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Spin-Transfer Torque

Spin and Charge Transport

The spin-transfer torque is the torque acting on the magnetization of a ferromagnetic electrode of a magnetic tunnel junction (MTJ), when an electrical current flows through the MTJ. The spin-transfer torque may cause either a magnetization precession in the ferromagnetic electrode or a reversal of the magnetization of the electrode. The spin-transfer torque is utilized as a recording mechanism in the spin-transfer-torque magnetic random access memory (STT-MRAM).


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 10 (pp.27-29).
 

 


 

It might be possible that in the future the STT-MRAM will be a universal memory, which may replace high-density non-volatile memory (hard disks, Flash memory) and high-density fast-speed memory (DRAM, SRAM).

wikipedia page on spin transfer torque is here

How to calculate the spin-transfer torque from the model of spin-up/spin-down bands See here

Terminology: spin-torque current induces a torque acting on the conduction electrons ; spin-transfer torque is the torque acting on the local d-electrons

STT-MRAM (Spin-transfer-torque random access memory)

Wikipedia page on STT-MRAM is here

STT-MRAM is a non-volatile memory. The merits of STT-MRAM are the high density and the fast operation speed.

Fig.1. Animated picture. Magnetic tunnel junction as a memory cell. Data is stored by means of magnetization of free layer. The electrical current induces the spin transfer torque in the "free" layer, which reverses the magnetization of the "free" layer. Polarity of the spin-transfer torque depends on the polarity of the electrical current.

 

The MTJ is a basic cell of the STT-MRAM memory. The MTJ consists of two ferromagnetic metals and a thin isolator (a tunnel barrier) between them. The magnetization of one ferromagnetic metal is pinned by an exchange field with an antiferromagnetic layer and the magnetization of this layer can not be reversed. This ferromagnetic layer is called the “pinned” layer. The magnetization of the second ferromagnetic layer may have two stable directions along its easy axis. This ferromagnetic region is called the “free” layer. A bit of data in the MTJ cell is stored by means of two opposite magnetization directions of the “free” layer.

 

Reading function: Tunnel magnetic resistance (TMR)

The resistance of the MTJ (TMR~ 100%) is different by a factor of 2 in cases of magnetization of the “free” layer parallel or anti parallel to the magnetization of the “pinned” layer. By measuring the MTJ resistance the data can be read.

 

Writing function: Spin-Transfer Torque

When sufficient current flows through the MTJ, the magnetization of the “free” layer may be reversed and the data is memorized. The direction, to which the magnetization of the “free” layer may be changed, depends on the polarity of the current (See Fig.1)

 

 


 

Physical origin of the spin-transfer torque

Fig 2. Animated figure. Precession of spins of the localized d-electrons (red arrow) and conduction electrons (blue arrows) due to the exchange interaction between them. The spin-transfer torque occurs only when there is non-zero angle between the spin direction of the spin-polarized conduction electrons and the spin direction of the local d-electrons.

 

Important Features:

The spin-transfer torque is a complex effect. It is the joint work of three different mechanisms. All these mechanisms should work effectively and they should combine their join efforts in order to achieve a high spin torque.

(step 1) Drift of the spin by spin-polarized conduction electrons from one electrode to another electrode. (mechanism 1)

(step 2) Rotation of spin direction of already-existed spin-polarized conduction electrons due to injection of spin-polarized conduction electrons from another electrode (mechanism 2)

(step 3) Rotation of the magnetization direction (the spins of localized d-electrons) following the spin rotation of the conduction electrons. (mechanism 3)

(Mechanism 1)

The transfer of the spin from one metal electrode to another metal electrode by an electrical current of spin-polarized conduction electrons.

Under a bias voltage, the conduction electrons move along a conductor transferring both the spin and the charge. They transfer spins from one place to another place and from one electrode to another electrode. There is a problem of this spin-transfer mechanism in a metal. In a metal, there are nearly-equal amounts of holes and electrons (See here and here ). The spins of a hole and an electron are in the same directions, but the electron and the hole move in opposite directions along a bias voltage. The spin transfer in forward direction by electrons is compensated by the spin transfer in the opposite direction by the holes. As a result, the spin transfer is not effective in a bulk of a metal.

The situation is improved when there is no balance between numbers of electrons and holes in a metal. It is the case for (1) a region near a contact between two metals; (2) a region near an interface or edge of a metal; (3) in a metal with large number of defects (a low-conductivity metal). In all these cases, the numbers of electrons and holes are substantially different (See here and here) and the spin transfer by a current of conduction electrons becomes effective (See here and here).

Additionally to unbalance between numbers of holes and electrons, there is an additional reason why the spin-transfer is effective in above-mentioned cases. The reason is the increase of the number of the standing-wave electrons. The standing-wave electron are a conduction electron, which is bounced between two defects or is bonded to the interface (See here and here). Therefore, it cannot move freely through the bulk of metal. The standing-wave electrons are not very effective to transfer the charge, but they still effective to transfer the spin (See here). As a result, for a given charge current the standing-wave electrons are effective for the spin-transfer than the conventional (running-wave) conduction electrons.

When and where the spin-transfer by conduction electrons are most effective? Why?

most effective: magnetic tunnel junction (MTJ)

A conduction electron cannot freely move through the tunnel barrier. It should tunnel through it. It means that a conduction electron should bond to one side of the burrier, wait for a while and tunnel through the barrier into the state on another site of barrier. This transport mechanism is very different from the transport by a free-moving conduction electron in the bulk of a metal. The type of current changes from the band current to the scattering current. The scattering current is very effective for spin transfer (See here)

moderate or low effectiveness : through a contact between two metals

Giant-magnetic resistance effect (GMR)

There is a barrier between two metals due the difference of their work functions (e.g. see here. (click to expand)). A conduction electron cannot freely move through it. Some conduction electrons are bounced back and some electrons are bonded to the contact interface. It improves the spin transport in comparison to the transport in the bulk of the metal. However, a low energy barrier between two metals cannot fully suppress the band current , which reduces substantially effectiveness of the spin transport.

most ineffective : in the bulk of a metal

Domain-wall movement by an electrical current

In the bulk of metal the major transport mechanism is the the band current , which ineffective for the spin transfer. It is because of balance of nearly-equal, but opposite the spin transfer by the electrons and the holes. In a metal of a low-conductivity with a high density of defect the effectiveness of the spin transfer is better due to the increased number of the standing-wave electrons

 

(Mechanism 2)

The rotation of spin direction of the existed spin-polarized conduction electrons due to the injection of spin-polarized conduction electrons from another electrode.

The number of injected spin-polarized conduction electrons is substantially smaller than the number of the spin-polarized conduction electrons, which already exists in the electrode. Therefore, a small amount of injected electrons should turn spins of substantially-larger amount of the existed electrons. It is done by the spin torque effect. It is very effective mechanism (See here)

(Mechanism 3)

The rotation of spin direction of the localized d-electrons due to the exchange interaction with the conduction electrons..

The spin direction of the the localized d-electrons rotates due to the sp-d exchange interaction with the conduction electrons. The torque occurs when the angle between the spin direction of spin-polarized conduction electrons and the localized d-electrons is not 0 deg or 180 degrees.

(1) -The first contribution to the sp-d exchange interaction is the scatterings between the conduction electrons and the localized electrons.

The spin-polarized conduction electrons are scattered into unoccupied places of localized d-electrons. As result, the spin direction of some d-electrons becomes parallel to the spin direction of the spin-polarized conduction electrons and their spin direction becomes different from spin direction of other localized d-electrons. Due to the exchange interaction between the localized d-electrons the spin of all localized d-electrons turn toward spin direction of the spin-polarized conduction electrons.

-The second contribution to the sp-d exchange interaction is the direct exchange interaction between spin-polarized conduction electrons and localized d-electrons.

There are several contributions to this kind of the sp-d exchange interaction

(a) antiferromagnetic due to the spin-dependent Coulomb interaction between electrons

(b) ferromagnetic due to the spin-dependent Coulomb interaction between electrons and atomic nuclears

-The third contribution is direct dipole interaction between magnetic moments of the spin-polarized conduction electrons and the localized d-electrons.



 

1.When a current flows through the MTJ, the angle between the spin direction of the spin-polarized conduction electrons and the spin direction of the local d-electrons becomes non-zero. Therefore, a torque starts to act on the localized d-electrons due to the sp-d exchange interaction

Note: In equilibrium the spin direction of spin-polarized conduction electrons is parallel or antiparallel to the spin direction of the local d-electrons (See here)

2. The magnitude of the spin-transfer torque is largest near the tunnel barrier and it exponentially decreases into the depth of each electrode. Therefore, the thickness of the "free" layer should be at least shorter than the spin diffusion length in the "free" layer.

The spin transfer torque is proportional to the spin-torque current. The spin-torque current is proportional to the gradient of the spin accumulation (See here), which is described as ~ exp(-x/λspin), where x is the diffusion distance into the electrode and λspin is the spin diffusion length.

3. The spin transfer torque is always accompanied by the spin precession of the local d-electrons and by the spin precession of the conduction electrons (Fig.2)

 


Explanation in short

Under an applied voltage, an electrical current of conduction electrons flows between electrodes of the MTJ. Due to the electrical current, some spin-polarized conduction electrons are injected from one electrode to another electrode. Due to the injection of spin-polarized electrons the spin direction of all spin-polarized conduction electrons turns away from equilibrium and the angle between spin-polarized conduction electrons and localized d-electrons becomes non-zero. In the case of a non-zero angle, the exchange interaction between the conduction electrons and the d-electrons causes a precession of spins of the d-electrons and a precession of spins of the conduction electrons around a common axis. In the case of a sufficiently large spin-torque current, the spin direction of the d-electrons may be reversed.


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Both the localized d-electrons and the conduction electrons experience several different torques of different origins. The Spin-Transfer Torque is combined effort and balance of all these torques.


Torques acting on conduction electrons


The spin-transfer torque is determined by a balance of torques of different origins acting on the conduction electrons and on the local d-electrons.

1. Torque due to the spin-torque current

Direction: toward the magnetization direction of the "pinned" electrode

(The spin-torque current is trying to align the spin directions of the spin-polarized conduction electrons of “pinned” and “free” layers in the same direction.)
(about spin-torque current see here)
The torque due to the spin-transfer current in the "free" layer depends on
(1) the angle between the spin directions of the spin-polarized conduction electrons in the "free" and "pinned" electrodes
(2) the magnitude of the drift current flowing between electrodes
(3) the diffusion distance from the tunnel barrier
(4) the spin diffusion length in the "free" electrode.

Torque due to the spin-transfer current exponentially decreases from the tunnel barrier into the depth of the "free" layer

The spin-torque current can be calculated as

See here for details

2. Torque due to the exchange interaction of the spin-unpolarized conduction electrons with the d- electrons

Direction: towards the spin direction of the d-electrons

It occurs due to the exchange interaction between the unpolarized conduction electrons and the local d-electrons. Due the exchange interaction with the d-electrons, some spin-unpolarized conduction electrons become spin-polarized with spin direction along spins of d-electrons. This effect is called the spin pumping. When the spin directions of the d-electrons and other spin-polarized conduction electrons is different, this causes the spin torque, which is trying to align the spin direction of all conduction electrons parallel to the spin direction of the d-electrons
(about the spin pumping due to the exchange with the d-electrons, see here)

The spin torque increases when the angle between the spin directions of the d-electrons and the spin-polarized conduction electrons increases ( when the angle is smaller than 62 deg). ( See Fig.4(left) here), For larger angles the spin torque decreases when the angle increases.

 

3.Torque due to the exchange interaction the spin-polarized conduction electrons with the d- electrons

Direction: toward the spin direction of the d-electrons

There is a spin precession of spin-polarized conduction electrons due to the sp-d exchange with the d- electrons. This torque occurs due to the damping of this precision.

The torque can be calculated as (See Eq. (4) and solution of Landau-Lifshiz equation here)

where theta is the angle between spin directions of the d-electrons and the spin-polarized conduction electrons and tλ is the precession damping time in the exchange field.

This torque is largest in the cases of theta =90 deg

 


Torques acting on the local d-electrons


 

1. Torque due to the exchange interaction with the conduction electrons of the spin-polarized conduction electrons

Direction: toward the spin direction of the spin-polarized conduction electrons

Fig 4. Damping of the spin precession of the local d-electrons (red arrow) and the spin-polarized conduction electrons (blue arrows) due to the sp-d exchange interaction. The damping causes a torque on both the d-electrons and the spin-polarized conduction electrons

There is a spin precession of the d-electrons due to the exchange interaction with the spin-polarized conduction electrons. The torque occurs due to the damping of this precision.

The torque can be calculated as (See Eq. (4) and solution of Landau-Lifshiz equation here)

where theta is the angle between the spin directions of the d-electrons and the spin-polarized conduction electrons and tλ is the precession damping time in the exchange field.

This torque is largest in the cases of theta =90 deg

 

 

2. Crystal anisotropy/shape anisotropy torque

Direction: toward the easy axis

Crystal anisotropy/shape anisotropy torque can be calculated by a similar method as is in the case of the Stoner–Wohlfarth model.

For example, in the case of uniaxial anisotropy the energy of the magnetic anisotropy of volume V can be calculated as

where Ku is the anisotropy parameter.

The torque can be calculated by differentiating Eq. (10)

3. Torque due to the exchange field between d-electrons

Direction: it is trying to align all d-electrons in the same direction.

It can be strong in the case when there is a large gradient of the spin transfer torque. This may be the case for materials with a shortest spin diffusion length in the region near the tunnel barrier. In this case the excitement of spin waves may be efficient.

 


 

 


Events leading to the spin-transfer torque:


The spin-transfer torque has contributions from several other effects. For better understanding, the effects leading to the spin-transfer torque may be divided into several events or steps.

 

Step 1. The flow of spin-polarized drift current from one electrode of the MTJ to the other electrode

Fig.5 Generation of diffusion spin current by a spin-polarized drift current. The drift current (yellow balls) flows between the source (Voltage polarity "-") and the drain (Voltage polarity "+"). At the tunnel barrier spin is accumulated and diffuses inside each ferromagnetic metal. Arrows show the magnetization of the metals. Green balls show the spin diffusion current, which exponentially decays with propagation length. Note: spin polarization of the drift current and the diffusion current continuously (not abruptly) changes from metal to metal, because of the spin-torque current.

 

 

Under an applied voltage the drift current flows from one ferromagnetic electrode to the other electrode. The spin polarization of each electrode is non-zero and the conduction electrons are in both the TIS and TIA assemblies. This means that the drift current flowing in the electrodes is spin-polarized (See here) and in the drift flow there are some electrons from both TIA and TIS assemblies. The direction of spin polarization of the drift current in each electrode is different.

About drift current See here.

Step 2. Generation of spin diffusion current, which flows from the tunnel barrier into each electrode

Spins are accumulated at a tunnel barrier and a spin diffusion current flows away from the tunnel barrier into each ferromagnetic layer. The spin diffusion current decays exponentially as it flows away from the tunnel barrier.

 

Step 3. Spin-torque current flows between electrodes

The spin direction of the TIA assembly is different in ferromagnetic electrodes. Also, there is a gradient of the spin accumulation. As it is shown in here, these two conditions are sufficient for a spin-torque current to flow between the electrodes. Since the spin-torque current is linearly proportional to the gradient of the spin accumulation, it is largest near the tunnel barrier and decays exponentially as it flows away from the tunnel barrier.

Step 4. The spin direction of the conduction electrons of the TIA assembly rotates away from the direction of the local d-electrons.

In the absence of current, the spin direction of the TIA assembly is along to the spin direction of the d-electrons. The spin torque current turns the spin direction of the TIA assembly away from the spin direction of the d-electrons.

Step 5. The spin direction of the local d-electrons rotates following the spin direction of conduction electrons of the TIA assembly, because of the exchange interaction between the d-electrons and the conduction electrons of the TIA assembly.

The exchange interaction between the d-electrons and the electrons of the TIA assembly leads to a spin precession of the d-electrons and a spin precession of the electrons of the TIA assembly. The damping of these precessions induces a torque acting on the d-electrons and a torque acting on the conduction electrons of the TIA assembly. These torques are directed toward each other and they are trying to align the d-electron and the conduction electrons of the TIA assembly back along each other. The torque acting on the d-electrons turns the spins of the d-electrons away from the easy axis direction.

Step 6. When the exchanged field exceeds the anisotropy field, the magnetization of the d-electrons is reversed. Otherwise, there is a spin precession of the local d-electrons and a spin precession of the conduction electrons of the TIA assembly.

In the case when this torque is sufficiently large, the spin direction of the d-electrons may be reversed. As was mentioned above, this effect is used for recording data in the STT-MRAM memory. In the case when the torque is not sufficient for the magnetization reversal, there is a stable precession of the d-electrons and the electrons of the TIA assembly. Since the resistance of the MTJ depends on the relative spin directions of electrons of the TIA assemblies at different sides of the tunnel barrier, the resistance of the MTJ may be modulated following the precession of electrons of the TIA assembly. The precession frequency is commonly in the microwave spectrum region and the DC current flowing through the MTJ can be modulated at a microwave frequency. This method is used to generate microwave oscillations in the microwave torque oscillator.

 


 

Question about thicknesses of free , pin and space layers of MTJ and GMR devices

Question about thicknesses of free and pin layers of MTJ and GMR devices. Click to expand

Q.

Hi sir, Good afternoon. Myself Eswar.H doing phd in the field of non linear dynamics especially in Spin Torque Nano Oscillator
I had one doubt sir? Typically Spin Torque Nano Oscillator consists of three layers (i.e.) 2 ferromagnetic layer separated by a non magnetic spacer. My doubt was
1)On what basis they fix the thickness of spacer,Pinned layer, and free layer (everything in nano-meter range)? Is there any physics behind that?
2)or Is there any ratio between the all three layers? (Example: 2:1 ratio between pinned and free layer) Likewise?
please tell me sir. Finally, Tanx in Advance sir.

A.

Dear Eswar,

The resistance of a tunnel barrier sharply increases with a slight increase of the thickness of the tunnel barrier.
For Fe/MgO/Fe See Fig.1 here
https://staff.aist.go.jp/v.zayets/spin3_46_MgO.html
For MgO thinner than 1 nm, there is no tunneling and MR.
For 3 nm of MgO the resistance is too high and it is even hard to measure the resistance.

The STO (spin-torque oscillator) needs a high current, therefore the thinnest-possible MgO thickness( about 1 nm) is used.

Thickness of the free layer is fixed about 2 nm.
There is an effective spin-torque only within thickness about 1 nm in the vicinity of MgO. For a thinner free-layer, it easier to make its magnetization to precess with smaller spin-torque and a smaller current.

The pin layer also experiences the spin-torque, but its magnetization should not precess. Therefore, it should be a thicker as possible. However, the magnetization of a thick film breaks into domains. ALso, the magnetic field from a thick magnetic layer undesirably affect the free layer. It is not good. Therefore, an antiferromagnetic layer is used for pinning.

This is " a long story in short". I hope it will be useful for you.


Q.

Now i understood the concept clearly. Sir but you explained it for a Tunneling Magneto Resistance. Please explain me what happens if we replace the insulating barrier by a conducting spacer (i.e) for Giant Magneto Resistance (Non-magnetic conducting spacer)
1) In case of GMR how we fix the thickness of free and fixed layer and spacer? Please explain the physics behind that?
2)Is there any ratio behind the thickness of all three layers?
Please explain me sir. Tanx in Advance.

A.

Dear Eswar,

For the GMR structure the story is almost the same as for the TMR structure.
The reasons for optimum thicknesses of pin and free layers are the same.
The physics the same:
The free layer:
Thickness should be smaller than a spin diffusion length in this layer (~ 1-2 nm). Therefore, whole layer should experience the spin transfer torque.

It should be very thin, but It could not be too thin.
The mobility of Fe atoms on the most of surfaces are high, a thin Fe film tends clustering and it is difficult to make a smooth continuous Fe layer thinner than 1 nm. For the case of amorphous FeB (or FeCoB) is better, but still it is difficult to make a thin continuous film.
The pin layer:
The thickness should be thicker than the spin diffusion length. Therefore, only a part of layer experiences the spin transfer torque and the magnetization of the pin layer should not be reversed by current.

It should be thick, but it could not be too thick.
The magnetic field of this layer should be minimized. It should not affect the free layer.
It should be a thin enough to be in a single-domain state.
The spacer layer:

The spacer layer separates the free and the pin layers. Therefore, there is no exchange interaction between them. The magnetization direction of each layer can be independent from another layer.
Generally speaking, the thickness of the spacer layer should be thicker than the length of interlayer exchange interaction, which is a few interatomic distances.
However, the thickness of the spacer layer should be substantially thicker.
It could not be thicker than the spin diffusion length in the spacer layer. Therefore, a metal with a long spin diffusion length (like Cu) is used as the spacer material.

The reasons why the spacer layer should be thick are:
1) GMR ratio should be high. Output of STO is proportional to GMR ratio.
2) Even though there is no direct exchange interaction between localized electron at distances longer than a few atomic monolayer, there is another longer-range exchange interaction, which mediated by conduction electrons. Such exchange coupling exists only in GMR structure, but not in TMR. The spacer of TMR structure is isolator, it does not have the conduction electrons. The range of such interaction is about the size of conduction electrons, which equals to the mean-free-path. For example, in Co:Ru:Co such interaction is strongest at 0.9 nm of Ru thickness

3) The spin torque is larger in the case of a thicker spacer layer if the thickness is thinner than spin diffusion length.
For higher spin torque, the conductivity type should be changed from bulk-type to the interface type.
In the bulk of a metal, there is almost no spin drift and no spin-torque.
In the case of TMR, the conductivity by the tunneling is different from the bulk conductivity.
For GMR with a thin spacer, the conductivity is still bulk like, therefore the spin torque is small.
----------------
There is no a "magic" ratio between thicknesses of the layers. However, for each choice of metals the thicknesses of each layers should be optimized, especially for the GMR structure.

Again it is a long story in short
I hope it is helpful for you.

Best regards

 


 

 

 

 

 

 

 

 

 

An explaination can be found in Slides 13 and 14 of this Audio presentation or here

 

 

 

 

 

I truly appreciate your comments, feedbacks and questions

I will try to answer your questions as soon as possible

 

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