electrons with different spin angles takes some time until they are fully transformed into one group of spinpolarized electrons of the same spin direction. We assign t_{2} as the effective time, after which the spin direction of electrons of the TIA1 group is fully rotated into its final direction. Even though a tiny oscillations around the final angle persists for a long time, the defined time t_{2} is the time, after which the deviation of the spin angle from its final position is less than 0.1 degrees. For examples of Fig.1 and Fig.2, , the time t_{2} corresponds to a time duration of ~57 scattering events 5. The number of spinpolarized electrons ( TIA2), which are injected during the time t_{2}, is calculated as where is the injection rate. Under a continuous and constant injection rate, Δn_{TIA2} is the constant number of spinpolarized electrons of the group TIA2, which spin is not yet aligned along spin direction of the existed spinpolarized electrons of group TIA1. These spinpolarized electrons of the group TIA2 are in the process of the conversion to the group TIA2 and to the group of spinunpolarized electrons. As sooner as they are converted, the same number of spinpolarized electrons TIA2 is injected. As a result, Δn_{TIA2} remains a constant. Equilibrium: during time t_{2} > (1) Δn_{TIA2} electrons were injected into group TIA2; _{} (2) Δn_{TIA2} electrons was converted from group TIA2 into group TIA1 and the group of spinunpolarized electronsWhen the Δn_{TIA2} exceeds 1 % of the number of spinpolarized electrons in the group TIA1 (the number of the existed spinpolarized electrons), the injected rate is defined as fast and the dependence of the coefficients A(φ) and B(φ) on the injection rate should be taken into calculations.
Can we mark or distinguish somehow a spinpolarized electron from different groups of different spin directions or from spinunpolarized electrons? For example, similar as an electron can be distinguished in a semiconductor between to be an "electron" or a "hole"?A. No. The electrons are frequently scattered between different groups. It is only average number of electrons in each group remains unchanged after a frequent scatterings (See here). It contrasts to the case of "holes" and "electrons" in a semiconductor. The electron scattering between a "hole" and "electron" is a very rare event, because of their different spacial symmetry (See here) and an electron stays as a "hole" and "electron" for a long time in a semiconductor. (Note, the case of a metal is different)
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.An explanation can be found in Slide 11 of this Audio presentation or here

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