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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.



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 8 (pp.22-25).
An explanation can be found in Slide 11 of this Audio presentation or here

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 injection rate of , the spin-polarized electrons of the electron gas experiences the spin torque

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

where nTIS is the number of electrons in the group of spin-unpolarized electrons.

 

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.

 

 


 

Examples

Example 1. Magnetic field is applied at some angle with respect to the spin direction of spin accumulated electrons.

Animated figure. Precession of spins of the local d-electrons (red arrow) and conduction electrons (blue arrows) because of the spin torque induced by a magnetic field in a ferromagnetic metal.

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 Gilbert damping and the spin torque

A magnetic field converts the electrons from the TIA assembly into the TIS assembly. The spin direction of the converted electrons is along the magnetic field. In the case when the spin direction of the main TIA assembly is different than the spin direction of the converted electrons, the electrons of the main TIA assembly experience a spin torque, which rotates their spin towards the direction of the magnetic field.

In addition to the spin torque, there is another torque , which rotates the main TIA assembly toward the magnetic field. There is a precession of the electrons of TIA assembly in the magnetic field. This torque is due to the damping of the precession.

Example: ferromagnetic metal in a magnetic field

For example, if a magnetic field is applied at some angle with respect to the easy axis of a ferromagnetic metal, the spin direction of the conduction electrons of the TIA assembly rotates towards the magnetic field, because of the spin torque. The metal magnetization (spin of the local d-electrons) may only slightly turn away from the easy axis. In this case there will be a non-zero angle between the spin directions of the local d- and conduction electrons. This causes the spin precession of the d- electrons and the spin precession of the conduction electrons of the TIA assembly.

Decrease or increase of the spin accumulation in a magnetic field

A magnetic field converts the electrons from the TIS into the TIA assembly. Due to this conversion the number of electrons in the TIA assembly increases. The spin torque is always accompanied by an additional spin relaxation. This means that due to the spin torque the electrons are converted back from the TIA into TIS assembly, which reduces the number of electrons in the TIA assembly.

Due to the balance of these two conversions the number of electrons in the TIA assembly either increases or decreases depending on the angle between the magnetic field and the spin angle of the TIA assembly. Compared to the case without magnetic field, the number of electrons in the TIA assembly increases when the angle is smaller than 67 degrees and the number decreases when the angle is greater than 67 degrees.

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 converts electrons from the TIS assembly into the TIA assembly with spin direction normal to the film (green balls with arrows). This induces a spin torque, which rotates the spin direction of conduction electrons of the TIA assembly away from the magnetization direction. Because of the exchange interaction the magnetization follows the spin direction of the conduction electrons. After the illumination has stopped, the magnetization returns to in-plane.

 

 

 

Example 2. illumination by circularly polarized light.

Circular -polarized light illuminating a metal or semiconductor may may convert the electrons from the TIS assembly into the TIA assembly. The spin direction of the converted electrons is along the incident direction of light and it may not coincide with the spin direction of the electrons of the main TIA assembly. In this case the spin direction of the main TIA assembly rotates toward the incident direction of light.

In this case, light converts electrons of the TIS assembly of the metal into the TIA2 assembly, whose spin direction is along the incident direction of light. The spin direction of TIA2 assembly might be different from the spin direction of electrons of the existing TIA1 assembly in a ferromagnetic metal. The electrons of the TIA1 assembly will experience a spin torque, which turns their spin direction along the incident direction of light.

 

Example: rotation of the magnetization of a soft ferromagnetic metal by illumination by circular -polarized light

For example, if circular polarized light incidents normally on a thin film of a soft ferromagnetic metal (See Fig.2), whose magnetization is in plane, it induces a spin torque, which rotates the spin direction of the conduction electrons of the TIA assembly 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 d-electrons, there are precessions of the conduction and d-electrons, because of the exchange interaction between them (the precessions are not shown in Fig.2). Because of the damping of these precessions, the magnetization turns towards the normal-to-plane direction.

 


All-Optical Magnetization Reversal

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

1) ferromagnetic or ferrimagnetic metals

Light interacts with delocalized electrons of the electron gas. The interaction is effective

Origin: Light-created spin polarization in the electron gas causes the spin torque in the electron gas, which turns the spin polarization of the electron gas.

Circularly-polarized light excites some spin-polarized electrons in the electron gas, which makes the electron gas spin-polarized. Spin direction of this light-excited spin polarization is different from the spin direction of the existed spin-polarization. At the same place in the electron gas two spin accumulations with different spin direction can not coexist. Because of the scatterings, the spins are mixed and only one spin accumulation remains with spin direction somewhere between two initial spin directions. As result the spin polarization of the electron gas turns out from the spin direction of the d-electrons. The exchange field between delocalized and localized electrons can be considered as an effective magnetic field. When spin of delocalized electrons turn out from the direction the localized d- electrons, spins of both localized and delocalized electrons starts to precess around common axis as shown in this figure 1. Because of this precession, the spin direction of the d-electrons may be reversed.

Since photons even in a focused laser beam are rather delocalized, light interacts strongly with delocalized electrons of the electron gas and light interaction with the localized d-electrons or sp-electrons is rather weak. For this reason this mechanism is efficient for the magnetization reversal.

2) Oxide and transparent dielectrics (for example, YIG)

Light interacts with localized electrons, which are responsible for the super-exchange interaction or the exchange interaction.

Origin: 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 and the magnetization start to precess or it may even be reversed.

 


 

 

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 the assemblies combine into one group of spin-polarized electrons and some electrons are converted into the group of the spin-unpolarized electron (additional spin relaxation).

The interaction of two TIA assemblies significantly depends on the relative number of electrons in each assembly. In the case of the interaction of two TIA assemblies with the same number of electrons, one scattering event 5 between electrons of the TIA assemblies is sufficient to combine the assemblies. In the case of the interaction of two TIA assemblies with a different number of electrons, it takes several scatterings until the two TIA assemblies relax into one TIA assembly.

In order to calculate the interaction of two or more TIA assemblies, the quantum nature of spin should be considered. However, the incorrect representation of spin as a 3D spacial vector gives a simple and easy-to-understand picture for the interaction of TIA assemblies. In contrast to the correct Quantum-mechanical calculations, where the interaction can only be numerically simulated, in the case of the spin representation as a 3D spacial vector it is possible to obtain an analytical formula. Click below to see the calculation of the interaction of two TIA assemblies in the case when spin is represented as a 3D spacial vector.

 

classical calculation of interaction of two TIS assemblies

If in the same time in a metal there are two TIS assemblies with different spin directions, they will quickly converts into one TIA assembly. Also, some electrons will be converted into TIS assembly.

The interaction of several TIS assemblies should be calculated considering the quantum nature of spin. Calculations below are based on a classical treatment of spin. Therefore, they are incorrect. However, they are helpful to understand basic properties of these interactions. In the case when spin may be considered as a simple 3D-vector, the number of electrons and the direction of TIA assembly, which the result of the interaction of several TIA assemblies, can be calculated as a vector sum of all spin states as

where is a unit vector directed along spin direction of the TIA assembly, is the number of spin states in TIA assembly, is the number of spin states with energy E and spin directed along vector r and D(E) is the density of states. The surface integral is over an unit sphere.

In the case of constant density of states, the TIA assembly (TIA3) resulting of the interaction of two TIA assemblies (TIA1 and TIA2) is calculated as

where r1, r2 are unit vectors directed along TIA1 and TIA assemblies and n_TIA1, n_TIA2 are the number of electrons in these assemblies.

For example, if the angle between TIA1 and TIA2 assemblies is phi2 and the TIA1 assembly is directed along the z-axis, Eq. (20.2) is simplified to

where the number of electrons in TIA3 assembly and its spin direction along the z-axis are

Next, I will consider two limited cases. The first case when the number of electrons in one of two TIA assemblies is significantly smaller than the number of the electron in the second assembly. That case is important for calculation of spin-torque current. The first limited case is the case of equal number of electrons in two assembly

Case 1

in the case of interaction of two TIA assemblies of equal magnitudes

 

Case 2. The interaction of two TIA assemblies of equal magnitudes

in the case of interaction of two TIA assemblies of equal magnitudes

The first Eq. of (20.6) is different from correct Eq (11) and Eq. (12), because above-calculations have ignored the quantum nature of spin.

 

 

Quantum-mechanical calculation of the interaction of two TIA assemblies.

In the case of the interaction of two TIA assemblies with an equal number of electrons, the number of electrons in the resulting TIA assembly will be (for proofs click here)

where n_TIA1 is the number of electrons in each TIA assembly before interaction and phi is the angle between them.

In the case of the interaction of two TIA assemblies with different numbers of electrons, it takes several scattering events 5 until they relax into one TIA assembly. For example, let us consider the interaction of TIA1 and TIA2 assemblies when the number of electrons in TIA1 is greater than the number of electrons in TIA2.

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

After the first scattering event 5 the spin direction of TIA2 will have the angle phi/2 with respect to the z-axis, the spin direction of TIA1 will not change. The number of electrons in TIA1, TIA2 and TIS assemblies after the first scatterings will be

After the second scattering event 5, the angle between the spin directions of the TIA assemblies will be phi/4. In the case when the number of electrons in the TIA2 assembly is greater than in the TIA1 assembly, the number of electrons in TIA1, TIA2 and TIS assemblies will be

Otherwise, when the number of electrons in TIA1 is greater than in TIA2, the spin direction of the TIA2 assembly rotates.

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

It should be noted that the probability of scattering event 5 is not constant, but it is proportional to the number of electrons in the TIA1 and TIA2 assemblies. The probability of one scattering event 5 can be calculated as

where are the number of “spin” states in TIA1, TIA2 and TIS assemblies, respectively. is the probability of a single scattering event 5.

Numerical simulation of interaction of two TIA assemblies

The interaction of two TIA assemblies was simulated numerically by the method described above

Figure 1 shows the case of the interaction of the TIA1 and TIA2 assemblies when initially 80 % of the conduction electrons are in the TIA1 assembly, 20 % of the conduction electrons are in the TIA2 assembly and there are no electrons in the TIS assembly. Figure 2 shows a similar case of the interaction of the TIA1 and TIA2 assemblies. However, initially 98 % of the conduction electrons are in the TIA1 assembly and 2 % of the conduction electrons are in the TIA2 assembly. The initial angle between the spin directions of the TIA assemblies was 145 deg in both cases. After each scattering event 5 the angle between the TIA assembles decreases and some electrons are converted into the TIS assembly. During the first 3-4 scattering events 5 there is a significant conversion of the electrons from the TIA1 assembly into the TIS assembly (spin relaxation), the number of electrons in the TIA1 assembly increases and the number of electrons in the TIA2 assembly decreases. Only the spin direction of the TIA1 assembly changes during the first scatterings. However, after 5-7 scatterings the number of electrons in the TIA assemblies becomes comparable and the spin direction of both assemblies rotates. After 10-15 scatterings the angle between the TIA assemblies becomes very small and the two assemblies can be considered as one TIA assembly.

 

Fig 1. The interaction of two TIA assemblies. (left). Animated polar figure. The radius of arrows corresponds to the probability of electrons to be in an assembly and the angle indicates the spin direction. Initial conditions for the spin direction and the occupation probability: TIA1 assembly (red arrow) probability=0.8 angle= 0 deg and TIA2 assembly (blue arrow) probability=0.2 angle= 145 deg. The height of the green bar on the left corresponds to the total number of electrons in the TIA assemblies and the height of the yellow bar on the left corresponds to number of electrons in both TIA assemblies. The animated parameter is the number of scattering events 5.

(right) Occupation probabilities of TIA1 (black line) and TIA2 assemblies (red line) as a function of number of scatterings. Green and blue lines show the probability of electrons to be in one of TIA assemblies and TIS assembly, respectively. Inset shows spin directions of TIA1 and TIA2 assemblies.

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig 2. The interaction of two TIA assemblies. This figure is similar to Fig 1, but the initial conditions for the TIA assemblies are different: TIA1 assembly (red arrow) probability=0.98, angle= 0 deg and TIA2 assembly (blue arrow) probability=0.02, angle 145 deg.

 

 

 

 

 

 

 

 

 

 

 

Click here to see another examples of interactions of TIA assemblies

 

Figure P11 shows the case when the angle between the TIA assemblies is 179 degrees. Even in this case most of the electrons of the TIA2 assembly are converted into the TIS assembly and after 1st scattering only a tiny amount of electrons remain in TIA2 assembly, still after 12 scatterings the amount of electrons in the TIA2 assembly becomes comparable with the amount of electrons in the TIA1 assembly.

Fig P11. The interaction of TIA assemblies. This figure is similar to Fig 1, but with different initial conditions for the TIA assemblies: TIA1 assembly (red arrow) p0=0.8 angle= 0 deg and TIA2 assembly (blue arrow) p0=0.2 angle 179 deg.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig.3 The interaction of TIA assemblies. (left) rotation angle of TIA1 assembly and (right) final occupation probabilities of TIA and TIS assemblies as a result of the interaction of TIA1 and TIA2 assemblies. Initial angle between TIA1 and TIA2 assemblies is 145 deg. Initially there were no electrons in the TIS assembly.

 

 

Figure 3 shows the rotation angle and the amount of electrons in the TIA and TIS assemblies as a result of the interactions of the TIA1 and TIA2 assemblies. When the initial amount of electrons of the TIA2 is less than 10 %, the rotation angle and the amount of electrons converted into the TIS assembly is linearly proportional to the initial amount of electrons in the TIA2 assembly.

 

 

 

To note: The spin rotation angle and the amount of electrons converted into the TIS assembly are linearly proportional to the amount of electrons in the TIS assembly, which has a smaller number of electrons.

 

Fig.4 The interaction of TIA assemblies. (left) Rotation angle and (right) the probability for an electron to be converted into the TIS assembly as a function of the initial angle between the TIA1 and TIA2 assemblies. Initially 99 % of electrons were in the TIA1 assembly and 1 % of electrons were in the TIA2 assembly. Initially there were no electrons in the TIS assembly.

 

Figure 4 shows the rotation angle and the amount of electrons converted into the TIS assembly as a result of the interactions of the the TIA1 and TIA2 assemblies, for the case when 99 % of conduction electrons are in the TIA1 and 1 % of conduction electrons are in the TIA2. There is no rotation in the cases when the spin directions of TIA1 and TIA2 are parallel or antiparallel. The rotation is largest in the case when the angle between the spin directions of the TIA assemblies is around 67 degrees.

The number of electrons converted into the TIS assembly increases when the angle between the assemblies increases. When the angle is 120 degrees or larger, the number of converted electrons is around 2%. This is twice the initial amount of TIA2 electrons. For angles smaller than 67 degrees, the amount of converted electrons is smaller than the initial amount of TIA2 electrons. Therefore, the number of electrons in the resulting TIA assembly is larger than the initial number in TIA1. This means that the number of electrons in the TIA assembly increases because of the injection. In contrast, in the case of angles greater than 67 degrees, the number of electrons in the TIA assembly decreases due to the injection.


 

Spin torque

In the case when a small amount of electrons of the TIA2 assembly is continuously injected (or converted from the TIS assembly) at a rate into the region where there are some electrons in the TIA1 assembly, the electrons of TIA1 experience a spin torque, which can be calculated as as

where n_TIA1 is the number of electrons in the TIA1 assembly, are unit vectors directed along the TIA1 and TIA2 assemblies, respectively, and A(phi) is the spin torque coefficient, which depends on the angle phi between the TIA assemblies.

Also, the injection(conversion) causes a conversion of electrons from the TIA assembly to the TIS assembly at a rate

where B(phi) is the TIS conversion coefficient, which depends on the angle phi between TIA assemblies.

The coefficients A and B may be found by the above-discussed method of calculations of the interaction of two TIA assemblies, assuming that the injection rate is small.

 

Fig.5 The interaction of TIA assemblies. (left) Spin torque coefficient A from Eq. (20) and (right) TIS conversion coefficient B from Eq. (21) as functions of the initial angle between TIA1 and TIA2 assemblies.

 

 

 

 

Figure 5 shows the spin torque coefficient A and the TIS conversion coefficient B as functions of the angle between the spin directions of the TIA1 and TIA2 assemblies. For small angles the coefficient A equals ~57 deg and the coefficient B equals ~2.

 

 

Dependence of spin torque on the injection (conversion) rate

From Eq. (20) the spin torque is linearly proportional to the injection (conversion) rate. It is only in the case of a slow injection (conversion) rate, when the coefficients A and B do not depend on the injection (conversion) rate.

 

In the calculations shown in Fig.5 a small injection (conversion) rate was assumed. In the case of a higher rate, the spin torque coefficient A and the TIS conversion coefficient B depend on the injection (conversion) rate.

 

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

As was shown in Fig.1 and 2, the interaction of two TIA assemblies takes some time. We assign t2 as the effective time after which the spin direction of the TIA1 assembly is rotated and most of the electrons are converted into the TIS assembly. As was shown above, the time t2 corresponds to a time duration of ~5-7 scattering events 5 , so it is known. The number of electrons of the TIA2, which are injected (converted) during the time t2 is

This number is the effective number of electrons of the TIA2 assembly in the metal. In the case when this number exceeds 1 % of the number of electrons in the TIA1 assembly, the dependence of the coefficients A and B on the injection (conversion) rate should be considered.

 

Fig.6 The interaction of TIA assemblies. (left) Spin torque coefficient A from Eq. (20) and (right) TIS conversion coefficient B from Eq. (21) as functions of the effective number of electrons of the TIA2 assembly (Eq.(25))

 

Figure 6 shows the spin torque coefficient A and the TIS conversion coefficient B as a function of for different angles between the TIA1 and TIA2 assemblies. It could be noticed that the value of the coefficient B is always between 1 and 2.

 

 

 

 

 

 

 

 

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