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
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: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 Pumping induced by a Magnetic field
Spin and Charge TransportA magnetic field generates a spin polarization in an electron gas (Spin pumping). The spin polarization of a ferromagnetic metal increases in a magnetic field. An electron gas, which is not spin polarized without magnetic field, becomes spinpolarized when an magnetic field is applied. There is a spin precession around the magnetic field. Due to the precession damping, the spins of spinunpolarized electrons are aligned along magnetic field. Therefore, the electron gas becomes spinpolarized.All calculations below fit very well to experimental measurements (See here)This effect is used to measure spin polarization of the electron gas (See here)
Results in short fact 1: In a magnetic field, conduction electrons from the group of the spinunpolarized electrons are converted into the group of the spinpolarized electrons. The magnetic field induces an additional spin pumping Mechanism of the spin pumping in magnetic field: In a magnetic field, the spins of spinunpolarized electrons aligns along the magnetic field due to the precession damping (See here). However, scatterings quickly re aligns spins of electrons into two groups of spin polarized (all spins are one direction) and spinunpolarized electrons (spins are equally distributed in all directions). (details See here) As a result, there are more spinpolarized electrons. fact 2: Spin polarization of electron gas increases in a magnetic field. (1) major reason: The magnetic field induces an additional spin pumping; (2) minor reason: The magnetic field partially suppresses (reduces) the spin relaxation.
Spin polarization sp of electron gas increases in a magnetic field and it is calculated as
where sp_{0 }is the spin polarization in absence of an external magnetic field (a material parameter), H_{pump} is the pumping magnetic field (a material parameter)
In a magnetic field, electrons from the group of spinunpolarized electrons are converted into the group of spinpolarized electrons. This process is called the spinpumping.. In case of a weak magnetic field when
the spinpumping rate can be calculated as
where n_{TIA} and n_{TIS} are the numbers of spinpolarized and spinunpolarized electrons, respectively , t_{scattering} is the effective time of scattering event 5,
is the precession damping time, λ lambda is a phenomenological damping parameter of LandauLifshitz Equations, g is the gfactor and is μ_{B} the Bohr magneton. In the case of a strong magnetic field when condition (13) is not satisfied, the spinpumping rate can be calculated as
where is the coefficient, which depends on the ratio of the scattering time to the spin damping time.
Effect of magnetic field on electron gas in a metalNonmagnetic metal Electron gas is notspin polarized without magnetic field. When a magnetic field is applied, the electron gas become spinpolarized. Direction of spin polarization is along the magnetic field. The magnetic filed converts the spinunpolarized electrons into spinunpolarized electrons, until this conversion is balanced by the reversed conversion due to spin relaxation mechanisms. The spin relaxation rate increases when the spin polarization increases. The spin polarization can be found from the condition that rates of the spin pumping and spin relaxation are equal. In the imaginary case of the absence of any spin relaxation mechanisms, in a magnetic field all electrons would become spinpolarized and spinpolarization (100 % spin polarization) in a magnetic filed.
Ferromagnetic metal Electron gas is spinpolarized in a ferromagnetic metal even without an external magnetic field. The spd exchange interaction and the intrinsic magnetic field of metal magnetization generate the spin polarization in the electron gas, which spin direction is along the magnetization. When an external magnetic field is applied along the metal magnetization, the spin polarization becomes larger. When an external magnetic field is applied at an angle in respect to magnetization direction, (1) there is a spin precession of spinpolarized electron (2) the magnitude of the spin polarization becomes either larger or smaller depending on the angle of magnetic field.. When the angle between magnetic field and magnetization is: smaller than 67 degrees > spin polarization becomes larger. larger than 67 degrees > spin polarization becomes smaller. See the interaction of two groups of spinpolarized electrons for more details.
Spin polarization of electron gas in a magnetic fieldThe spin polarization sp of the electron gas is defined as a ratio of the number of spinpolarized electrons to the total number of the spinpolarized and spinunpolarized electrons (see more here): where n_{TIA} and n_{TIS} are the numbers of spinpolarized and spinunpolarized electrons, respectively. The amounts of spinpolarized and spinunpolarized electrons is determined by a balance between the spin pumping and the spin relaxation. The spin pumping is the conversion of electrons from groups of spinunpolarized electrons into the group of the spinpolarized electrons. The spin relaxation is the conversion in the opposite direction. Why TIS and TIA abbreviations are used for the spin unpolarized and spinpolarized electrons?TIS means "timeinverse symmetrical". TIA means "timeinverse asymmetrical"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.Spinrelaxation rate: The spin damping describes the conversion of electrons from the group of the spinpolarized electrons into the group of spinunpolarized electrons(See here). The conversion rate of spinrelaxation is described as: where t_{relax} is the spin relaxation time. Spinpumping rate: The spin pumping describes the conversion of electrons from the group of the spinpolarized electrons into the group of spinunpolarized electrons (See here). The conversion rate of the spinpumping is described as where t_{pump} is the spin pumping time. Spinpumping rate induced by a magnetic field: In a magnetic field, the spins of spinunpolarized electrons aligns along the magnetic field due to the precession damping (See here). However, scatterings quickly re aligns spins of electrons into two groups of spin polarized (all spins are one direction) and spinunpolarized electrons (spins are equally distributed in all directions). (details See here) As a result, there are more spinpolarized electrons. The spinpumping (See Eq.19 below) where t_{H,pump} is the spin pumping time in a magnetic field. In the case that magnetic field is not large, so following condition is satisfied (see below for details) The spin pumping time in a magnetic field can be calculated as Spin polarization
The spin polarization sp of electron gas can be found from the condition that in an equilibrium there is a balance between the spin pumping and the spin relaxation, which is described from Eqs (1.2),(1.5) (1.6) by the condition: Substitution of Eq.(1.4) in Eq. (1.6a) gives where H_{pump} is the pumping magnetic field (a material parameter), which is calculated as Substitution of Eq.(1.1) into Eq.(1.21) gives the spin polarization sp of electron gas in a magnetic field as where sp_{0}_{ }is the spin polarization in absence of an external magnetic field (a material parameter) , which can be calculated as main partSpinPumping induced by a magnetic field.Conversion of electrons from group of spinunpolarized electrons into group of spinpolarized induced by a magnetic fieldHow calculations are done:(step 1): The LandauLifshitz Equations are solved to find the angle, at which the electron spin turns over time t toward the magnetic field due to the precession damping. (step 2): The modification of the distribution of spin directions in the group of spinunpolarized electrons due precession damping are calculated. (step 3): Spin pumping rate is calculated. The modified distribution are decomposed into a sum of the distribution of spinpolarized electrons (all spins are in one direction) and the distribution of spinunpolarized electrons (spins are equally distributed in all directions). Note, the electron scatterings quickly (near immediately) re distribute electrons of any spin distribution into two group of spin polarized and spinunpolarized electrons (details See here).
Does the spin precession induces spin pumping?No. the spin precession itself does not induce any spin pumping. Only the precession damping induces the spin pumping. Both the spinpolarized and spinunpolarized electrons precess in the magnetic field. The precession itself does not affect the time inverse symmetry of both groups of electrons and it does not induce any spin pumping ( the conversion from group of spinunpolarized electrons into the group of spinpolarized electron).
During the spin precession, the direction of spin slowly turns in the direction of the magnetic field. This effect is called the precession damping and there are many physical reasons for it (some of them are described here). (step 1): Solution of LandauLifshitz EquationsThe spin precession and the precession damping are described by the LandauLifshiz equation where gamma is the electron gyromagnetic ratio, lambda is a phenomenological damping parameter, is the magnetic moment of an electron, g is the gfactor and is the Bohr magneton. Eq. (1) can be simplify to In the case when a magnetic field is applied along the zaxis, a solution of the LandauLifshiz equation for the z component of a vector directed along the direction of electron spin is
where theta is the angle between the direction of the magnetic field and the spin direction. Theta can be found from the differential equation: where is the precession damping time. The solution of Eq.(4) is Eq. (5) describes the rotation of the spin towards the direction of a magnetic field. When , the direction of spin approaches the direction of the magnetic field (The magnetic moment of electron will be opposite to the magnetic field). If at time t0 the angle between spin and magnetic field is theta0, Eq. (5) will be to see how to find a solution (4), (5) of the LandauLifshiz equation (2), click here
Noticing that and introducing new unknowns Eq. (1) is simplified to or using definition (5a) where is the Larmor frequency. A solution of Eqs. (1.4) can be found as where Sxy is the projection of the spin vector onto the xyplane. Solution (1.5) describes the precession of the spin in the xyplane with the Larmor frequency. It can be noticed that the Larmor frequency does not depend on the angle between the spin vector and the magnetic field. Substituting Eqs.(1.5) into Eqs.(1.4) gives The solution of Eqs. (1.6) can be found as where theta is the angle between the spin direction and the zaxis. S is the magnitude of the spin vector, which equals ½. Theta is an unknown function of time. Substituting (1.7) into (1.6) gives Two equations of (1.8) are the identical and they can be express as Integration of Eq. (1.9) gives the solution of Eq. (1.10) is
(step 2): The modification of the distribution of spin directions in the group of spinunpolarized electronsA magnetic field converts the electrons from the group of spinpolarized electrons (group is timeinverse symmetrical (TIS) and its total spin is zero) into the group of the spin polarized electrons (group is timeinverse asymmetrical (TIA) and its total spin is proportional to the number of electron in this group) . The conversion occurs due to the precession damping. In the group of spinunpolarized electrons, the spins are distributed equally in all directions. However, the precession damping aligns the spins along the direction of the magnetic field (see Fig.4) and the distribution becomes different. In the following the temporal evolution of spin directions in the group of the spinunpolarized electrons are calculated. At this step, the scatterings are not included into the calculations What would happen if there were no scattering in the electron gas?After some time, the spins of all spinunpolarized electron would align along the magnetic field and the spin polarization of the electron gas would become 100%.
Note: Since the electron energy is smallest, when spin parallel to the magnetic field, and it is largest when spin is opposite to the magnetic field, the precession damping could be understood as a process to minimize electron energy.In a magnetic field, spins precess around the direction of the magnetic field. Also, spins turn toward the direction of the magnetic field with the angular rotation speed V_{θ}_{} (See Eq (4)) where θ is the angle of the spin direction with respect to the zaxis and t_{ λ } is the effective damping time for spin precession (Eq. 5a) It is assumed that the magnetic field is applied along the zaxis. The angular distribution of spin direction describes the probabilities for a “spin” state to have the spin direction in the interval between θ and θ+dθ with respect to the zaxis. Assuming that we known the angular distribution of the spin directions at time t, it is possible to calculate the distribution at time t+dt. For example, if some states had a spin angle in the interval at time t, due to precession damping their spins turn toward the zaxis and at time t+dt their spin angle will be in the interval . Assuming that time dt is sufficiently short, theta1 and d_theta1 can be calculated as The precession damping does not change the number of spins, it only changes their directions. Therefore, the number of “spin” states is the same in the interval as in the interval . Using this fact and Eq (8), the angular distribution of spin direction at time t+dt is calculated as where is the probability for a “spin” state to have spin direction in the interval and is the angular spin distribution at time t
Substituting Eq.(8) into Eq.(9) gives
Substituting Eq (7) into Eq. (10) gives the equation, which describes the temporal evolution of the angular distribution of spin directions as
To see a supporting example, which may help to understand Eq. 11, click here to expand
extended spring (simpler task to simplify understanding of solution (11))For example, a 1D spring extends so that each point of the spring has different extension speed. In this task it is necessary to find the density of the spring as a function of time. At time t, the density of the spring at point x0 can be calculated as where dm is the spring mass of the spring between points . Each point of the spring is moving with different speeds V(x), after a short time delta_t the position of points will be , where The amount of spring mass between points at time t is same as the amount of spring mass between points at time t+dt, because of the conservation of the spring mass. However, the distance between the points does change. Therefore the spring density at time t+dt can be calculated as Substituting Eq. (2.1) into Eq. (2.3) gives the equation, which describes the spring density at time t+dt
In order to solve Eq. (11), an initial condition should be used. For example, let us assume that at time t=0 a magnetic field has been applied, and the spins starts to rotate toward the magnetic field. At the time t=0 before the magnetic field is applied. Therefore, at time t=0, in the group of spinunpolarized electrons the spins are distributed equally in any direction. This conditions gives the angular spin distribution at time t=0 as Figure 5 shows the time evolution of the spin angular distribution obtained by solving numerically Eq (11) using the initial condition (12). For time longer than , spins are mainly directed along the magnetic field. After an even longer time, eventually all spins will be aligned along the magnetic field (See Fig.4). This could only happen in the case when there are no scatterings between electron states. Why dos not spins of all spin unpolarized electrons align along the magnetic field?The magnetization of a ferromagnetic material always aligns itself along along a sufficientlystrong magnetic field. Similarly, the arrow of a compass aligns itself along direction of external magnetic field. Why the spins of spinunpolarized electrons does not do the same?A. Frequent electron scatterings prevent such alignment. In fact, the spins are trying to align themselves along the magnetic field, but they are scattered back. However, after such scatterings some electrons become spinpolarized. The scattering event 5 always converts electrons of any spin distribution into two groups of spinpolarized and spinunpolarized electrons. Therefore, the scattering event 5 does not give a sufficient time to spinunpolarized electrons to turn their spins fully toward the magnetic field. Even after a slight rotation toward the magnetic field, the electrons are scattered back into two groups of spinpolarized and spinunpolarized electrons. Most of electrons remain spinunpolarized and spins of these electrons are distributed equally in all directions. A small amount of electrons, which represent "the excess of spins” in one direction towards the magnetic field, is converted into the group of the spinpolarized electrons.
Schematically, this process of the conversion from the spinunpolarized electrons into the spinpolarized electrons is shown in Fig.6. Even “spin” states experience the precession damping and the scattering event 5 simultaneously, for clearer understanding these two processes are shown separately in time. Explanation of Fig.6. Before the magnetic field has been applied, all spins of all spinunpolarized electrons were distributed equally in all directions (represented by the yellow sphere). After the magnetic field has been applied, spins start to turn towards the direction of the magnetic field, which corresponds to the elongation of the yellow sphere along the magnetic field. The scattering event converts the spin distribution back into the distributions of spinpolarized and spinunpolarized electrons. As the result of such conversion, some electrons remain with spin directed along magnetic field (shown by the blue surface) and therefore they become spin polarized. Since some electrons were converted to be spinpolarized, the number of spinunpolarized electron decreases and therefore the radius of the yellow sphere decreases as well. The spinunpolarized electrons are continuously converted into the group of the spinpolarized electrons by the magnetic field. Eventually, all spinunpolarized electrons would be converted into the spinpolarized (This corresponds to 100 % of spin polarization), but there is a reverse conversion of spinpolarized electrons back to the spinunpolarized, which is called the spin relaxation (See above). In an equilibrium, the conversion rates in both directions are equal and there are both the spinpolarized and spinunpolarized electrons.(step 3): Calculation of spin pumping rateIn the following, the conversion rate from the group of spinpolarized to the group of spinpolarized electrons is calculated.How the calculations are done ? 1. At previous step 2, we have calculated how an applied magnetic field modifies the distribution of spin directions of spinunpolarized electrons 2. In the following, the modified distribution are decomposed into a sum of the distribution of spinpolarized electrons (all spins are in one direction) and the distribution of spinunpolarized electrons(spins are equally distributed in all directions). It gives the conversion rate of spinunpolarized to spin polarized electrons or the spin pumping rate Note: The similar calculations were done to find the equilibrium distribution of spin directions for spinpolarized and spinunpolarized electrons in an electron gas. (See Eqs. (60), (61) here)How to distinguish between the spin unpolarized and spinpolarized electrons?The distinguish property of the spinunpolarized electrons is that it is timeinverse symmetrical (TIS). The group of spinpolarized electrons is timeinverse asymmetrical (TIA). This important property is used to distinguish in the calculation between the spinpolarized and spinunpolarized electrons.The scattering event 5 is rather frequent. The precession damping time is inversely proportional to the intensity of the magnetic field. In the case of a small or moderate magnetic field, should be significantly longer than the scattering time This means that the rotation angle of spins towards the magnetic field is very small between scatterings. In this case it is possible to solve analytically Eq. (11) and to find the conversion rate. Using condition (13), Eq. (11) is simplified to The angular spin distribution can be expressed as where is the angular spin distribution of the group of the spinunpolarized electrons (Eq. (12)) and For time significantly shorter than , can be used in Eq. (16) instead of (See Eq. (15)). Using Eq. (12), Eq. (16) can be further simplified to
The scatterings quickly convert any inhomogeneous distribution of “spin” states into one distribution of the spinpolarized electrons and one distribution of the spinunpolarized electrons. The probability for “spin” states to be in the group of the spinpolarized electrons can be calculated from Eq. (17) using Eq. (60) from here as Substituting Eqs. (16),(17) into (18), the integration gives the spin pumping (in case of scattering time significantly shorter than ) as where n_{TIA} and n_{TIS} are the numbers of spinpolarized and spinunpolarized electrons, respectively , t_{scattering} is the effective time of scattering event 5, The spin damping time decreases when the intensity of the magnetic field increases. In the case of a large magnetic field, the condition (13) may be not satisfied, in this case the conversion rate can be calculated as where is is the conversionefficiency coefficient, which depends on the ratio of the scattering time to the spin damping time. Figure 7 shows the conversionefficiency coefficient k ( See Eq (20)) , which was evaluated from a numerical solution of Eq.(11). The conversionefficiency coefficient k becomes smaller than 4/3 for a stronger magnetic field..
Magnetization of spinpolarized conduction electrons
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 5 (pp.1419).An explanation can be found in Slides 10 and 11 of this Audio presentation or here
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.

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