The energy distributions of all electrons (spin-polarized electrons+spin-unpolarized electrons) is described by the Fermi-Dirac distribution. The individual distribution of the spin-polarized and spin-unpolarized electrons is described by the spin statistics (see below). The spin-polarized and spin-unpolarized electrons have different energy distributions, because of their different scattering probabilities. The spin statistics is critically important for description of different spin-dependent effect in a solid. For example, the different distributions of the spin-polarized and spin-unpolarized electrons is a reason of a difference of conductivities of the spin-polarized and spin-unpolarized electrons.

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Content

click 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: Spin-unpolarized electron gas

(5).Spin distribution. Case 2: Spin-polarized electron gas

(6). spin pumping & spin relaxation vs mean-free path λ_mean

(7).Total number of spin-polarized and spin-unpolarized electrons

(8). Spin distribution within approximation of spin-up/spin-down bands

(8a).Non-degenerated semiconductor. Approximation of spin-up/spin-down bands

.........

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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.9-14).

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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 spin-polarized and spin-unpolarized 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 spin-polarized electrons are directed in the same direction, there is no direct scatterings between states occupying by spin-polarized electrons. In contrast, spins of spin-unpolarized electrons are directed in different directions. There is a probability of scatter ins between states, which are occupied by spin-unpolarized electrons. The difference in the scattering probabilities makes different the distributions of spin-polarized and spin-unpolarized 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.

 

Non-magnetic material

Calculation details is here SpinStatisticNonMagneticMetal.pdf

There are two groups of conduction electrons in a non-magnetic 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 spin-polarized electrons:

Probability F_TIA(E), that a conduction electron of energy E is spin-polarized, 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 spin-unpolarized electrons

Probability F_TIS(E), that a conduction electron of energy E is spin-unpolarized, 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 spin-inactive electrons

The spin-inactive electrons occupies the states, which are filled by two electrons of opposite spins

Probability F_full(E), that a conduction electron of energy E is spin-inactive, is calculated as

The energy distribution of all electrons (the Fermi-Dirac distribution)

A conduction electron can be in one of three groups: (1) group of spin-polarized electrons; (2) group of spin-unpolarized electrons; (3) group of spin-inactive 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 spin-polarized and spin-unpolarized electrons?

A. It is because the formulas were obtained by integrated over a sphere of possible spin-directions of the spin unpolarized electrons (See Eq.(17)). Therefore, the number is the product of π (See Eq.(17)).

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SpinStatistic.m -Matlab function, which calculates spin statistics (number of spin-polarized and spin-unpolarized electrons)

 

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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 spin-polarized and spin-unpolarized 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 spin-polarized electrons are directed in the same direction, there is no direct scatterings between states occupying by spin-polarized electrons. In contrast, spins of spin-unpolarized electrons are directed in different directions. There is a probability of scatter ins between states, which are occupied by spin-unpolarized electrons. The difference in the scattering probabilities makes different the distributions of spin-polarized and spin-unpolarized 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 spin-polarized and spin-unpolarized electrons causes the difference in electrical conductivity for spin-polarized and spin-unpolarized 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).

 

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How the distributions for spin-polarized and spin-unpolarized electrons are calculated???

Firstly, the electron scatterings probabilities between groups of spin-polarized and spin-unpolarized 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 spin-polarized electrons, and the number of states, which are occupied by one electron of group of spin-unpolarized electrons.

Secondly, the energy distributions are calculated from the condition that the sum of 3 probabilities (an electron is either in the group of spin-polarized electrons or in the group of spin-unpolarized electrons or in the group of spin-inactive electrons (which fill the full-filled states) is described by the Fermi-Dirac distribution.

 

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Why electron scattering depends whether the electron belong to the group of spin-polarized or spin-unpolarized electrons??

The electron scatterings are spin-dependent 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 spin-up spin direction can be scattered into a quantum state, which is already filled by a spin-down electron.

 

Scattering probabilities:

(case 1) Scattering probability=0

(a) A scatter of a spin-polarized electron into a state, which already filled by one spin-polarized electron

Spins of all electrons in the group of spin-polarized electrons are in the same direction. For example, all spins are up. Therefore, all unfilled empty place in these state are spin-down and only spin-down electrons can be scattered into these empty places. Since all electrons are spin-up, 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 spin-polarized electron into a state, which already filled by one spin-unpolarized electron
(b) A scatter of a spin-unpolarized electron into a state, which already filled by one spin-unpolarized electron
(c) A scatter of a spin-unpolarized electron into a state, which already filled by one spin-polarized 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 spin-unpolarized electrons the spin directions are distributed equally in all directions. Therefore, there is always non-zero 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
(b) A scatter of an electron from a full-filled state, which is occupied by two electrons of opposite spins, into a state, which is occupied by one electron.

Total spin of a full-filled 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 full-filled state into a half-filled state, always has spin opposite to the spin of the electron, which is already occupying the half-filled state.

 

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The energy distribution of spin-polarized and spin-unpolarized electrons (Spin Statistic) is important for understanding any spin-transport effect and phenomena.

The Spin Statistic is also important to understand properties of electron transport even in the case of spin-unpolarized electron gas. For example, it describes properties of electrons and holes in a metal and a semiconductor (See here for details)

 

 

 

 

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Main part

 

 

Distribution of spin-directions in the group of spin-polarized and spin-unpolarized electrons.

 

Challenge: There is a significant exchange of electrons between groups of spin-polarized and spin-unpolarized electrons, because of frequent spin-independent scatterings between electron states. All these scatterings should be taken into calculation!!!

A little help: The spin-independent scatterings do not change the number of electrons in each assembly. The total spin of all spin-unpolarized electrons remains zero. The spin-independent scatterings do not change the total spin of spin-polarized 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 Fermi-Dirac statistic

 

All conduction electrons consist of groups of spin-polarized, spin-unpolarized and spin-inactive electrons. Each group is described by its own distribution: (1) F_TIS is the distribution of spin-polarized electrons; (2) F_TIA is the distribution of spin-unpolarized electrons; (3) F_full is the distribution of full states. Each full state is occupied by two spin-inactive 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 (Fermi-Dirac distribution)= spin-polarized electrons + spin-unpolarized electrons +spin-inactive electrons

Electron gas in the absence of a spin accumulation consists of (1) spin-unpolarized electrons; (2) spin-inactive electrons. The condition 1 are described as

 

Electron gas in the spin-polarized electron gas consists of (1) spin-polarized electrons; (2) spin-unpolarized electrons; (3) spin-inactive electrons. The condition 1 are described as

Scatterings and Energy distributions

The 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 spin-up electron can be scattered into a "spin" state, which is already occupied by one electron, only if the spin-up place is available or the same if the state is occupied by a spin-down electron. As a result, the scattering probability of spin-up electron into spin-down "spin" state is the largest (= one). The scattering probability of spin-up electron into spin-up "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 spin-unpolarized 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

 

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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: Spin-unpolarized electron gas.

Spin Distribution in the absence of a spin accumulation

In this case there are no spin- polarized electrons. There are only spin-unpolarized and spin-inactive electrons. The distribution of electrons in each of these two group is calculated below. The energy distribution of all electrons (spin-unpolarized + spin-active) is described by the Fermi-Dirac distribution !!

In the case when the electron gas is not spin-polarized, 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 spin-unpolarized electron gas, the electrons occupying the "spin" states are called spin-unpolarized electrons. Electrons occupying the "full" states are called spin-inactive electrons. (See here)

The energy distribution of all electrons (independently of their spin direction) is described by the Fermi-Dirac distribution !!

 

Main idea of the calculation:Firstly, the scattering probabilities between spin-unpolarized electrons and spin-inactive 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 Fermi-Dirac 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 spin-unpolarized electrons; Probability F_full that a quantum state is occupied by two electron or probability, that a conduction electron belong to the group of spin-inactive electrons.

Why the scattering probabilities are different for spin-polarized electrons, spin-unpolarized electrons and spin-inactive 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 spin-up electron can be scattered only ether into a fully empty state or a state, which is occupied by one spin-down electron and in which spin-up 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 spin-polarized electrons are directed in one direction. Spins of spin-unpolarized electrons are distributed equally in all directions. The spin-inactive electrons have no defined spin direction. Therefore, their scatterings is not limited by the spin direction.

Main part:

Calculation of scatterings

if the electron gas is spin-unpolarized why conduction electrons should be divided into the groups?

A. If in the case when the electron gas is not spin-polarized, 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 scatterings

In 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 z-axis, the probability of scattering of "spin" states with angles φ₁ , φ₂ 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 spin-unpolarized 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 φ₁ , φ₂ as

Integration of Eq.(16) gives

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 spin-unpolarized electrons in spin-unpolarized electron gas can be calculated as

and distribution F_full of spin-inactive electrons in spin-unpolarized electron gas is calculated as

 

Figure 2 shows the distribution of "spin", "full" and "empty" states in a spin-unpolarized 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)

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The total number of “spin” states in the group of spin-unpolarized 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.

 

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Spin Distribution. Case 2: spin-polarized electron gas

 

 

 

 

 

Additional calculations (comparing to the above case of spin-unpolarized gas): The scatterings of spin-polarized electrons are taking into account; The rate equation for the conversion between spin-polarized and spin-unpolarized electrons are solved.

Main idea of the calculation:Firstly, the scattering probabilities between spin-polarized electrons, spin-unpolarized electrons and spin-inactive 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 Fermi-Dirac 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 spin-polarized electrons; Probability F_TIS that a quantum state is occupied by one electron or probability, that a conduction electron belong to the group of spin-unpolarized electrons; Probability F_full that a quantum state is occupied by two electron or probability, that a conduction electron belong to the group of spin-inactive electrons.

 

Are scatterings of spin-polarized and spin unpolarized electrons independent?

A. Not at all. It is even opposite. For example, the spin-polarized can be scattered into a quantum state of spin-polarized electrons. E.g. if the spins of all spin-polarized electrons are up, a spin-polarized electron can be scattered only into a quantum state where spin-up position is empty. However, in the case of spin-polarized electrons, the spin-up position is occupied and only spin-down position is empty. Therefore, there is no scatterings between spin-polarized electrons.

Do the spin- independent scatterings change the number of the spin-polarized and spin-unpolarized electrons?

A. No. The numbers of spin-polarized and spin-unpolarized electrons remain the same. The total spin of whole electron gas remains the same as well.

What is a spin-independent scattering? What is a difference between a spin-independent and spin-dependent 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 spin-dependent scattering is the interaction of the electron gas with an external spin particle. The spin-dependent scattering may change the total spin of the electron gas. The effect, which occur due to the spin-dependent scatterings, are the Anomalous Hall effect, the Spin Hall effect, the spin relaxation.

main part

Note:Since spin-independent scatterings intermix all electrons, including the electrons of spin-polarized, spin-unpolarized and spin-inactive electrons, the energy distribution of all groups of electrons is described by a single Fermi-Dirac distribution with the same Fermi energy.

rate equations:

More details about spin polarization and rate equations are here

Spin-pumping rate:

The spin pumping describes the conversion of electrons from the group of the spin-polarized electrons into the group of spin-unpolarized electrons. The conversion rate of the spin-pumping is described as (details are here)

where t_pump is the spin pumping time.

Spin-relaxation rate:

The spin damping describes the conversion of electrons from the group of the spin-polarized electrons into the group of spin-unpolarized electrons. The conversion rate of spin-relaxation 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 spin-polarized electrons and the distribution F_TIS of spin-unpolarized electrons as

The spin polarization sp of the electron gas is defined as a ratio of the number of spin-polarized electrons to the total number of the spin-polarized and spin-unpolarized 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 spin-polarized electrons and the distribution F_TIS of spin-unpolarized electrons as

Approximation (soft): It can be assume that sp(E) of Eq.(20e) is energy-independent 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 spin-polarized electrons, or group of spin-unpolarized electrons or group of spin-inactive electrons. This condition gives

Substituting the Fermi-Dirac 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 spin-polarized F_TIA and spin-unpolarized electrons F_TIS as

where sp is the spin polarization of the electron gas

 

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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 zero-order 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 spin-pumping on the electron energy E:

Details on different mechanisms of the spin pumping is here

spin-pumping mechanism (1): sp-d exchange interaction. Dependency on electron energy E: moderate.

The strength of the sp-d exchange interaction depends on the overlap of wave functions of a conduction electron and a localized d-electron. When λ_mean becomes larger, the overlap becomes smaller.

spin-pumping mechanism (2): sp-d scatterings; Dependency on electron energy E:moderate.

The probability of a scattering between of a localized d-electron into a state of conduction electron depends on the overlap of wave functions of a conduction electron and a localized d-electron. When λ_mean becomes larger, the overlap becomes smaller.

spin-pumping 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.

spin-pumping mechanism (4): spin-dependent scatterings & spin accumulation due to Spin Hall effect; Dependency on electron energy E:moderate.

A spin-dependent 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 spin-relaxation on the electron energy E:

Details on different mechanisms of the spin relaxation is here

spin-relaxation mechanism (1): spin-dependent scatterings; Dependency on electron energy E:moderate.

A spin-dependent 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.

spin-relaxation 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.

spin-relaxation 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.

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Total number of spin-polarized and spin-unpolarized electrons

(number of all electrons excluding the spin-inactive 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 spin-polarized n_TIA and spin-unpolarized n_TIS electrons as

where D(E) is the density of states, F_TIA , F_TIS are energy distributions of spin polarized and spin-unpolarized 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 spin-polarized electrons and the number n_TIS of spin-unpolarized electrons and the their total number (all conduction electrons except spin-inactive 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 spin-polarized and spin-unpolarized electrons are important.

A. It is because only the spin-polarized and spin-unpolarized electrons determine the property of charge and spin transport (See here and here). The contribution of spin-inactive 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.

 

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end of main part

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Since the approximation of spin-up/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 spin-up/spin-down bands is

good for a non-degenerated semiconductor

bad for a metal

 

Spin distribution within approximation of spin-up/spin-down bands

Classic model of spin-up/spin-down bands

The model of spin-up/spin-down bands ignores that the spin rotates during the scattering events N4, N5. It assumes that spin-up and spin-down electrons do not interact (or there is only a very weakly interaction between them). Therefore, the classical model assumes that spin-up and spin-down electrons are staying inside their own bands for a relatively long time.

Since there is no interactions between the bands, the energy distributions of spin-up and spin-down electrons are independent. The distribution of electrons in each band is assumed to be described by its own Fermi-Dirac distribution and the Fermi energy for each band can be different. Thus, the total amount of electrons may not be described by the Fermi-Dirac distribution.

For example, in the case of a spin accumulation, one band (in the case of Fig. 1(a), it is the spin-up band) is filled by a greater number of the electrons. The Fermi energy for the spin-up band is larger than the Fermi energy for the spin-down band and the distribution of all electrons is not the Fermi-Dirac distribution.

 

Correct description

All conduction electrons can be divided into the groups of spin-polarized and spin-unpolarized electrons (See Fig. 1 (b)). There is a frequent exchange of electrons between the groups, because of frequent spin-independent scatterings (scattering events N4 and N5). However, the spin-independent scatterings do not change the number of electrons in each group.

Because of the significant exchange of electrons between the groups of spin-polarized and spin-unpolarized electrons, the Fermi energy should be the same for both spin-polarized and spin-unpolarized electrons

 

 

The electron distribution in the model of spin-up/spin-down bands.

In the model spin-up/spin-down bands assumes that there are two independent distributions for spin-up and spin-down electrons.

Note: in the classical model, the spin-up and spin-down directions are not specified. Usually, they are chosen from the geometrical considerations (direction of a magnetic field, the magnetization direction)

 

Both the distribution of spin-up electrons and the distribution of spin-down electrons are described by the independent Fermi-Dirac distribution (See Fig. 1a):

where are the Fermi energies for the spin-up and spin-down electrons, respectively.

 

 

 

 

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it is the case when energy distribution calculated from the approximation of the spin-up/spin-down bands is correct

Even in this case, some spin properties of electron gas are described correctly by the approximation of the spin-up/spin-down bands

 

Non-degenerated semiconductor. Approximation of spin-up/spin-down bands

A semiconductor, in which the Fermi energy is in its band gap and does not cross either the conduction or valence band, is called a non-degenerated semiconductor.

In the case of a non-degenerated semiconductor, the electron energy distribution is described by the Boltzmann statistic.

In the case of a non-degenerated semiconductor, the approximation of spin-up/spin-down bands can be used for the description of the spin transport. The use the approximation of the spin-up/down band for the spin transport is described here

In the case when electron energy is sufficiently greater than the Fermi energy

the energy distribution of the spin-polarized and spin-unpolarized electrons (Eqs. (24),(25)) can be simplified as

The analysis of the spin transport using the model of spin-up/spin-down bands may be simpler and sometimes more understandable.

Eqs. (27) can be rewritten as

where the effective chemical potentials for spin-polarized and spin-unpolarized electrons are defined as

In the case of the model of spin-up/spin-down bands, the distribution of electrons in the spin-up and spin-down bands of a non-degenerated semiconductor are described as

The similarities of the distributions (53) and (55) implies that the the spin transport in a non-degenerated semiconductor can be approximated by the classical model of spin-up/spin-down bands by utilizing the individual chemical potential Eqs. (54) for spin-polarized and spin-unpolarized 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 spin-polarized 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 spin-polarized and spin-unpolarized electrons are linearly proportional to number of electrons in each group of spin-polarized and spin-unpolarized electrons. Therefore, generally they are different.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Possible confusion!!: from 2014 to 2017 I have used names TIA and TIS for groups of spin-polarized and spin-unpolarized 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.9-14).

An explanation can be found in Slides 7 and 8 of this Audio presentation or here