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: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 & Answers 
Spin Statistic
Magnetism of electron gasThe energy distributions of all electrons (spinpolarized electrons+spinunpolarized electrons) is described by the FermiDirac distribution. The individual distribution of the spinpolarized and spinunpolarized electrons is described by the spin statistics (see below). The spinpolarized and spinunpolarized electrons have different energy distributions, because of their different scattering probabilities. The spin statistics is critically important for description of different spindependent effect in a solid. For example, the different distributions of the spinpolarized and spinunpolarized electrons is a reason of a difference of conductivities of the spinpolarized and spinunpolarized electrons.Contentclick on the chapter for the shortcut
(1). Explanation in short(2). Why energy distribution depends on the electrons spin?(3). Calculation of the spin distribution. Method(4). Spin distribution. Case 1: Spinunpolarized electron gas(5).Spin distribution. Case 2: Spinpolarized electron gas(6). spin pumping & spin relaxation vs meanfree path λ_{mean}_{}(7).Total number of spinpolarized and spinunpolarized electrons(8). Spin distribution within approximation of spinup/spindown bands(8a).Nondegenerated semiconductor. Approximation of spinup/spindown bands.........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 4 (pp.914).
Why the spin statistic is important?The spin statistic is important for: (1) solving the Boltzmann transport equation for analyzing the spin transport; (2) calculating the spin life time; (3) calculating the spin torque and spin torque current; (4) calculating the Pauli paramagnetism and for many other calculations, where the properties of spin in an electron gas is relevant.Why energy distributions are different for spinpolarized and spinunpolarized electrons?It is because of different electron scattering probabilities between states, which is occupied only by one electron.When an electron scattered into a state, which is already occupied by one electron, the spin of the scattered electron should be opposite to the spin of the electron, which is already occupying one of two places in the quantum state.Since spins of all spinpolarized electrons are directed in the same direction, there is no direct scatterings between states occupying by spinpolarized electrons. In contrast, spins of spinunpolarized electrons are directed in different directions. There is a probability of scatter ins between states, which are occupied by spinunpolarized electrons. The difference in the scattering probabilities makes different the distributions of spinpolarized and spinunpolarized electrons.
Explanation in short
The energy distributions of conduction electrons in a quantum states, which occupied by a single electron and by two electrons of opposite spins, are different, because of difference of their scattering probabilities (reason wthy there is a difference of scattering probabilities) The electron spin influences the probability of the electron scattering. An electron can be scattered into a quantum state, which is already occupied by one electron, only if the electron spin is opposite to spin of other electron. In contrast, when an electron is scattered in a fully unoccupied state, there is no limitation on spin direction.
Nonmagnetic material
Calculation details is here SpinStatisticNonMagneticMetal.pdfThere are two groups of conduction electrons in a nonmagnetic material: (group 1) Conduction electrons in states, which are occupied by one electron; (group 2) Conduction electrons in states, which are occupied by two electrons of opposite spins. (group 1) Conduction electrons in states, which are occupied by one electron. Their distribution is calculated as where E is the electron energy (group 2) Conduction electrons in states, which are occupied by two electrons of opposite spins. Their distribution is calculated as the number of empty places in states, which is not occupied either by one or two electrons is cuclculated as .
The energy distribution of spinpolarized electrons: Probability F_{TIA}(E), that a conduction electron of energy E is spinpolarized, is calculated as where sp is the spin polarization of the electron gas, E is electron energy and E_{F} is the Fermi energy.The energy distribution of spinunpolarized electrons Probability F_{TIS}(E), that a conduction electron of energy E is spinunpolarized, is calculated as
where sp is the spin polarization of the electron gas, E is electron energy and E_{F} is the Fermi energy.The energy distribution of spininactive electrons The spininactive electrons occupies the states, which are filled by two electrons of opposite spinsProbability F_{full}(E), that a conduction electron of energy E is spininactive, is calculated as The energy distribution of all electrons (the FermiDirac distribution) A conduction electron can be in one of three groups: (1) group of spinpolarized electrons; (2) group of spinunpolarized electrons; (3) group of spininactive electrons.Probability F_{all}(E) that a conduction electron has energy E is calculated as Why there is a strange uneven number like 0.3832.. in the above formulas for the distributions of spinpolarized and spinunpolarized electrons?A. It is because the formulas were obtained by integrated over a sphere of possible spindirections of the spin unpolarized electrons (See Eq.(17)). Therefore, the number is the product of π (See Eq.(17)).SpinStatistic.m Matlab function, which calculates spin statistics (number of spinpolarized and spinunpolarized electrons)
Q. Why quantum states filled by one and two electrons should be distinguished???
A quantum state, which is filled by two electrons of opposite spins, has zero spin. In contrast, a quantum state, which is filled only by one electron, has spin 1/2. The spin makes a big difference for energy distributions of these states. Q. Why energy distribution depends on the electrons spin? Why spinpolarized and spinunpolarized electrons have different energy distributions? How the spin makes the difference?It is because of different electron scattering probabilities between states, which is occupied only by one electron. When an electron scattered into a state, which is already occupied by one electron, the spin of the scattered electron should be opposite to the spin of the electron, which is already occupying one of two places in the quantum state. Since spins of all spinpolarized electrons are directed in the same direction, there is no direct scatterings between states occupying by spinpolarized electrons. In contrast, spins of spinunpolarized electrons are directed in different directions. There is a probability of scatter ins between states, which are occupied by spinunpolarized electrons. The difference in the scattering probabilities makes different the distributions of spinpolarized and spinunpolarized electrons.
Q. Why we should care about the energy distributions? They are critically important for a description of charge and spin transports. For example, the difference in energy distribution for spinpolarized and spinunpolarized electrons causes the difference in electrical conductivity for spinpolarized and spinunpolarized electrons (See here). Another example is the hole. Only using the spin statistics it is possible to explain the charge and spin transport properties of holes in metals and semiconductors (See here).
How the distributions for spinpolarized and spinunpolarized electrons are calculated???Firstly, the electron scatterings probabilities between groups of spinpolarized and spinunpolarized electrons are calculated for each electron energy. It fixes the ratio between the number of states, which are occupied by two electrons, the number of states, which are occupied by one electron of group of spinpolarized electrons, and the number of states, which are occupied by one electron of group of spinunpolarized electrons. Secondly, the energy distributions are calculated from the condition that the sum of 3 probabilities (an electron is either in the group of spinpolarized electrons or in the group of spinunpolarized electrons or in the group of spininactive electrons (which fill the fullfilled states) is described by the FermiDirac distribution.
Why electron scattering depends whether the electron belong to the group of spinpolarized or spinunpolarized electrons??
The electron scatterings are spindependent because an electron only of a fixed spin direction may be scattered into a quantum state, which is already filled by another electron. For example, only an electron with spinup spin direction can be scattered into a quantum state, which is already filled by a spindown electron.
Scattering probabilities:(case 1) Scattering probability=0 (a) A scatter of a spinpolarized electron into a state, which already filled by one spinpolarized electron Spins of all electrons in the group of spinpolarized electrons are in the same direction. For example, all spins are up. Therefore, all unfilled empty place in these state are spindown and only spindown electrons can be scattered into these empty places. Since all electrons are spinup, the electrons can not be scattered between states of spin polarized electrons.(case 2) Scattering probability is between 0 and 1. (a) A scatter of a spinpolarized electron into a state, which already filled by one spinunpolarized electron An electron can be scattered into state, which is already filled by one electron, only if spin of scattered electron is opposite to the spin of of the electron, which is already occupying the state. In the group of the spinunpolarized electrons the spin directions are distributed equally in all directions. Therefore, there is always nonzero probability that an electron has required direction of the spin and the electron can be scattered into a state, which is already occupied by one electron. (case 3) Scattering probability = 1 (a) A scatter of an electron into a state, which is not occupied by any electron Total spin of a fullfilled state, which is filled by two electrons of opposite spins, is zero and there is no spin direction for this state. Therefore, an electron, which scattered from a fullfilled state into a halffilled state, always has spin opposite to the spin of the electron, which is already occupying the halffilled state.
The energy distribution of spinpolarized and spinunpolarized electrons (Spin Statistic) is important for understanding any spintransport effect and phenomena.The Spin Statistic is also important to understand properties of electron transport even in the case of spinunpolarized electron gas. For example, it describes properties of electrons and holes in a metal and a semiconductor (See here for details)
Main part
Distribution of spindirections in the group of spinpolarized and spinunpolarized electrons.
Challenge: There is a significant exchange of electrons between groups of spinpolarized and spinunpolarized electrons, because of frequent spinindependent scatterings between electron states. All these scatterings should be taken into calculation!!!A little help: The spinindependent scatterings do not change the number of electrons in each assembly. The total spin of all spinunpolarized electrons remains zero. The spinindependent scatterings do not change the total spin of spinpolarized electrons.
What is the energy distribution?A. The energy distribution F(E) is a probability to find electron at energy E. The energy distributions of electrons and holes in a metal are determined by the FermiDirac statistic
All conduction electrons consist of groups of spinpolarized, spinunpolarized and spininactive electrons. Each group is described by its own distribution: (1) F_{TIS} is the distribution of spinpolarized electrons; (2) F_{TIA} is the distribution of spinunpolarized electrons; (3) F_{full} is the distribution of full states. Each full state is occupied by two spininactive electrons; (4) F_{empty} is the distribution of states, which is not occupied.; Condition 1: The sum of three distributions of each group of electrons gives the distribution of all electrons (Eq.1): All electrons (FermiDirac distribution)= spinpolarized electrons + spinunpolarized electrons +spininactive electronsElectron gas in the absence of a spin accumulation consists of (1) spinunpolarized electrons; (2) spininactive electrons. The condition 1 are described as
Electron gas in the spinpolarized electron gas consists of (1) spinpolarized electrons; (2) spinunpolarized electrons; (3) spininactive electrons. The condition 1 are described as Scatterings and Energy distributionsThe symmetry of scatterings fixes the the relations between F_{TIS} , F_{TIA} and F_{full}. Substitution them into Eqs.(10) or Eqs.(10s) gives the system of equations from which F_{TIS} , F_{TIA} and F_{full} are calculated How the scatterings are calculated?A. Only dependencies of scattering probabilities on the spin direction is calculated. For example, an spinup electron can be scattered into a "spin" state, which is already occupied by one electron, only if the spinup place is available or the same if the state is occupied by a spindown electron. As a result, the scattering probability of spinup electron into spindown "spin" state is the largest (= one). The scattering probability of spinup electron into spinup "spin" state is the smallest (= zero). The scattering probability for other directions is between zero and one. The spin direction of spin unpolarized electrons are distributed equally in all directions. In order to calculate the scattering probability of a spinunpolarized electron, the scattering probability is integrated over all possible spin directions. Details about the dependence of scattering probability on the spin direction can be found here
At first, the spin distribution is calculated in the simplest case when when there is no spin accumulation in the metal. Next, the case, when there is a spin accumulation, is calculated.Spin Distribution. Case 1: Spinunpolarized electron gas.Spin Distribution in the absence of a spin accumulationIn this case there are no spin polarized electrons. There are only spinunpolarized and spininactive electrons. The distribution of electrons in each of these two group is calculated below. The energy distribution of all electrons (spinunpolarized + spinactive) is described by the FermiDirac distribution !!In the case when the electron gas is not spinpolarized, a conduction electron may occupy ether a "spin" state or "full" states. There are also "empty" states, which are not occupied.. The ratio between the numbers of these 3 states is determined by the condition that the rate of conversion of "spin" states into "full" states and "empty" states (scattering event 4) is equal to the rate of back conversion of "full" states and "empty" states into "spin" states (scattering event 5). In the case of the spinunpolarized electron gas, the electrons occupying the "spin" states are called spinunpolarized electrons. Electrons occupying the "full" states are called spininactive electrons. (See here)The energy distribution of all electrons (independently of their spin direction) is described by the FermiDirac distribution !!
Main idea of the calculation:Firstly, the scattering probabilities between spinunpolarized electrons and spininactive electrons are calculated. It gives the ratio between numbers between electrons in each group.Secondly, the distribution for each group is calculated from the condition that the energy distribution of all electrons is described by the FermiDirac distribution. What is calculated below?:Probability F_{TIS} that a quantum state is occupied by one electron or probability, that a conduction electron belong to the group of spinunpolarized electrons; Probability F_{full} that a quantum state is occupied by two electron or probability, that a conduction electron belong to the group of spininactive electrons.
Why the scattering probabilities are different for spinpolarized electrons, spinunpolarized electrons and spininactive electrons?A. It is because an electron with a fixed direction of spin can be scattered only to an unoccupied quantum state in which this spin direction is allowed. For example, a spinup electron can be scattered only ether into a fully empty state or a state, which is occupied by one spindown electron and in which spinup position is not occupied. How the calculations have been done?A. Firstly, the scattering probability is calculated for a fixed spin angle of a scattered electron. Secondly, the probability is integrated over all possible spin angles of a scattered electron and spin angles of possible quantum state, to which the electron is scattered. Spins of spinpolarized electrons are directed in one direction. Spins of spinunpolarized electrons are distributed equally in all directions. The spininactive electrons have no defined spin direction. Therefore, their scatterings is not limited by the spin direction. Main part: Calculation of scatteringsif the electron gas is spinunpolarized why conduction electrons should be divided into the groups?A. If in the case when the electron gas is not spinpolarized, a quantum state can be occupied either by one or by two electrons. The spin properties of these two kinds of electrons are very different. Their equilibrium concentration is calculated from the condition that the scattering rate from one group to another group is equal to the scattering rate in the opposite direction. Main idea beyond the calculations of the spin statistics: take into account spin features of all possible scatteringsIn the scattering event 4 an electron from a “full” state is scattered into an “empty” state. As a result, the number of “full” and “empty” states decreases and the number of “spin” states increases. In the scattering event 5 an electron from a “spin” state is scattered into another “spin” state. As a result of this scattering, the number of “full” and “empty” states increases and the number of “spin” states decreases. The average number of “spin”, “full” and “empty” states in a metal is determined by the condition, that scatterings of “spin” states into “full” + “empty” states are balanced by the scatterings of “full” + “empty” states back into “spin” states. The ratio of "spin" states to "empty"/"full" states is determined by the balance of the conversion of "spin" states into "full"/"empty" states (scattering event 5) and the reverse conversion of the "full"/"empty" states into "spin" states (scattering event 4). The result of an electron scattering out of a “full” state into an “empty” is two “spin” states. Therefore, scattering event 4 causes an increase of the number of “spin” states with the rate where P_{scattering }is is the probability of an electron scattering event per unit time.
A scattering of an electron out of a “spin” state into an empty place of another “spin” state (the scattering event 5) reduces the number of “spin” states. The result of such a scattering is either two "spin" states or a "full" state +a "empty" state. The probability of a scattering of spin states into "empty" +"full" states by this scattering event depends on the angle between the spin directions of “spin” states where φ is the angle between the spin directions of the spin states. If F_{spin}(φ) is the distribution of “spin” states, which have the angle φ with respect to the zaxis, the probability of scattering of "spin" states with angles φ_{1} , φ_{2} is calculated as Since the spin of a spin unpolarized electron can have any direction with equal probability, the angular distribution F_{spin}(φ) of the spinunpolarized electrons can be calculated as where F_{spin} is the number of “spin” sates at energy E. From Eqs. (14) ,(15), the reduction rate of “spin” states due to the the scattering event 5 can be calculated by integrating Eq.(14) over all possible angles φ_{1} , φ_{2} as
In equilibrium, the conversion rate of “spin” states into “full” states Eq.(17) is equal to the rate of the back conversion Eq.(12) Substituting Eq. (11) and Eq. (17) into Eq. (17a) gives Solving the system of Eqs. (10), (17b) and using Eqs. (1) and (2), the distribution F_{TIS} of spinunpolarized electrons in spinunpolarized electron gas can be calculated as and distribution F_{full} of spininactive electrons in spinunpolarized electron gas is calculated as
Click here to see details how to obtain Eq. (18) from Eqs (10) and (17)
The system of Eqs. (10) and (17b) is
Substituting the 2d and 3d Eqs. (18.1) into the 1st Eq. gives The solution of the last equation of (18.2) gives From Eqs. (1) and (2) we have Substituting (18.4) into (18.3) gives
Figure 2 shows the distribution of "spin", "full" and "empty" states in a spinunpolarized electron gas calculated from Eqs. (18). The "spin" states are mainly distributed near the Fermi energy. The "full" states are mainly distributed at lower energies and "empty" states are mainly distributed at higher energies. Figure 3 shows the ratio of electrons in “spin” states to the total number of electrons (red line) and the ratio of holes in “spin” states to the total number of holes (black line). It should be noticed that for energies larger than ~2 kT above the Fermi energy the number of "spin" states is significantly greater than the number of "full" states. This means that in this case almost all electrons are in "spin" states. Similarly, for energies smaller than ~2 kT below the Fermi energy, all holes are in "spin" states. (more about the holes is here)
The total number of “spin” states in the group of spinunpolarized electrons can be calculated as where D(E) is the density of states In the case of a metal, which has a nearly constant density of states near the Fermi energy, the total number of “spin” states can be calculated as Therefore, the number of “spin” states increases linearly with temperature.
Spin Distribution. Case 2: spinpolarized electron gas
Additional calculations (comparing to the above case of spinunpolarized gas): The scatterings of spinpolarized electrons are taking into account; The rate equation for the conversion between spinpolarized and spinunpolarized electrons are solved. Main idea of the calculation:Firstly, the scattering probabilities between spinpolarized electrons, spinunpolarized electrons and spininactive electrons are calculated. It gives the ratio between numbers between electrons in each group.Secondly,conversion between groups due to the spin pumping and the spin relaxation is calculated. Thirdly,the distribution for each group is calculated from the condition that the energy distribution of all electrons is described by the FermiDirac distribution. What is calculated below?:Probability F_{TIA} that a quantum state is occupied by one electron or probability, that a conduction electron belong to the group of spinpolarized electrons; Probability F_{TIS} that a quantum state is occupied by one electron or probability, that a conduction electron belong to the group of spinunpolarized electrons; Probability F_{full} that a quantum state is occupied by two electron or probability, that a conduction electron belong to the group of spininactive electrons.
Are scatterings of spinpolarized and spin unpolarized electrons independent?A. Not at all. It is even opposite. For example, the spinpolarized can be scattered into a quantum state of spinpolarized electrons. E.g. if the spins of all spinpolarized electrons are up, a spinpolarized electron can be scattered only into a quantum state where spinup position is empty. However, in the case of spinpolarized electrons, the spinup position is occupied and only spindown position is empty. Therefore, there is no scatterings between spinpolarized electrons. Do the spin independent scatterings change the number of the spinpolarized and spinunpolarized electrons?A. No. The numbers of spinpolarized and spinunpolarized electrons remain the same. The total spin of whole electron gas remains the same as well. What is a spinindependent scattering? What is a difference between a spinindependent and spindependent scatterings?A. The spin independent scattering is a scattering inside of the electron gas, which does not change the total spin of the electron gas. Such scattering can be considered as occurring inside the close system of the electron gas. In contrast, a spindependent scattering is the interaction of the electron gas with an external spin particle. The spindependent scattering may change the total spin of the electron gas. The effect, which occur due to the spindependent scatterings, are the Anomalous Hall effect, the Spin Hall effect, the spin relaxation. main partNote:Since spinindependent scatterings intermix all electrons, including the electrons of spinpolarized, spinunpolarized and spininactive electrons, the energy distribution of all groups of electrons is described by a single FermiDirac distribution with the same Fermi energy.rate equations: More details about spin polarization and rate equations are hereSpinpumping rate: The spin pumping describes the conversion of electrons from the group of the spinpolarized electrons into the group of spinunpolarized electrons. The conversion rate of the spinpumping is described as (details are here) where t_{pump} is the spin pumping time. Spinrelaxation rate:
The spin damping describes the conversion of electrons from the group of the spinpolarized electrons into the group of spinunpolarized electrons. The conversion rate of spinrelaxation can be described as (details are here) where t_{relax}_{} is the spin relaxation time. In equilibrium there is a balance between spin relaxation and spin pumping, which gives Eq.(19e) gives the ratio between distribution F_{TIA} of spinpolarized electrons and the distribution F_{TIS} of spinunpolarized electrons as The 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. At each electron energy E, the spin polarization can be defined as: At each energy the spin polarization sp(E) is determined by a balance between the spin pumping and spin relaxation. Substitution of Eq.(19f) into Eq.(20s) gives
Eq.(19e) gives the ratio between distribution F_{TIA} of spinpolarized electrons and the distribution F_{TIS} of spinunpolarized electrons as Approximation (soft): It can be assume that sp(E) of Eq.(20e) is energyindependent and equal to the spin polarization sp of the electron gas. Discussion on the validity of this approximation is below. All calculations below are valid also for the case when sp(E) depends on the electron energy E A conduction electron can be either in the group of spinpolarized electrons, or group of spinunpolarized electrons or group of spininactive electrons. This condition gives Substituting the FermiDirac distribution (1) and (2) and Eq.(23) into Eq.(22) gives The ratio between F_{full}, F_{empty} and F_{TIA} , F_{TIS} are found from calculation of scatterings (check on extended menu below). Solution of Eqs.(22e) gives distributions of the spinpolarized F_{TIA} and spinunpolarized electrons F_{TIS} as where sp is the spin polarization of the electron gas To see how to derive Eqs. (24),(25) , click here
What is calculated below?The scatterings probabilities between spinpolarized, spinunpolarized and spininactive electrons Integrating of possible spin direction of the spinunpolarized electrons
Note: x is used as the notation for the spin polarization in this inset (sp is used in main part)
Substituting Eq. (23) into Eq. (22) gives Therefore, the system to be solved consists of Eqs.(17b) and (24.1) Substituting 2d and 3d equations of (24.2) into the 1st equation gives Solving the last Eq. of (24.3) gives
Do the spin pumping and spin relaxation rates depend on electron energy? How good is the approximation that the spin polarization does not depend on electron energy?A. Both the spin pumping and spin relaxation depend on the electron energy. The approximation of these dependencies to the constant average values of the spin pumping and spin relaxation is a zeroorder approximation. Taking into account these dependencies are straightforward (See Eqs.(24),(25))
Why the spin pumping depends on λ_{mean} and correspondingly on the electron energy E?Each mechanism of the spin pumping has its own reason for its dependency on E. Dependency of each mechanism of spinpumping on the electron energy E: Details on different mechanisms of the spin pumping is herespinpumping mechanism (1): spd exchange interaction. Dependency on electron energy E: moderate. The strength of the spd exchange interaction depends on the overlap of wave functions of a conduction electron and a localized delectron. When λ_{mean }becomes larger, the overlap becomes smaller. spinpumping mechanism (2): spd scatterings; Dependency on electron energy E:moderate. The probability of a scattering between of a localized delectron into a state of conduction electron depends on the overlap of wave functions of a conduction electron and a localized delectron. When λ_{mean} becomes larger, the overlap becomes smaller. spinpumping mechanism (3): precession damping in a magnetic field; Dependency on electron energy E:moderate. The spin damping is not a spin conserving mechanism. The spin damping occurs due to an electron interaction with another spin particle (See here for details). Such interaction depends on the electron size and therefore on λ_{mean}. spinpumping mechanism (4): spindependent scatterings & spin accumulation due to Spin Hall effect; Dependency on electron energy E:moderate. A spindependent scattering depends on which side of scattering center the electron is (See here). When λ_{mean} becomes larger, the size of conduction electron becomes larger, the conduction electron overlaps more scattering centers and the spin dependency of the scatterings becomes weaker. Why the spin relaxation depends on λ_{mean} and correspondingly on the electron energy E?Each mechanism of the spin relaxation has its own reason for its dependency on E. Dependency of each mechanism of spinrelaxation on the electron energy E: Details on different mechanisms of the spin relaxation is herespinrelaxation mechanism (1): spindependent scatterings; Dependency on electron energy E:moderate. A spindependent scattering depends on which side of scattering center the electron is (See here). When λ_{mean} becomes larger, the size of conduction electron becomes larger, the conduction electron overlaps more scattering centers and the spin dependency of the scatterings becomes weaker. spinrelaxation mechanism (2): incoherent spin precession in a spatially inhomogeneous magnetic field: Dependency on electron energy E:moderate. When λ_{mean} becomes larger, the size of conduction electron becomes larger and influence of the spatial inhomogeneities of magnetic field becomes weaker. spinrelaxation mechanism (3): local fluctuation of magnetization direction:Dependency on electron energy E:moderate. When λ_{mean} becomes larger, the size of conduction electron becomes larger and influence of the local fluctuations of magnetic field becomes weaker.
Total number of spinpolarized and spinunpolarized electrons
(number of all electrons excluding the spininactive electrons)For the case when there is no spin accumulation See Eq. (18.b)The integration over all quantum states gives the total numbers of spinpolarized n_{TIA} and spinunpolarized n_{TIS} electrons as where D(E) is the density of states, F_{TIA} , F_{TIS} are energy distributions of spin polarized and spinunpolarized electrons (See Eqs. (24),(25)) In the case of a metal, which has a nearly constant density of states near the Fermi energy, the number n_{TIA} of spinpolarized electrons and the number n_{TIS} of spinunpolarized electrons and the their total number (all conduction electrons except spininactive electrons) can calculated as
where D is the density of states at the Fermi energy and sp is the spin polarization of the electron gas
The number of in a metal linearly increases with temperature.
Why the total number of spinpolarized and spinunpolarized electrons are important.A. It is because only the spinpolarized and spinunpolarized electrons determine the property of charge and spin transport (See here and here). The contribution of spininactive electrons is very weak. The properties of the "full" state is nearly identical to the properties of "empty" states. Both these types of quantum states do not contribute much to the transport.
end of main partSince the approximation of spinup/down bands has been used in many classical research works, it is important to verify the conditions for validity of this approximation.Approximation of of spinup/spindown bands is good for a nondegenerated semiconductorbad for a metalSpin distribution within approximation of spinup/spindown bands
Classic model of spinup/spindown bandsThe model of spinup/spindown bands ignores that the spin rotates during the scattering events N4, N5. It assumes that spinup and spindown electrons do not interact (or there is only a very weakly interaction between them). Therefore, the classical model assumes that spinup and spindown electrons are staying inside their own bands for a relatively long time. Since there is no interactions between the bands, the energy distributions of spinup and spindown electrons are independent. The distribution of electrons in each band is assumed to be described by its own FermiDirac distribution and the Fermi energy for each band can be different. Thus, the total amount of electrons may not be described by the FermiDirac distribution. For example, in the case of a spin accumulation, one band (in the case of Fig. 1(a), it is the spinup band) is filled by a greater number of the electrons. The Fermi energy for the spinup band is larger than the Fermi energy for the spindown band and the distribution of all electrons is not the FermiDirac distribution.
Correct descriptionAll conduction electrons can be divided into the groups of spinpolarized and spinunpolarized electrons (See Fig. 1 (b)). There is a frequent exchange of electrons between the groups, because of frequent spinindependent scatterings (scattering events N4 and N5). However, the spinindependent scatterings do not change the number of electrons in each group. Because of the significant exchange of electrons between the groups of spinpolarized and spinunpolarized electrons, the Fermi energy should be the same for both spinpolarized and spinunpolarized electrons
The electron distribution in the model of spinup/spindown bands.In the model spinup/spindown bands assumes that there are two independent distributions for spinup and spindown electrons. Note: in the classical model, the spinup and spindown directions are not specified. Usually, they are chosen from the geometrical considerations (direction of a magnetic field, the magnetization direction)
Both the distribution of spinup electrons and the distribution of spindown electrons are described by the independent FermiDirac distribution (See Fig. 1a): where are the Fermi energies for the spinup and spindown electrons, respectively.
To See details for electron distribution in case of model of spinup/spindown bands (Eq. (4)), click here
The energy distributions of "spinup", "spindown", "full" and "empty" states may be also calculated within the classical model. In the following the spin statistics is described using the classical model of spinup/spindown bands. For this purpose the probability of each state is calculated using the assumption that the energy distribution of the electrons of each band is the FermiDirac distribution The probability of a "spinup" state, in which only one of the two places is occupied by electrons, will be a product of the probabilities that the spinup place is occupied and the spindown place is not occupied The probability of a "spindown"state, in which the spindown place is occupied and the spinup place is not occupied, will be The probability of a "full" state, in which both the spinup and spindown places are occupied, will be The probability of a "full" state, in which both the spinup and spindown places are occupied, will be
In the classical model, the number of accumulated spins is determined by the spin chemical potential as
it is the case when energy distribution calculated from the approximation of the spinup/spindown bands is correct Even in this case, some spin properties of electron gas are described correctly by the approximation of the spinup/spindown bands
Nondegenerated semiconductor. Approximation of spinup/spindown bandsA semiconductor, in which the Fermi energy is in its band gap and does not cross either the conduction or valence band, is called a nondegenerated semiconductor. In the case of a nondegenerated semiconductor, the electron energy distribution is described by the Boltzmann statistic.In the case of a nondegenerated semiconductor, the approximation of spinup/spindown bands can be used for the description of the spin transport. The use the approximation of the spinup/down band for the spin transport is described hereIn the case when electron energy is sufficiently greater than the Fermi energy the energy distribution of the spinpolarized and spinunpolarized electrons (Eqs. (24),(25)) can be simplified as to see how Eq. (27) is derived from (24),(25), click here
Using condition (26), Eq. (24) can be simplified as
The analysis of the spin transport using the model of spinup/spindown bands may be simpler and sometimes more understandable. Eqs. (27) can be rewritten as where the effective chemical potentials for spinpolarized and spinunpolarized electrons are defined as In the case of the model of spinup/spindown bands, the distribution of electrons in the spinup and spindown bands of a nondegenerated semiconductor are described as The similarities of the distributions (53) and (55) implies that the the spin transport in a nondegenerated semiconductor can be approximated by the classical model of spinup/spindown bands by utilizing the individual chemical potential Eqs. (54) for spinpolarized and spinunpolarized electrons. It is important to emphasize that the simple form of the distribution (53) leads to a simple description of the spin transport . When the spin direction of the spinpolarized electrons is the same over whole sample, the electron transport of each assembly is independent and it satisfies the Ohm law: where the conductivities for spinpolarized and spinunpolarized electrons are linearly proportional to number of electrons in each group of spinpolarized and spinunpolarized electrons. Therefore, generally they are 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.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 4 (pp.914).An explanation can be found in Slides 7 and 8 of this Audio presentation or here

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