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Spin Torque

Magnetism of electron gas


The spin torque occurs when the electron gas is already spin-polarized and a small amount of spin-polarized electrons of different spin direction are injected. As result, the spin direction of all spin-polarized electrons rotates toward the spin direction of the injected electrons.

The spin torque is always accompanied by an additional spin relaxation.



Can two groups of spin-polarized electrons with different spin directions coexist together. For example, is it possible that the spin direction of one group of spin-polarized electrons is along the x-axis and another group of spin-polarized electrons is along the y-axis?

interaction of two groups of spin-polarized electrons

same spin direction

perpendicular spin directions opposite spin directions
   
the spin direction does not change. The spin-unpolarized electrons are not created.    
number of spin-polarized electron in both groups is the same.
click on image to enlarge

A. No. At any point of space only one group of spin-polarized electrons and one group of spin-unpolarized electrons can coexist. At one point of space, the spins of all spin-polarized electrons are in one direction. At different points of space, the spin-polarized electrons may have different spin directions (See here and here).

What happens when there are two spin-pumping sources and the first source creates spin-polarized electrons with spin direction along x-axis and the second source creates spin-polarized electrons with spin direction along y-axis?

A. The conduction electrons will be spin-polarized. The spin direction of the spin-polarized electrons will be in the xy-plane.


 

This page describes the interactions of two (or more) groups of spin-polarized electrons of different spin directions


The same content can be found in V. Zayets JMMM 356 (2014)52–67 (click here to download pdf) or (http://arxiv.org/abs/1304.2150 or this site) . Chapter 8 (pp.22-25).

Result in short

When the number of spin-polarized electrons in electron gas there is nTIA1 and spin direction of spin-polarized electrons is along normal vector , and spin-polarized electrons of different spin direction are injected with the injection rate of , the spin-polarized electrons of the electron gas experiences the spin torque, which is calculated as

where A(φ) is the spin torque coefficient, which depends on the angle φ between the spin directions of injected and existed electrons. (See calculations of A(φ) below)

The spin torque causes an additional spin relaxation or conversion from group of spin-polarized electrons into group of spin-unpolarized electrons at the rate

where nTIS is the number of spin-unpolarized electrons, R(φ) is the spin-relaxation coefficient, which depends on the angle φ between the spin directions of injected and existed electrons. (See calculations of R(φ) below)

 

Properties of spin torque:

- Spin rotation of the spin-polarized electrons is directed towards the spin direction of the injected electrons

- The spin torque is largest in the case when the angle between the spin directions of existing and injected spin-polarized electrons is 67 degrees

- There is no spin torque in the cases when the spin directions of existing and injected spin-polarized electrons are parallel or anti parallel in respect to each other.

- The spin torque is linearly proportional to the injection rate of spin-polarized electrons

- Spin torque is always accompanied by spin relaxation (a conversion from the group of spin-polarized electrons into the group of spin-unpolarized electrons). The conversion rate is greater for a larger angle between the spin directions of existing and injected spin-polarized electrons.

- Injection of spin-polarized electrons increases the total number of spin-polarized electrons in the electron gas only when the angle between the spin directions of existing and injected spin-polarized electrons is smaller than 67 degrees. When the angle is larger than 67 degrees, the number of spin-polarized electrons decreases, because the spin relaxation, which is induced by the spin torque, becomes larger than the supplying rate of new spin-polarized electrons.


Cases when there are two or more spin pumping sources of different spin directions

case 1. magnetic field applied to a ferromagnetic metal

spin-pumping source (1): magnetic field (see here). Spin Direction: along magnetic filed

spin-pumping source (2): sp-d exchange interaction and sp-d scattering (see here and here); Spin Direction: along magnetization

case 2. a ferromagnetic metal is illuminated by circular-polarized light

spin-pumping source (1): spin of a photon (see here).Spin Direction: along light propagation direction

spin-pumping source (2): sp-d exchange interaction and sp-d scattering (see here and here); Spin Direction: along magnetization

case 3. Spin-orbit torque (SOT) in a ferromagnetic metal (see here)

spin-pumping source (1): spin Hall effect (see here) due to scatterings of spin-unpolarized electrons (see here). Spin Direction: along current

spin-pumping source (2): spin Hall effect (see here) due to scatterings of spin-polarized electrons (see here). Spin Direction: perpendicularly to the magnetization

spin-pumping source (3): sp-d exchange interaction and sp-d scattering (see here and here); Spin Direction: along magnetization

case 4. spin-torque current in a ferromagnetic metal (see here)

it is the case when the spin direction of spin-polarized electrons is different at different points of the sample. It is the case ,for example, magnetization direction is different in different magnetic domains.

spin-pumping source (1): spin diffusion from a neighbor region

spin-pumping source (2): sp-d exchange interaction and sp-d scattering (see here and here);

case 5. antiferromagnetic , compensated ferromagnetic, ferrimagnetic metals (see here)

it is the cases of antiferromagnetic , compensated ferromagnetic, ferrimagnetic metals, the magnetization of localized electrons are different from a atom to a atom. For example in FeTb the coupling between Fe atoms and Tb atoms is antiferromagnetic. Therefore if magnetization of Fe atom is up, the magnetization of Tb atom is down.

spin-pumping source (1): sp-d exchange interaction (scatterings) with spin-up localized electrons Direction: along magnetization

spin-pumping source (2): sp-d exchange interaction (scatterings) with spin-down localized electrons; Direction: opposite to magnetization


The spin torque occurs as a result of the interaction of two groups of spin-polarized electrons.

The features of this interaction define the properties of the spin torque.

The interaction of two groups of spin-polarized electrons with different spin directions.

It is possible that at some moment in time in a metal there are two or more groups of spin-polarized electrons. However, within a very short time all spin-polarized electrons of two groups combine into one group of spin-polarized electrons and some electrons are converted into the group of the spin-unpolarized electron (additional spin relaxation).

How long does it take for two groups of spin-polarized electrons with different spin directions to combine into one group with the same spin direction?

A. Between very quickly and almost immediately.

The time of interaction of two groups of spin-polarized electrons significantly depends on the relative number of electrons in each group. (almost immediately): In the case of the interaction of two groups with the same number of spin-polarized electrons, only (!!!) one scattering event 5 between each electron of two groups is sufficient to combine spin-polarized electron into one group with the same direction of spin. (very quickly): In the case of the interaction of two groups with a different number of spin-polarized electrons, one spin-polarized electron should experience several scattering event 5 until the two groups fully combine into one group with the same direction of spin. (See calculation below)

 


Spin torque. Case 1. Spin-polarized electron gas in a magnetic field

. Magnetic field is applied at some angle with respect to the spin direction of spin accumulated electrons.

Precession of spins of the local d-electrons (red arrow) and spin-polarized conduction electrons (blue arrows)

Note: in an equilibrium there is no any precession of magnetic moment (See text)

 

Case: External magnetic field applied to a ferromagnetic metal at an angle to its magnetization direction. There are two spin-pumping sources to create the spin-polarized conduction electrons: (1) magnetization (localized d-electrons); (2) external magnetic field

The spin torque affects the spin-polarized conduction electrons both in equilibrium and during the precession damping.

spin torque in equilibrium: (a) magnitude of the spin polarization slightly changes (b) spin direction of spin-polarized electron slightly changes

spin torque during precession damping: Pressing damping of spin-polarized conduction electrons increases during the precession damping

Why it is important to understand and to calculate the spin-polarization of conduction electron in the case when a magnetic field is applied at an angle with respect to the magnetization direction?

It is important for measurement of anisotropy field and the strength of perpendicular magnetic anisotropy (PMA) (See here)

Does the magnetization always align along direction of magnetic field?

A. Not always. When an external magnetic field is applied, the magnetization aligns itself very quickly (within 10-100 ns) along an effective magnetic field due to the precession damping. However, the direction of the effective magnetic field may be different from the direction of external and internal magnetic fields due to influence of the spin-orbit interaction and the exchange interaction:

(influence 1) spin-orbit interaction (SO)

Due to some relativistic effect (see here), an electron experience an additional magnetic field. The magnitude and direction of this magnetic field depends on orbital deformation. As a result, the conduction electrons and localized d-electrons may experience the SO magnetic field of different directions and magnitude

(influence 2) exchange interaction

Due to the quantum nature of electron, there is spin-dependent field between two or more electrons. Since the exchange interaction depends on the electron spin, it may align the electron spin along a direction different from the direction of the applied magnetic field. Note, in contrast to the SO interaction, the change interaction does not induce any magnetic field.

Origin of the spin torque in a magnetic field

A magnetic field converts the electrons from the group of spin-unpolarized to the group of spin-polarized electrons. This effect is called the spin-pumping (See here and here). The spin direction of the converted electrons is along the magnetic field. In the case when the spin direction of the existed spin-polarized electrons in electron gas is different than the spin direction of the converted electrons, the existed spin-polarized electrons experience a spin torque, which rotates their spin towards the direction of the magnetic field.

Example: a magnetic field is applied in-plane to a ferromagnetic metal with a perpendicular magnetic anisotropy (PMA). This configuration is used to measure the anisotropy field (See here)

In the this case, in-polarization linearly depends on the magnetic field. The linear dependence makes easier to measure the strength of the PMA with a high precision (See here). The influence of the spin torque on this measurement should be taking into calculations.

Fact: Decrease or increase of the spin polarization in an external magnetic field, depending on angle between the magnetic field and spin of existed spin-polarized electrons.

A magnetic field constantly creates the spin-polarized electrons (spin pumping). The spin direction of these created spin-polarized electrons is along the magnetic field. When direction of the magnetic field is along the spin direction of existed spin-polarized electrons, the spin polarization of the electron gas in the magnetic field increases (experiment is here). When direction of the magnetic field is opposite to the spin direction of existed spin-polarized electrons, the spin polarization of the electron gas in the magnetic field decreases. For other angles of the magnetic field, the spin polarization either increases or decreases.

67 degrees: the margin angle between applied magnetic field and spin direction of existed spin-polarized conduction electron

smaller than 670 Spin polarization increases

smaller than 670 Spin polarization decreases

The spin torque is always accompanied by an additional spin relaxation. When existed spin-polarized electrons combine with spin-polarized electron of a different spin angle created by the magnetic field, the total number of spin-polarized electrons decreases. This decrease can be large so resulting number spin-polarized electrons becomes smaller than prior-existed number and the spin polarization decrease in the magnetic field. Or the decrease can be small so resulting number spin-polarized electrons becomes larger than prior-existed number due to newly created spin-polarized electrons and the spin polarization increase in the magnetic field. The margin angle between these two cases is 670(see calculations below)

 

Influence of spin torque on precession damping of spin-polarized electrons

In a magnetic field there is a precession of spins of spin-polarized electron around the magnetic field. Also, the spin direction of the spin-polarized electrons slowly aligns itself along the direction of the magnetic field. The alignment of spin direction of spin-polarized electrons along the magnetic field is joint work of precession damping and the spin torque.

Why the spin torque affects the precession damping of spin-polarized conduction electrons?

In a magnetic field the spin-polarized electrons are constantly created (spin pumping). The spin direction of these created is along the magnetic field. The spin of existed spin-polarized conduction electrons precess along the magnetic field. Both groups of the spin-polarized electrons combine into one group, which spin direction becomes closer and closer to the direction of the magnetic field.

Can the spin precession exist in an equilibrium (for conduction electrons or localized d-electrons)?

A. No. Any spin precession means that the magnetization changes its direction in time. According to the Maxwell's equations, any change of magnetization generates an electromagnetic wave (See here). The emission of photons removes both the energy and spin from the electron gas. Therefore, the electron gas is not in an equilibrium. The spin precession may exist in an equilibrium only if there is an additional source of the energy and the spin, which compensates the energy and spin loss due to the magnetization precession. Note, the emission of photons is one of origins of precession damping (See here).

 

 


Spin torque. Case 2. Spin-polarized electron gas is illuminated by light

. Illumination by circularly polarized light, which is spin-polarized. The transfer of the photon spin to an electron gas

Fig.2 Animated picture. Changing the magnetization direction of a ferromagnetic metal by circularly polarized light. Without light illumination the magnetization of ferromagnetic film is in the plane (yellow arrows). Circularly polarized light creates the spin-polarized electrons with spin direction along light incident direction (normal to the film) The green balls with arrows show the light-created spin-polarized electrons. The spin direction of existed spin-polarized electrons is along magnetization (yellow arrows). The interaction of existed and light-created spin-polarized electrons induces a spin torque, which rotates the spin direction of existed spin-polarized conduction electrons away from the magnetization direction. Because of the exchange interaction between conduction and d- electrons, the magnetization rotates following the rotation of the spin direction of the conduction electrons. After the illumination has stopped, the magnetization returns to in-plane.

When absorbed, a circularly- polarized photon may convert a conduction electron from the group of spin-unpolarized electrons or from group of spin-inactive electrons to the group of spin-polarized electrons (See here about spin distributions in different groups of electrons). The spin of circularly- polarized photon is one (See here) and it transformed to the electron gas when a photon is absorbed by a conduction electron. The spin direction of the optically-created spin-polarized electrons is along the incident direction of light.

In the case when the electron gas is already spin-polarized (e.g. in a ferromagnetic metal), the spin-direction of photo-excited spin-polarized electrons may not coincide with the spin direction of already-existed spin-polarized electrons. In this case, light induces the spin torque, which turns the spin direction of the existed spin-polarized electrons toward the incident direction of light.

Circularly- polarized light may change the magnetization direction in a ferromagnetic metal. For example, when circular polarized light incidents normally on a thin film of a soft ferromagnetic metal (See Fig.2), which magnetization is in plane, it induces a spin torque. The spin torque rotates the spin direction of the spin-polarized conduction electrons from the in-plane direction to the normal-to-plane direction. Since the spin direction of the conduction electrons is rotated away from the spin direction of the localized d-electrons, the d-electrons experience a torque due to the sp-d exchange interaction and their spins are rotated following the spin rotation of the conduction electrons.

Note: The interaction of light with electron spin is very complex. It depends on the electron spin, orbital moment, orbital quenching, etc. See here for details

All-Optical Magnetization Reversal

There are several possible physical origins of the all-optical magnetization reversal. The mechanism of reversal depends on a material where it occurs:

(case 1 of All-Optical Magnetization Reversal): a ferromagnetic metal

Origin: spin torque

Light interacts with delocalized conduction electrons. Light transfer its spin and creates the spin-polarized conduction electrons. These electrons creates the spin torque and the spin direction of the spin-polarized electrons turns away from its equilibrium direction. Due to the sp-d exchange interaction and scattering the spin direction of the localized d-electrons follows the turn of the spin of conduction electrons

Two steps of spin rotation by light in a ferromagnetic metal:

(step 1) Turn of spin direction of conduction electrons

Circularly-polarized light excites spin-polarized electrons in the electron gas. Spin direction of the light-excited spin-polarized electrons is along light propagation direction and this spin direction is not necessarily the same as the spin direction of already existed spin-polarized electrons. Since the spin-polarized electrons of two different spin directions cannot coexist at the same place, all spin-polarized electrons quickly (nearly immediately) combine into one group of the spin-polarized electrons. The spin direction of this combined group of the spin-polarized electrons is between the spin directions of existed and light-excited spin-polarized electrons. As a result, the spin polarization of the spin-polarized turns away from its equilibrium direction toward the spin-direction of photo-excited electrons.

(step 2) Turn of spin direction of localized d-electrons

The sp-d exchange interaction and scatterings between delocalized conduction electrons and localized d-electrons aligns spins of both types electrons in one same direction. When spin of conduction electrons turn out from the its equilibrium direction and the direction the localized d- electrons, the spin of localized d-electrons follows the rotation and turns to be along the spin of the conduction electrons.

With which electrons light interacts more effectively? With conduction electrons or with localized d-electrons?

A. Light interacts more effectively with conduction electrons, because their larger size.

Interaction between two particles are most effective when they have the same size. The size of a photon is about 0.1 mm -1 mm (from a light bulb) and 1 mm -1 m (from a laser). The size an electron is substantially smaller. The length of a conduction electrons is called the mean-free path λ mean. The size of a conduction electrons ( λ mean) is about 30-200 nm in a semiconductor and 0.5~20 nm in a metal. The size of a localized d-electron is about the size of atomic orbital ~0.1 nm. As a result, the interaction of a photon with a conduction electron is more effective than with a localized electron.

(case 2 of All-Optical Magnetization Reversal): an oxide and transparent dielectric (for example, YIG)

Origin: excitement of bonding electrons. As a result-> distortion of the bonding and spin alignment

Light excites the electrons, which are responsible for the super-exchange interaction or the exchange interaction, to a higher energy level. The properties (spin, orbit and so forth) of excited electrons are different from the properties they had at unexcited state. It breaks the equilibrium spin arrangement.

mechanism: Light excites an electron responsible for the super-exchange (exchange) interaction on a higher energy level. This locally reduces or turns off the super-exchange (exchange) interaction between them and the magnetization start to precess or it may even be reversed.

Note: in this case the light-induced magnetization reversal mechanism is not related to the spin torque

Calculations

What is calculated below?

The case is calculated when there are two groups of spin-polarized electrons with different spin directions. It is calculated how these two groups are mixed up by scatterings to combine into a single group of the spin-polarized electrons of a single spin direction.

How are calculations done?

(calculation 1) The final number of spin-polarized electrons and their direction is calculated from the conservation law of the time-reversal symmetry and the total spin of the electron gas.

The spin is a quantum-mechanical object. The spins cannot be simply integrated or sum up. The quantum-mechanical rules have to be used. It is very difficult to do in the case of many electrons with different directions of spins. Additional difficulties is that the spin direction of each electron may be changed after a scattering (See here). The conservation law of the time-reversal symmetry greatly helps such calculations. In an equilibrium, there is only one group of spin-polarized electrons of the same spin direction. Additionally there are spin-unpolarized and spin-inactive electrons (See here). In the case when the electron gas does interact with any external "spin" objects (close system), the electron scatterings do not change the total spin of the electron gas and degree of its time- inverse symmetry. Such scatterings are called spin-independent scatterings(See here). Since the time-inverse symmetry does not change during mixing of two groups of spin-polarized electrons by scatterings, the final spin direction and spin polarization is evaluated from comparison of initial degree of time- inverse symmetry of two groups of spin-polarized electrons with final degree time- inverse symmetry of one group of spin-polarized electrons.

Note: The calculation of the degree of time- inverse symmetry is very similar to the calculation of the total spin of the electron gas, but simpler (see here).

(calculation 2) Tracing of the evaluation of spin distribution in time

In this case, the change of spin distribution is calculated after each scattering event. The calculation traces step by step evolution of spin distribution from initial state, which is the sum of two distributions with different spin angles, to the final distribution of a single spin angle. The spin properties of all possible scatterings between spin-polarized, spin

 

Calculation Results

(simple case 1): Two groups of spin-polarized electrons with the same number of electrons

resulting spin angle: exactly between initial two angles

+=

resulting number of spin-polarized and spin-unpolarized electrons:

where nTIA1 is the number of spin-polarized electrons in each initial group of before interaction and phi is the initial angle between spin directions of two groups of spin-polarized electrons.

(for proofs click here)

 

+=200%

All initial spin-polarized electrons remain spin-polarized. None of them become spin-unpolarized

+=0 %

All initial spin-polarized electrons become spin-unpolarized. None of them remain spin-polarized

+=100 %

A half of initial spin-polarized electrons remain spin-polarized. Another half become spin-unpolarized.

(simple case 2): Number of electrons in one group spin-polarized electrons much smaller than in other group

resulting spin angle: spin angle of the larger group slightly rotates

(general case): Two groups of spin-polarized electrons with different numbers of electrons

In the case of the interaction of two groups of different numbers of spin-polarized electrons, it takes several scattering events 5 until they relax into one group of one spin direction. For example, let us consider the groups named TIA1 and TIA2 when the number of spin-polarized electrons in group TIA1 is greater than the number of spin-polarized electrons in group TIA2.

The spin direction of TIA1 is along the z-axis and the spin direction of TIA2 has angle φ with respect to the z-axis.

After the first scattering event 5 the spin direction of TIA2 group will have the angle φ/2 with respect to the z-axis, the spin direction of TIA1 will not change. The number of spin-polarized electrons nTIA1 , nTIA2 in TIA1 and TIA2 groups and number of spin-unpolarized electrons nTIS after the first scatterings are calculated as (See scattering event 5):

After the second scattering event 5, the angle between the spin directions of two groups of spin-polarized electrons will be φ/4 and the numbers of spin-polarized and spin-unpolarized electrons can be calculated as

where the number of spin-polarized electrons in the group TIA2 is assumed to be larger than in group TIA1: nTIA2> nTIA1

Therefore, after each scattering event 5, the spin angle between the two groups of spin-polarized electrons is reduced by a factor of 2. The spin direction of the group, in which there were fewer spin-polarized electrons, rotates and the number of electrons in this group increases. The spin direction of the group, in which there were more spin-polarized electrons, does not rotate and the number of electrons in this assembly decreases.

How long time does it take to convert two spin-polarized electrons into one equilibrium group?

The most of the spin-polarized electrons make one group of the same spin direction very quickly (See the numerical simulation below).

Do the spin and energy electron distributions changed during interactions of two groups of spin-polarized electrons?

 

It should be noted that the probability of scattering event 5 is not constant, but it is proportional to the number of spin-polarized electrons in the TIA1 and TIA2 groups. The probability pscat5 of one scattering event 5 in the electron gas can be calculated as (See scattering event 5 )

where nTIA1 , nTIA2 are numbers of spin-polarized electrons in TIA1 and TIA2 groups, nTIS is the number of spin-unpolarized electrons and pscat5,0is the probability of a single scattering event 5.

You calculate the time evaluation in unit of number of scattering event 5. How does it relate to the real time?

The most of the spin

 

 

Why TIS and TIA abbreviations are used for the spin unpolarized and spin-polarized electrons?
TIS means "time-inverse symmetrical". TIA means "time-inverse asymmetrical". TIA1 and TIA2 describe two group of spin-polarized electrons with different directions of spin..
In fact, the group of the spin-unpolarized contains the electrons with a defined direction of spin, but their spins are distributed equally in all directions. There are third group of electrons, which is called spin-inactive or deep-level electrons. These electrons do not a defined direction of spin. (See details here). The distinguish property of the spin-unpolarized electrons is that it is time-inverse symmetrical. The group of spin-polarized electrons is time-inverse asymmetrical. This important property is used to distinguish in the calculation between the spin-polarized and spin-unpolarized electrons.

Fig 1. The interaction of two groups of spin-polarized electrons. (left): Animated polar figure. The radius of arrows corresponds to the probability of electrons to be in one of groups of the spin polarized electrons and the angle indicates the spin direction of each group. The animated parameter is the number of scattering events 5. (right): Occupation probabilities of TIA1 (black line) and TIA2 (red line) groups of spin-polarized electrons as a function of number of the number of scattering events 5. Green line shows the probability for the total number of spin-polarized electrons (TIA=TIA1+TIA2). The yellow line show the probability for the spin-unpolarized electrons (TIS). (Inset) Inset shows spin directions of electrons of TIA1 and TIA2 groups.

Initial conditions: Occupation probability is 0.8 for TIA1 and 0.2 for TIA2. It means that 80 % of spin-polarized electrons are in group TIA1 and their spin angle is 0 deg.. 20 % of spin-polarized electrons are in group TIA2 and their spin angle is 145 deg.. There is no spin-unpolarized electrons.

 

 

Numerical simulation of interaction of two groups of spin-polarized electrons having different spin angles

Initial conditions: Figure 1 shows the calculated interaction of two groups TIA1 and TIA2 of spin-polarized electron when initially all electrons are spin-polarized (sp=100 %), 80 % of the spin-polarized electrons belongs to group TIA1 (80 % of electrons have the same spin direction), 20 % of the spin-polarized electrons belongs to group TIA1 (20 % of electrons have a different spin direction from electrons of TIA1 group), and there are no spin-unpolarized electrons. The initial angle between the spin directions of two groups is 145 deg. Figure 2 shows a similar case of the interaction of the TIA1 and TIA2 groups of the spin-polarized electrons . However, initially 98 % of the spin-polarized electrons are in the TIA1 group and only 2 % of the spin-polarized electrons are in the TIA2 group. Similarly, the initial angle between the spin directions of two groups is 145 deg.

Number of spin-polarized electrons in each group: After each scattering event 5 the angle between the spin directions of two groups of spin-polarized electrons decreases and some electrons become spin-unpolarized. During the first 3-4 scattering events 5 , a significant number of spin-polarized electrons becomes spin-unpolarized (there is a significant spin relaxation). Additionally, the number of spin-polarized electrons in the TIA1 group decreases and the number of spin-polarized electrons in the TIA2 group increases until the numbers of spin-polarized electrons of both spin angles become nearly equal. After that, the numbers oscillate approaching the same number.

Spin direction: During a few first scatterings, the spin direction of the spin-polarized electrons of TIA2 group (a smaller number of electrons) changes substantially. In contrast, the change of spin direction of the spin-polarized electrons of TIA1 group (a larger number of electrons) is very small (nearly no change). The change becomes larger and substantial after 5-7 scatterings, when the numbers of spin-polarized electrons in both groups of different spin angles become comparable.

Fig 2. The interaction of two group of spin-polarized electrons of different spin angles. This figure is similar to Fig 1, but the initial conditions are different: There are 10 times smaller electrons in group TIA2

Initial conditions: 98 % of spin-polarized electrons are in group TIA1 and their spin angle is 0 deg.. 2 % of spin-polarized electrons are in group TIA2 and their spin angle is 145 deg.. There is no spin-unpolarized electrons.

Interaction time: After 10-15 scatterings, the angle between the spin directions of spin- polarized electrons of two groups becomes very small and the two groups may be considered as one group of spin-polarized electrons of one spin direction..

 

 

 

 

 

 

 

 

 

 

Fig P11. The interaction of two group of spin-polarized electrons of different spin angles. This figure is similar to Fig 1, but the initial conditions are different: The angle between spins of two groups of spin-polarized electrons is 179 deg.. Theretofore, spin directions is nearly opposite.

Initial conditions: 80 % of spin-polarized electrons are in group TIA1 and their spin angle is 0 deg.. 20 % of spin-polarized electrons are in group TIA2 and their spin angle is 175 deg.. There is no spin-unpolarized electrons.

When the initial spin directions of two group of spin polarized electrons are opposite, the resulting remaining number of spin-polarized electrons is difference between spin-polarized electrons of two groups. The resulting spin-angle is the along the spin direction of the group of the larger amount of electrons.

Figure P11 shows the case when the angle between spin directions of two is 179 degrees. It is near opposite.

In this case, the spin-polarized electrons of group TIA2 become spin-unpolarized very quickly. Even the most of the electrons of the TIA2 group are converted into spin-unpolarized already after 1st scattering, a tiny amount of electrons still remains in TIA2 group. Afterwards, after each scattering the electron amount in this group sharply increases. After 12 scatterings, the amount of electrons in the TIA2 group becomes comparable with the amount of electrons in the TIA1 group.

 

 

 

 

 

 

 

Dependence on a relative amounts of electrons in each group

Fig.3 The final amounts of spin-polarized and spin-unpolarized electrons after interaction of two groups of spin-polarized electrons of different spin angles as a function of a relative amounts of electrons in group TIA2. (left) final spin angle of spin-polarized electron with respect to the initial spin angle of spin-polarized electrons of TIA1 group. (right) final amounts of spin-polarized and spin-unpolarized electrons.

Initial conditions: The angle between spins of electrons of TIA1 and TIA2 groups is 145 deg. . There is no spin-unpolarized electrons. Click on image to enlarge it

 

 

Figure 3 shows the rotation angle and the amount of electrons converted from both groups of spin-polarized electrons to the group of spin-unpolarized electrons as a function of related amounts of spin polarized electrons in each group. The probability of 0.5 means that there is an equal amount of electrons in each group.

To note: The spin rotation angle and the amount of spin-polarized electrons, which become spin-unpolarized (spin relaxation), are linearly proportional to the amount of spin-polarized electrons in group, which contains a smaller amount of electrons.

It is the case, when the number of spin-polarized electrons of one group is significantly smaller than amount in another group (less than 10 %),

 

 

Figure 4 shows the rotation angle and the amount of electrons converted from both groups of spin-polarized electrons to the group of spin-unpolarized electrons as a function of the angle between spin directions of electrons of two groups.

no spin rotation: There is no spin rotation in the cases when the spin direction of spin-polarized electrons of group TIA1 and TIA2 are parallel or antiparallel of spin direction of electrons of the group TIA2.

Dependence on angle between spins of two groups
Fig.4 The interaction of two groups of spin polarized electrons as a function of angle φ between the spin directions of injected (TIA2) and existed (TIA1) spin-polarized electrons .(left) final spin angle of spin-polarized electron with respect to the initial spin angle of spin-polarized electrons of TIA1 group. (right) final amounts of spin-polarized and spin-unpolarized electrons.
Initial conditions: The 99 % of spin-polarized electrons were in the group TIA1 and 1 % of spin-polarized electrons were in group TIA2. There is no spin-unpolarized electrons. Click on image to enlarge it

The largest spin rotation: The rotation is largest in the case when the angle between the spin directions is around 67 degrees.

Does the final amount of spin-polarized electrons increases or decreases after interaction of two groups of spin-polarized electrons?

It always decreases comparing to the total amount of initial spin-polarized electrons (TIA1+TIA2).

The final amount of spin-polarized electrons may either decrease or increase comparing to the initial amount in the larger group (TIA1). It depends on the spin directions in two groups.

The number of spin-polarized electrons, which become spin-polarized, increases when the angle between the spin directions of electrons of two groups increases. When the angle is 120 degrees and larger, the number of converted spin-unpolarized electrons is about 2%. This is twice the initial amount of spin- polarized electrons in the group TIA2. For angles smaller than 67 degrees, the amount of converted spin-unpolarized electrons is smaller than the initial amount of spin-polarized electrons in the group TIA2.

For angles smaller than 67 degrees, the final number of spin-polarized electrons is larger than the initial number in the group TIA1 (the group of the larger initial number of electrons). Therefore, we may say that the number of spin-polarized electrons in the TIA1 increases as a result of interaction with group of TIA2. In contrast, in the case of angles greater than 67 degrees, the number of spin-polarized electrons in the group TIA1 decreases due to the interaction.



 

Spin torque

The spin torque describes the case when a very small amount of spin-polarized electrons is continuously injected into the existed spin-polarized electrons of a significantly larger amount. The spin direction of the injected electrons is different from the spin direction of the existed electrons. As a result of the injection, the spin direction of the existed spin-polarized electrons rotates towards the spin direction of the injected electrons. The effective torque of the rotation is called the Spin Torque.

Calculations of the Spin Torque

How the calculations are done?

The spin torque is calculated from the condition of an equilibrium of injected rate of spin-polarized electrons (TIA2) and the conversion rate of spin-polarized electrons from group TIA2 into group TIA1 of the existed spin-polarized electrons and the group of spin-unpolarized electrons (See Eq.(25)). Even though the calculations are done numerically using the method of calculations of Fig.1 and Fig.2 , the results are expressed analytically with calculated parameters A(φ) and R(φ), which are defined as the spin torque coefficient and the spin-relaxation coefficient. The dependence of the spin torque on different material and injection parameters can be calculated from calculated dependencies of A(φ) and R(φ) on those parameters.

Results:

In the case when a small amount of spin-polarized electrons of the group TIA2 is continuously injected at a rate into the region where there are spin-polarized electrons of the TIA1 group, the spin-polarized electrons of TIA1 group experience a spin torque, which can be calculated as as

where nTIA1 is the number of spin-polarized electrons in the group TIA1, are unit vectors directed along spin directions of electrons of the TIA1 and TIA2 groups,, respectively, and A(φ) is the spin torque coefficient, which depends on the angle φ between the spin directions of injected and existed electrons.

The conversion rate (spin relaxation rate) of spin-polarized electrons into spin-unpolarized can be calculated as

where R(φ) is the spin-relaxation coefficient, which depends on the angle φ between the spin directions of injected and existed electrons.

old form:

Fig.5 (left) Spin torque coefficient A(φ) (See Eq. (20)) and (right) spin-relaxation coefficient R(φ) (See Eq. (21)) as a function of angle φ between the spin directions of injected (TIA2) and existed (TIA1) spin-polarized electrons. 1 % of spin-polarized electrons in TIA2 group (injected) , 99% of spin-polarized electrons in TIA1 group (existed). Click on image to enlarge it

 

 

 

 

Figure 5 shows the spin torque coefficient A(φ) and the spin-relaxation coefficient B(φ) (TIS conversion coefficient) as a function of angle φ between the spin directions of injected and existed electrons (angle between spins of TIA1 and TIA2 groups).

Note: For small angles φ (φ<30 deg) , A(φ) ~57 deg and R(φ) ~0.

Figure 6 shows the spin torque coefficient A(φ) and the spin-relaxation coefficient B(φ) (TIS conversion coefficient) as a function of the amount of injected spin-polarized electrons (TIA2) with respect to the number of existed spin-polarized electrons (TIA1) for different angles φ . The number is scanned from 0 to 40%. Note: the value of the coefficient B(φ) is between 1 and 2 (the variation is small).

In the calculations shown in Fig.5 and Fig.6, a slow injection rate was assumed. In the case of a faster injection rate, the spin torque coefficient A(φ) and the spin-relaxation coefficient B(φ) become dependent on the injection rate.

Property of spin torque:

Fig.6 (left) Spin torque coefficient A(φ) (See Eq. (20)) and (right) spin-relaxation coefficient R(φ) (See Eq. (21)) as a function of the amount of injected spin-polarized electrons (TIA2) with respect to the number of existed spin-polarized electrons (TIA1). Click on image to enlarge it

the spin torque is linearly proportional to the injection rate .

Note: It is only the case of a slow injection rate, when the spin torque coefficient A(φ) and the spin-relaxation coefficient B(φ) do not depend on the injection rate (the case of a slow injection rate).

 

How to determine whether the injection rate is slow or fast?

As was calculated in Fig.1 and Fig.2, the interaction of two groups of spin-polarized electrons with different spin angles takes some time until they are fully transformed into one group of spin-polarized electrons of the same spin direction. We assign t2 as the effective time, after which the spin direction of electrons of the TIA1 group is fully rotated into its final direction. Even though a tiny oscillations around the final angle persists for a long time, the defined time t2 is the time, after which the deviation of the spin angle from its final position is less than 0.1 degrees. For examples of Fig.1 and Fig.2, , the time t2 corresponds to a time duration of ~5-7 scattering events 5. The number of spin-polarized electrons ( TIA2), which are injected during the time t2, is calculated as

where is the injection rate.

Under a continuous and constant injection rate, ΔnTIA2 is the constant number of spin-polarized electrons of the group TIA2, which spin is not yet aligned along spin direction of the existed spin-polarized electrons of group TIA1. These spin-polarized electrons of the group TIA2 are in the process of the conversion to the group TIA2 and to the group of spin-unpolarized electrons. As sooner as they are converted, the same number of spin-polarized electrons TIA2 is injected. As a result, ΔnTIA2 remains a constant.

Equilibrium: during time t2 -> (1) ΔnTIA2 electrons were injected into group TIA2; (2) ΔnTIA2 electrons was converted from group TIA2 into group TIA1 and the group of spin-unpolarized electrons

When the ΔnTIA2 exceeds 1 % of the number of spin-polarized electrons in the group TIA1 (the number of the existed spin-polarized electrons), the injected rate is defined as fast and the dependence of the coefficients A(φ) and B(φ) on the injection rate should be taken into calculations.

Can we mark or distinguish somehow a spin-polarized electron from different groups of different spin directions or from spin-unpolarized electrons? For example, similar as an electron can be distinguished in a semiconductor between to be an "electron" or a "hole"?

A. No. The electrons are frequently scattered between different groups. It is only average number of electrons in each group remains unchanged after a frequent scatterings (See here). It contrasts to the case of "holes" and "electrons" in a semiconductor. The electron scattering between a "hole" and "electron" is a very rare event, because of their different spacial symmetry (See here) and an electron stays as a "hole" and "electron" for a long time in a semiconductor. (Note, the case of a metal is different)

 

 

 

 

 

 

 

 


 

Possible confusion!!: from 2014 to 2017 I have used names TIA and TIS for groups of spin-polarized and spin-unpolarized electrons, respectively. The reasons are explained here.
An explanation can be found in Slide 11 of this Audio presentation or here

 

 

 

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