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
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

Scatterings. Distribution of spin directions in electron gas Magnetism of electron gasA spin direction of an electron may not be conserved after a spinindependent scattering. The spin rotations occur due to a quantummechanical fact that, when a quantum state is occupied by two electrons of opposite spins, the total spin of the state is 0 and the spin direction of each electron in this state can not be determined.Because the timeinversion symmetry is conserved after a spinindependent scattering, the electrons in an electron gas can be divided into groups of spinpolarized and spinunpolarized electrons. Spinindependent scatterings do not change the number of electrons in each group. This is why it is possible to treat the groups of spinpolarized and spinunpolarized electrons as independent objects in the electron gas.
Results in short
Important fact: The rotation of the electron spin after a spinindependent scattering!! (1) Spin direction rotates after frequent spinindependent scatterings. (2) The timeinverse symmetry is conserved after the Why it is important??Because of these frequent scatterings: (1) it is possible to divide electrons into groups of spinpolarized and spinunpolarized electrons. (Proof is here) (2) Only one group of spinpolarized electrons can exists in electron gas. Two groups of spinpolarized electrons of different spin direction can not coexist at same place at the same time. (See spin torque) (3) Spinpolarized and spinunpolarized electrons have different energy distributions. (see spin statistics) (4) The spinpolarized and spinunpolarized electrons have a different conductivities. (see here) Note: A conduction electron experience a spin independent scatterings each 10100 femtoseconds. It is very frequent. What is the spinindependent scattering?It is a scattering, after which the total spin of electrons are conserved. Could be a scattering spindependent.Yes. It is the case when the conduction electrons also interact with localized electrons. These scatterings are very seldom. The spindependent scatterings is the origin of the Anomalous Hall effect and the Spin Hall effect. Reason why spin is rotated after a spinindependent scattering?
When an electron state is filled up by two electrons of opposite spin, the spin direction of each electron is not distinguishable. Because of this fact, the spin direction of the electrons may not be conserved even in the case of spinindependent scatterings.
The result of scattering of two halffilled states with angle phi between their spin directions can be fullfilled +"empty" states with probability
or it can be two halffilled states of the same spin direction with probability This is an important fact, which is used in calculations of the spin statistics and the spin torque.Electronics states in solid
The spin properties of an electron gas in a metal depend on its density of states near its Fermi energy. Each of the states is distinguished by a direction of its wave vector k in the Brillouin zone, its energy E and its spacial symmetry. Due to the Pauli exclusion principle, each state can be occupied maximum by two electrons of opposite spins. The energy distribution of electrons in the metal is the FermiDirac distribution. The states of a higher energy (at least 5kT above the Fermi energy), are not occupied by electrons and almost all of them are “empty” states with no spin (See Fig. 1). Almost all states of a lower energy (at least 5kT below the Fermi energy) are occupied by two electrons and almost all of them are the “full” states with no spin. A state, energy of which is near the Fermi energy, may be filled by one electron and have spin ½. Also, there is a probability that the state is not filled or it is filled by two electrons of opposite spins (See Fig. 1).
Spin rotation after spinindependent scatterings. Example of an elastic scattering.
Let us consider elastic spinindependent scatterings between 3 states, which all have the same energy and this energy is close to the Fermi energy. The states are labeled as “A”, “B” and “C” (See Fig.2) and they are only distinguished by the direction of their wave vector k in the Brillouin zone. Let us assume that at time t0 the state “A” is filled by one electron, which spin is up, the state “B” is filled by one electron, which spin is down, and the state “C” is not filled. At time t1 the electron of the state “A” experiences an elastic scattering. Since the spinup place of the state “B” is not occupied, the electron may be scattered into state “B”. In this case, at time t1 state “B” becomes occupied by two electrons, its spin is zero and it does not have a spin direction. At time t2 an electron of the state “B” experiences again an elastic scattering into an unoccupied state “C”. Since there was no electron in state “C” at time, the electron scattered into state “C” may have any spin direction. For example, it may have spinright direction. Than, spin of the electron, which remains on the state “B”, is spinleft. With equal probability the electrons of states “B” and “C” may have spinfront and spinback directions. Therefore, the electrons have experienced a random spin rotation after two consecutive spinindependent scatterings. The random spin rotation is because of a quantum nature of an electron. At time t1, the spin of state “B” is zero and inside of the electrons there is no record, which spin directions they had at time t0. For example, the three states shown in Fig. 1(b) are equivalent, indistinguishable and all of them equally represents one “full” state. Therefore, when an electron is scattered out off state “B”, its spin can have any direction, only the spin directions of the scattered and remaining electrons should be opposite.
Detailed example. Spin rotation in a degenerated GaAs.Click here to expand and to see the Detailed example. Spin rotation in a degenerated GaAs.
In this example I intentionally have used many unnecessary details in order to avoid possible questions.Let me take as an example a degenerated ntype GaAs, which Fermi level is 10 kT above the conduction band. The conduction band is isotropic. The temperature is room temperature. The density of state of GaAs at this Fermi energy is about 5.5E20 1/cm3 /eV. Each of state is distinguish by a direction of its wave vector k in the Brillouin zone and its energy E. Due to the Pauli exclusion principle, each state can be occupied maximum by two electrons of opposite spins. Let me assume that mobility mu of the electrons in the GaAs is 2800 cm2/V/s. Than, the average time between electrons collision can be estimated as τ_{collision}= μ* m_{eff}/q=0.28*0.067*9.1E31/1.6E19 s =10 ps, where m_{eff}=0.067*m_{electron} is the effective mass of a conduction electron in GaAs and q is the elementary charge. Let me consider an elastic scattering between 3 states labeled as A, B, C. The energy of states is the same and it is near the Fermi energy. Only the states are distinguished by their direction of the wave vector. For example, the wave vectors of states are directed as
Important part:
Let me assume that at time t1=0 ps:
At time t2=10 ps the electron of state A is experience an elastic scattering (for example, on defect). During the scattering, electron spin and electron energy do not change. Only the electron wave vector is changed. Since the spinup place of state B is not occupied, the electron may be scattered into state B. In this case, at time t2=10 ps the state B becomes occupied by two electrons, its spin is zero and it does not have a spin direction. At time t3=20 ps, one electron of the state B is scattered into state C due a spinindependent elastic scattering. Since there was no electron in state C at time t2=10 ps, the electron scattered into state C may have any spin direction. For example, it may have spinleft direction. Than, spin of the electron, which remains on the state B, is spinright. Therefore, the electrons, which had spinup/down directions at time t1=0 ps, rotates the spin to spinleft/right directions during 20 ps because of two elastic spinindependent scatterings. The random spin rotation is because of a quantum nature of electrons. At time t2=10 ps, the spin of state B is zero and inside of the electrons there is no record, which spin direction they had at time t1=0 ps. Therefore, when they are scattered out off state B their spin can have any direction, only the spin directions of both electrons should be opposite. During about a half of all scatterings that an electron experiences, its spin direction is randomly rotates. If we suggest that spin directions of all electrons are only in spinup and spindown directions, within a short time all directions will be mixed and all electrons are redistributed into the TIS assembly (spins are distributed in all directions with equal probabilities). As was shown in the Chapter 8 (Fig. 9), it takes about 1530 scatterings for two spin directions to be within 1 degree. Therefore, for above example the short time means 150300 ps.
General facts about the timeinverse symmetry breaking.To see general facts about the timeinverse symmetry breaking, Click here to expand
The breaking of the time inversion symmetry of an electron gas is essential for any spin manipulation and spin transfer. Only in the case when the time inversesymmetry is broken, it is possible to manipulate and store the spin data in a electron gas.
Figure 1 shows two different angular distributions of spins of conduction electrons. One distribution corresponds to the case when the spacial inversion symmetry is broken and the other distribution corresponds to the case when it is not broken. In both cases there is no spin accumulation and the timeinversion symmetry is not broken. However, the distributions are significantly different. When the spacial symmetry of an object is broken , it is possible to store a bit of information in this object. Figure 3 shows an example of how to store a bit of information in an electron gas by means of a direction of the spacial symmetry breaking without breaking of the timeinverse symmetry. Note: Spin distribution of Fig.1(a) corresponds to two TIA assemblies (See below), which quickly relax into one TIA assembly.
Difference between the classical and proposed models
In the classical model (of spinup/spindown bands) it is assumed that a fullfilled state is timeinverse symmetrical, but is not symmetrical for an arbitrary spacial rotation. The classical model ignores the fact that spin of a fullfilled state is zero!
In the proposed model (of TIA/TIS assemblies) a fullfilled state is timeinverse symmetrical and it is symmetrical for an arbitrary spacial rotation. This is because the spin of a fullfilled state is zero!
The timeinverse symmetry of an electron state
In a solid an electronic state can be occupied either a) by two electrons of opposite spin; b) one electron c) no electrons. The properties of an electron state are different in each case. These 3 states are defined as halffilled, "full' and "empty" states (See Fig 5). For details, see here. 1. fullfilled states and "empty" states have spin zero, they do not have a spin direction and they are timeinverse symmetrical 2. halffilled states have spin = 1/2, they do have a spin direction and they are timeinverse asymmetrical An important fact :this fact is due to the quantum nature of spinSince spin of a fullfilled state is zero, the fullfilled state does not have a spin direction. The spin directions of individual electrons of the fullfilled state are not distinguishable. This is because a wavefunction of a particle with zero spin is invariant under rotation.
Spinindependent scatteringsBy a spinindependent scattering we define a scattering, which does not depend on the electron spin. This means that there is no spin operator in the Hamiltonian describing the scattering event.During the spinindependent scattering the spin direction may not be conserved (see below), but the timeinversion symmetry/asymmetry is always conserved after any spinindependent scattering. 1. Scatterings, which do not change the spin direction A scattering of one electron from a halffilled state into an "empty" state (Fig 6) does not change the spin direction. A scattering of one electron from a fullfilled state into an halffilled state (Fig 7) does not change the spin direction . These two processes are responsible for establishing a thermoequilibrium.
2. Scatterings, which change the spin directiona) Scattering of two spin states into fullfilled +"empty" states. (scattering event N3)
During this scattering, an electron from a halffilled state is scattered into an unoccupied state of another halffilled state. The result is one fullfilled state and one "empty" state. During this scattering, two timeinverse asymmetric halffilled states, both of which have spin directions, are converted into two timeinverse symmetric states, which do not have any spin direction. Because of the quantum nature of spin, the information about spin direction of electrons before the scattering is fully lost after this scattering Note: Before the scattering, a system of two halffilled states is not invariant under axis rotation. After the scattering, the system of fullfilled and "empty" states is invariant under axis rotation.About the rotational invariance see Wikipedia's explanation here
b) The generation of a pair of halffilled states with an arbitrary spin direction. Scattering event N4.
For this scattering an electron from a fullfilled state is scattered into an "empty" state. The result is two halffilled states with arbitrary, but opposite spin directions. During the scattering, two timeinverse symmetric states, which do not have any spin direction, are converted into two timeinverse asymmetric halffilled states, which have spin directions.
c. Scatterings of two halffilled states with different spin direction. Scattering event N5
The scattering of two “spin” states , which have the angle phi between their spin directions, results either two “spin” states or a “full” state + a “empty” state. Below I the probabilities for the scattered electrons to be either in halffilled states or in fullfilled plus "empty" states are found. Let us consider a scattering of two “spin” states. The spin directions of the “spin” states are in the seaplane and have an angle phi /2 and phi /2 with respect to the xaxis (See Fig.10). These “spin” states are described by the spinors Two spin states can be considered as a closed system, the wavefunction of which is described by a product of the spinors S1 and S2. A spinindependent scattering can be considered as a perturbation inside the closed system and it does not change the overall wavefunction of the whole closed system. Therefore, even after the scattering the two electrons are described by the same overall wavefunction as before the scattering. Even though the wavefunction of each electron may change, the overall wavefunction of two electrons does not change. It is described by a product of the spinor S1 and S2.
The spin of a “full” and an “empty” state is zero. Therefore, the wavefunction of a “full” or an “empty” states is a scalar with respect to an axis rotation. There is only one possible scalar from a binary product of spinors S1 and S2. It is called the scalar product of spinors S1 and S2 (see Ref.1). Therefore, the wavefunction of a “full” + an “empty” state is the scalar product of spinors S1 and S2 and the probability that the scattered electrons will be in a “full” + an “empty” state is calculated as Since the result of a scattering of two halffilled states can be either “full” +”empty” states or two “spin” states, from Eq. (11) the probability of the scattered electrons to be in “spin” states is given as [1] L.D. Landau and E.M. Lifshitz. “Quantum mechanics. Nonrelativistic theory”, 3d ed., Pergamon, 1991, pp. 207208.To see details about the spinor of halffilled states , click here to expand
The wavefunction of two noninteracting electrons is the vector product of corresponding spinors: The spinor describing the “spin” state, in which the electrons are scattered, is defined by the binary form (12b), from which the scalar product of spinors S1 and S2 is excluded. It is the dot product of spinors S1 and S2* and the vector product of S1 and S2 In the case when spinors S1 and S2 are given by Eqs.(10), Eq. (12c) is simplified as The probability that the scattered electrons will be in “spin” states is calculated from Eq.12d as
d) The rotation of spin direction. Scattering event N6.This event consists of two consecutive spinindependent scatterings: event N3 and event N4 or could be also N5 and N4. The result of these consecutive spinindependent scatterings is a spin rotation. The rotation angle can be any angle with an equal probability.
Groups of spinpolarized and spinunpolarized conduction electronsEven though the spin direction of one electron is not conserved after scattering event N6, the total spin of all conduction electrons is conserved. Due to this fact it is possible to divide all conduction electrons into two groups. In the first group contains “spin” states with the same spin direction. The total spin of the first group is nonzero. In the second group there are “spin” states, “full” states and “empty” states. The total spin of the electrons of the second group is zero. Even though there is a frequent exchange of electrons between the groups, the number of electrons in each group does not change due to spinindependent scatterings. This division of conduction electrons into two groups significantly simplifies the analysis of the magnetic properties and spindependent transport of an electron gas.
group of spin unpolarized electronsIt consists of fullfilled states, "empty" states and halffilled states. The balance between the conversion of halffilled states into fullfilled and "empty" states (scattering event N5) and the reverse conversion (scattering event N4) determines the number of fullfilled, "empty" and halffilled states in the group of spin unpolarized electrons. Details see here. The fullfilled and "empty" states do not have any spin direction. There is an equal number of 'spin' states with all spin directions. Therefore, the group of spin unpolarized electrons does not have any spin direction and the total magnetic moment of the group of spin unpolarized electrons is zero. The overall wavefunction, which describes all electrons in the group of spin unpolarized electrons, is a scalar. In equilibrium in a nonmagnetic metal all conductions electrons are only in the group of spin unpolarized electrons (there is no spinpolarized electrons and there is no spin accumulation). Distribution of spin directions in the group of spinpolarized electrons can be described as where is the probability of an electron to be in the group of the spinunpolarized electrons.
group of spin polarized electrons
The group of spin polarized electrons contains only states filled by one electron. Spins of all electrons in this group are in the same direction. The group of spin polarized electrons has a direction, along which the timeinverse symmetry is broken (spin direction). In the case when the spin direction of the spinpolarized electrons is along the zaxis (theta=0), the distribution of spin directions of the group of spinpolarized electrons is described as where delta is the deltafunction.
There is a frequent exchange of electrons between the groups of spinpolarized and spinunpolarized electrons due to the scattering event 5. However, the spinindependent scatterings do not change the number of electrons in each group. Because of the frequent exchange of electrons, the spinpolarized and spinunpolarized electrons are in the same thermal equilibrium (they have the same temperature). In a magnetic field, the electrons from the group of spinunpolarized electrons are converted into the group of spinpolarized electrons (a spin accumulation is created). (See the electron gas in a magnetic field here)
Figure 13 shows an example of the distribution of spin directions in the group of spinpolarized and spinunpolarized electrons for two cases of the different spin accumulations. The distribution of the spin directions of spinunpolarized electrons is a sphere. It is because in this group the spins are equally distributed in all directions.. The distribution of the spin directions of spinpolarized electrons is a deltafunction. It is because all spins are directed in the same direction.
The equilibrium number of electrons in groups of spinpolarized and spinunpolarized electronsIn the following the equilibrium number of spinpolarized and spinunpolarized electron is calculated. Also, it is proved that spinindependent scatterings do not change the number of electrons in each group (for the proof, click the bar bellow to expand the calculations). For more details See Spin statistics. let us imagine that it is possible to create the electron gas with an arbitrary distribution of spin directions. Below I will show that within a very short time the electrons of the electron gas are scattered and redistributed into one group of spinpolarized electrons (all spin are in one direction) and one group of spinunpolarized electrons (spins are equally distributed in all directions). from any given arbitrary distribution of spin directions in the electron gas, it is possible to calculate the final spin polarization of the electron gas. Let us assume that at some moment in time the angular distribution of spin directions is , where is the probability for a “spin” state to have an angle between their spin direction and the zaxis in an interval . After the scatterings have redistributed the states into one group spinpolarized electrons and one group spinunpolarized electrons, the probability of an electron is spinpolarized can be calculated as The probability that electron is in the group spinunpolarized electrons can be calculated as Bellowcalculations also prove that the numbers of electrons in the groups of spinpolarized and spinunpolarized electrons do not change due to spinindependent scatterings. To see the calculations, which were used to obtain Eqs. (60) ,(61) , click here to expand
Even though a spinindependent scattering may rotate the spin of an individual electron, spinindependent scatterings do not affect the total spin of all conduction electrons. This is because, all conduction electrons can be considered as a closed system. The spinindependent scatterings are events occurring inside this closed system and only a source from the outside can change its spin. The conservation of the total spin during spinindependent scatterings also means the conservation of the projection of the total spin on any axis. The projection of the total spin on some axis can be calculated by summing up the spin projections of each electron on this axis. It should be noticed that the spin is not a conventional 3D geometric vector. For summing of spins the quantum mechanical rules should be used instead of the rules for geometrical vectors.Note: here we are calculated only halffilled states, which are filed only by one electron. The fullfilled states, which are filled by two electrons of opposite spins, have zero spin and they have no influence on this calculations. The number of the fullfilled states can be calculated by the Spin statistics. If the spin direction of an electron has angle theta with respect to the zaxis, the probability that this state has spin direction along (spinup) and opposite (spindown) to the zaxis is (For proof see Eq. (p.7) here (click to expand "Spin projection")) For example, if is the probability of an electron to have an spin angle in the interval with respect to the zaxis, the probability that an electron has spin direction along (spinup) and opposite (spindown) to the zaxis is calculated as The difference in probabilities of an electron to have either spinup or spindown direction is calculated from Eq. (41) as or the same In the group of the spinunpolarized electrons, the spins are distributed equally in all directions. It means that probabilities to find electrons with opposite spins are equal for any direction. Or the same the sum of the projections of spin onto any axis over all states should not depend on the axis direction. Spinup, spinleft, and spindown and any other projection should be the same. For example, the spinup and spindown projections are the same: Substituting Eq.(51) into Eq. (50), we obtain where is the probability for a spinunpolarized electron to have an spin angle in the interval All spins of spinpolarized electrons are directed in the same direction. Therefore, the probability to find a spinpolarized electron with spin directed opposite to the spin direction of all spinpolarized electrons is zero: From Eq.(50) the probability to find spinpolarized electron with spinup direction is where is the probability to find a spinpolarized electron with an spin angle in the interval Since all spinpolarized electrons are spinup, Eq.(54) simply describe the probability for an electron to be spinpolarized. Main part If at some moment in time the electron gas has some arbitrary distribution of spin directions , after a short time electrons are scattered and redistributed into the group of spinpolarized electrons and the group of spinunpolarized electrons. The spinindependent scatterings do not change the total spin of the electron gas. Therefore, the spin projection Eq.(41) does change as well. It means that does not matter how many spinindependent scatterings occur, the value of the integral of Eq.(50) remains the same. But after a sufficient number of scatterings all electrons are redistributed into the groups of spinpolarized and spinunpolarized electrons. Therefore, the integral of Eq.(54) gives the probability for an electron to be spinpolarized. Summingup: Knowing the angular spin distribution prior to equilibrium, in equilibrium the probability that an electron is spinpolarized can be calculated from Eqs.(44) and (45) as
The probability that electron is spinunpolarized can be calculated from Eq. (47) as It should be noticed that Eq (47) is asymmetrical for time reversal (the integral changes its sign when time is reversed) and Eq. (48) is symmetrical for time reversal. Since the timeinverse symmetry is conserved after a spinindependent scattering, the numbers of electrons in the groups of spinpolarized and spinunpolarized electrons do not change due to spinindependent scatterings.
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 2 (pp.47) is about the rotation of the spin direction after a spin independent scattering. Chapter 3 (pp.79) is about the assemblies.An explanation can be found in Slides 3,4 and 5 of this Audio presentation or here

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