My Research and Inventions

click here to see all content or button bellow for specific topic

 

Spin-Orbit Torque (SOT)

Spin and Charge Transport

Abstract: An electrical current generates the spin-polarized electrons due to the Spin Hall effect. These spin-polarized changes the magnetic properties of the nanowire. This effect is called the effect of spin-orbit torque (SOT). The change of magnetic properties by an electrical current may be substantial. For example, the magnetization can be reversed. The SOT effects describes the dependence of a magnetic properties of a magnetic nanowire on polarity and magnitude of electrical current.


Magnetic parameters, which depend on polarity of current in a nanomagnet:

"damp-like" torque"field-like" torque; Anisotropy field Hanis; Energy of perpendicular magnetic anisotropy (PMA) Eanis;Coercive field Hc;Spin polarization;Logarithm of magnetization switching time; Logarithm of Retention time; Delta Δ;Hall angle;size of nucleation domain for magnetization switching

 

All these parameters depend linearly on the gate voltage. However, the dependencies are not clearly depend on each other. It indicates that there are several independent contributions to the SOT effect.

High precision measurements of current dependencies of these magnetic parameters are described below.

 


Origin of Spin-Orbit Torque

click on image to enlarge it

The creation of spin-polarization of conduction electrons by an electrical current is the origin of the Spin-Orbit Torque.

How an electrical current can spin- polarize conduction of electrons?

The spin-dependent scatterings spin-polarize conduction electrons (See here)

 

Two types of Spin-Orbit Torque

Interface-type

The spin-polarization is created due to the spin-dependent scatterings across an interface. Typically spin-dependent scattering occurs at an interface between a non-magnetic heavy metal (like Pt, Ta, W) and ferromagnetic metal (like Fe,Co, FeCoB).

Bulk-type

The spin-polarization is created due to the spin-dependent scatterings in the bulk of ferromagnetic metal. Typically the spin-dependent scattering occurs in a ferromagnetic metal containing a heavy metal (like FeBTb)

 

 


 

Transformation of Hysteresis loop due SOT effect

The position of loop is shifted due to current, but width and height of the loop are constant.
The x-axis is applied out-plane magnetic field. Click on image to enlarge it.

Magnetic parameters affected by SOT effect

creation of damping -like torque

This SOT effect is similar to the effect produced by an usual magnetic filed HDL , which is applied perpendicularly to the electrical current and perpendicularly to the magnetization.

The direction of the magnetic field HFL depends on the magnetization direction. When magnetization rotates along the z-axis. The magnetic field HDL rotates as well.

 

creation of field-like torque

This SOT effect is similar to the effect produced by an usual magnetic filed HFL , which is applied along the electrical current.

The magnetic field HFL does not depend on the magnetization direction.

 

modulation of the anisotropy field Hanis and the energy of perpendicular magnetic anisotropy EPMA

The bias current generates a spin-polarized electrons. The spin-polarized electrons at may affect the magnetization near film interface and consequently the the strength of the perpendicular magnetic anisotropy (PMA)

 

modulation of coercive field

Change of magnetization switching time under current due to SOT

dependence is opposite for spin-down to up and spin-down to up switching

switching from spin-down to spin-up

switching from spin-up to spin-down

switching time increases under a negative current and decreases under a positive current switching time increases under a positive current and decreases under a negative current
The magnetization switching time at a different current density. Sample ud30 Volt53B Ta(2.5):FeCoB(1):MgO Nanowire width: 1000 nm, length 200 nm. Measurements date is 10. 2018
Click on image to enlarge it

Under a bias current, the hysteresis loop are shifted from its center position (See Fig). It looks similar as an additional magnetic field applied perpendicularly to the film.

The switching field from spin-up to down state became different from switching field from spin-down to up state

modulation of delta Δ and retention time

The Δ and retention time characterize stability of the magnetization against a thermo-activated reversal.

The modulation of the Δ changes the probability thermo-activated magnetization switching

modulation of effective magnetization Meff and effective size of nucleation domain

The Meff is magnetization of first magnetic domain (nucleation domain), which triggers the magnetization reversal.

The bias current may move domain wall due to the spin-transfer torque. As a result the size of the nucleation domain becomes smaller or larger. Consequently, the Meff becomes smaller or larger.

modulation of the Hall angle

The Hall angle or the Hall resistance depends on the magnetization of the ferromagnetic metal, spin-polarization of the conduction electrons and the strength of the spin-orbit (SO) interaction. The bias current generates a spin-polarized electrons. As a result, the spin polarization of electron gas and its distribution across film changes. It causes the change of the Hall angle.


Experiment

Experiment

Fig.2 Anomalous Hall Effect is used for all SOT measurements
Click on image to enlarge it.

All SOT measurements were done using the Anomalous Hall Effect (AHE).

Fabrication of FeB, FeCoB and FeTbB nanomagnets connected to a Hall probe

The FeB, FeCoB and FeTbB films were grown on a Si/SiO2 substrate by sputtering. A Ta layer was used as a non-magnetic adhesion layer. The thickness of the Ta was between 2 and 10 nanometers and wafers of different Ta thickness were tested. A nanowire of different width between 100 and 1000 nm with a Hall probe was fabricated by the argon milling. The width of the Hall probe is 50 nm. The FeB and FeCoB layers were etched out from top of the nanowire except a small region of the nanomagnet, which was aligned to the Hall probe. The nanomagnets of different lengths between 100 nm to 1000 nm were fabricated.

 

 

 

 


Major obstacle of a SOT measurement is the heating

The SOT becomes substantial at the current of about 10-100 mA/um2. The heating of nanowire is substantial at this current. It is hard to remove the heating even when a pulse mode is used. For example, in my standard measurements an electrical pulse of 300 ms following 5 s cooling is used. However, there is still a substantial heating in this pulse mode (see below).

In any SOT measurements the sample heating should be taking into account.

How to distinguish effects from the SOT from the effects from heating?

A. The SOT effect is linearly proportional to current, but heating ~I2, at a relatively-small current the SOT dominates, but at a higher current the heating dominates.

How to minimize the influence of the heating?

1) Sweep polarity of current

Usually (but not always) the SOT changes its polarity when the polarity of current is reversed. The heating does not dependent on the current polarity

2) use a narrower and shorter nanowire.

The dissipation of heating is more effective in this case.


Measurement method of "field -like" torque HFL

Dependence of in-plane component of magnetization on in-plane plane magnetic field. The magnetic field is applied in-plane and along the current. HFL is the the offset magnetic field, which is proportional to the current. The HFL is evaluated by linear fitting of the dependence, which measured in Hall configuration.
Click on image to enlarge it

Method to measure "damp-like" torque and "field-like" torque

This method is similar to the method of measurement of anisotropy field Hanis (See here). In this measurement the in-plane component of the magnetization is measured as a function of an external magnetic field Hext. The dependence is linear (See here). Therefore, it can be measured with a high precision. The anisotropy field Hanis is defined as the in-plane magnetic field, at which initially-perpendicular magnetization turns completely into the in-plane direction.

Under a sufficient bias current, additionally to Hext., there are two more additional magnetic fields: 1) effective field of "field-like" torque HFL in direction of current and 2) effective field of "damp-like" torque HDL in direction perpendicularly to the current. Therefore, the magnetization experiences two magnetic field:

In-plane magnetic field along bias current (the x-axis) Htotal,x=Hext,x+HFL ,

In-plane magnetic field perpendicularly to bias current (the y-axis) Htotal,y=Hext,y+HDL ,

 

As a result, the dependence M vs Hext is shifted on HFL (Hext is applied along x-axis) or HDL (Hext is applied along y-axis). From the shift the HFL are HDL are evaluated.

Note: The same measurements can be done by two method

Method 1. The direct measurement using a nano voltmeter (See Fig.2 above)

Method 2. The 2nd-harmonic measurement using a lock-in amplifier (See here)

Both measurements give the same values of HFL and HDL and nearly the measurement precision. However, the usage of the direct measurement is preferable for the following reasons. From the direct measurement, the dependence of HFL and HDL on bias current can be evaluated. From measured dependence of M vs H, the contributions of the "field-like" and "damp-like" torque can be separated in the case when they are in the same direction. In contrast to the 2nd-harmonic measurement, in the direct measurement the undesired contribution due to sample heating can be removed.

I use AHE measurement to evaluate the in-plane component of the magnetization. The tunnel magneto-resistance (TMR) of magnetic tunnel junction (MTJ) can be used as well (See here).

1st type of SOT effect: "Damp -like" torque

"Damping-like" torque

The electrical current creates a magnetic field HDL, which is directed along current and perpendicularly to the magnetization M.. Due to HDL, the magnetization M is inclined to the front direction. When external magnetic field Hext is applied, the magnetization M turns in-plane. Following M, HDL turns as well. From measurements of magnetization M vs Hext, HDL can be evaluated.

Click on image to enlarge it

 

This SOT effect is similar to the effect produced by an usual magnetic filed HDL , which is applied perpendicularly to the electrical current and perpendicularly to the magnetization.

The "damping-like" torque is described as

where the effective magnetic field of the "damping-like" torque is defined as

(See Landau–Lifshitz equation)

 

 

 

How to measure it?

Measurement method of "damping -like" torque HDL

Dependence of in-plane component of magnetization on in-plane plane magnetic field. The magnetic field is applied in-plane and perpendicularly to the current. The center of line is shifted, but its ends are at the same position. The HDL is the the offset magnetic field, which is proportional to the current. The HDL is evaluated by linear fitting of the dependence, which measured in Hall configuration.
Click on image to enlarge it

It can be measured by the same measurement, which is used to measure the anisotropy field (See here). The in-plane component of magnetization is measured as a function of in-plane magnetic field. The in-plane magnetic field, which is applied perpendicularly to the electrical current.

The HDL gives the field offset for such measurement (See right Fig). From a linear fitting of measured dependence, the HDL is evaluated.

In the case of the "damp-like" torque the dependence M vs H is not linear. Even in the case the fitting gives a high precision.

 

Effective magnetic field HDL of the "damping -like" torque

Slope: -0.21811 Oe/(mA/um2)

Slope: -0.3371 Oe/(mA/um2)

as Nov. 2018
Sample R64A Volt58B Ta(5):FeCoB(1):MgO Sample: L58B Volt58A . It is the same wafer as the left one, but device position on the wafer is different

Measured sample distribution of the current- modulation of the effective field HDL of "damp- like" torque in FeB and FeCoB samples.

Measurements date is 10. 2018
Click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

What is the "damp-like" torque?

Effective magnetic field of "damp-like" and "anti damp-like" torque

damp-like torque

anti damp-like torque

The damp-like torque align the spin along the external magnetic field. The anti damp-like torque align the spin opposite to the external magnetic field.
Red arrow shows the spin (the magnetization). Blue arrow show the magnetic field Hext. Green arrow shows the effective magnetic field of the damp Hdamp (anti damp Hanti damp) torque.
Click on image to enlarge it

 

 

Difference between "damp-like" torque and "field-like" torque

 

 

 

 

Properties of effective magnetic field of "damp-like" torque

Its direction changes, when the spin direction (magnetization direction) changes.

Its magnitude changes, when the spin direction (magnetization direction) changes. The magnitude is the largest, when the spin is perpendicular to Hext and the magnitude is the smallest (equals to 0), when the spin is parallel to Hext.

 

 

 

 

What is the direction of the "damp-like" torque?

3 components of the "damp-like" torque can be distinguished. They are labeled as Hdamp,x , Hdamp,y and Hdamp,z.

Since the direction and magnitude of the effective magnetic field of "damp-like" torque changes when the magnetization direction is changed, the following definition is used:

Hdamp,x aligns magnetization along the x-axis (along bias current)

Hdamp,y aligns magnetization along the y-axis. (in-plane and perpendicularly to bias current)

Hdamp,z aligns magnetization along the z-axis. (perpendicularly to plane)

 

Direction of effective magnetic field of "damp-like" torque

magnetization rotates in yz-plane

magnetization rotates in xz-plane

Hdamp,x and Hdamp,z change their magnitude. Hdamp,y changes its direction Hdamp,y and Hdamp,z change their magnitude. Hdamp,x changes its direction

Red Arrow :M is the magnetization (the spin).
Gold Arrow: Hdamp,x is magnetic field of damping towards the x-axis
Blue Arrow: Hdamp,y is magnetic field of damping towards the y-axis
Green Arrow: Hdamp,z is magnetic field of damping towards the z-axis

Click on image to enlarge it

most probable direction of the "damp-like" torque is Hdamp,x

It is because of the following reason: The bias current breaks the time-reversal symmetry along the x-axis. Similarly, the time-reversal symmetry breaks in this direction, when a magnetic field is applied along the x-axis. Then, damp-like" torque is Hdamp,x aligns the magnetization along this field

Note: The existence of Hdamp,y and Hdamp,z is also allowed by the symmetry.

 

 

How Hdamp,x , Hdamp,y and Hdamp,z change their magnitude and direction when magnetization is rotated in the yz-plane and the xz-plane

It is important because from measurements of such rotation both is "field-like" torque and "damp-like" torque are evaluated.

 

 

 

 

 

 

 


2nd type of SOT effect: "Field-like" torque

"Field like" torque

The electrical current creates a magnetic field HFL, which is directed along current. Due to HFL, the magnetization M is inclined to the right direction. When external magnetic field Hext is applied, the magnetization M turns fully in-plane at a smaller field Hext= Hanisotropy - HFL,in the left direction. The magnetization M turns field in-plane at a larger field Hext= Hanisotropy + HFL,in the right direction. Measuring the difference between two field, HFL can be evaluated
Click on image to enlarge it

 

This SOT effect is similar to the effect produced by an usual magnetic filed HFL , which is applied along the electrical current.

The "field-like" torque is described as

where the effective magnetic field of the "field-like" torque is defined as

 

 

 

 

 

How to measure it?

Measurement method of "field -like" torque HFL

Dependence of in-plane component of magnetization on in-plane plane magnetic field. The magnetic field is applied in-plane and along the current. HFL is the the offset magnetic field, which is proportional to the current. The HFL is evaluated by linear fitting of the dependence, which measured in Hall configuration.
Click on image to enlarge it

It can be measured by the same measurement, which is used to measure the anisotropy field (See here). The in-plane component of magnetization is measured as a function of in-plane magnetic field. The in-plane magnetic field is applied along the electrical current.

The HFL gives the field offset for such measurement (See right Fig). From a linear fitting of measured dependence, the HFL is evaluated.

The Hall measurement is used to evaluate the in-plane component of the magnetization. The TMR can be used as well (See here).

 

The dependence of HFL on the current is rather linear. All FeB and FeCoB samples, which I have measured by Nov. 2018, shows the same sign (positive) of HFL.

 

Effective magnetic field HFL of the "field like" torque induced by SOT

Slope: 0.22093 Oe/(mA/um2)

Slope: 0.42035 Oe/(mA/um2)

as Nov. 2018
Sample R64A Volt58B Ta(5):FeCoB(1):MgO Sample: L58B Volt58A . It is the same wafer as the left one, but device position on the wafer is different Measured sample distribution of the current- modulation of the effective field HFL of "field- like" torque in FeB and FeCoB samples.
Measurements date is 10. 2018
Click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


3d type of SOT effect: SOT modulation of anisotropy field Hanis and PMA energy

Change of anisotropy field Hanis & PMA energy due to the SOT effect

case 1: SOT modulation of Hanis

case 2: nearly no SOT modulation

Current- (SOT-) induced change of Hanis Sample distribution of ΔHanis

Due to SOT effect, the Hanis increases at a negative current and decreases at a positive current. Additionally, the electrical current heats the sample. The heating causes the decreases of Hanis. Since SOT effect is linearly proportional to current, but heating ~I2, at a small negative current the Hani increases, but at a higher current effect of the heating dominates. Under a higher current the heating is stronger and temperature increase. When T increase, Hanis sharply decreases (see here). The decrease is symmetrical for opposite polarities of current. The of anisotropy field Hanis for two opposite directions of current as function of current as Nov. 2018
Sample R64A Volt58B Ta(5):FeCoB(1):MgO Sample: L58B Volt58A . It is the same wafer as the left one, but device position on the wafer is different Sample: ud49 Volt53, Ta(2.5) FeCoB(1):MgO, wire width 400 nm, nanomagnet length 500 nm Measured sample distribution of the current- modulation of the anisotropy field ΔHanis in FeB and FeCoB samples.
Measurements date is 10. 2018
Click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Measurement method of anisotropy field

Dependence of in-plane component of magnetization on in-plane plane magnetic field. The slope depends on the electrical current. The anisotropy field is evaluated from the slope of the line.
Click on image to enlarge it

 

 

 

 

 

 

 

 

 

 


 

4th type of SOT effect:SOT modulation of the coercive field

Change of coercive field due to the SOT effect (case 1:stronger heating)

Transformation of Hysteresis loop due SOT effect. The position of loop is shifted due to current, but width and height of the loop are constant.

Switching fields (coercive field) between spin-down to up and spin-up to down states. At a positive current, the coercive field of spin-down to up switching is larger. At a negative current, the coercive field of spin-up to down switching is larger. Such dependence is due to the SOT effect. For both current polarities, the coercive field decrease when the bias current increases. It is due to heating and the temperature rise.

 

Difference of switching fields between spin-down to up and spin-up to down states. The dependence is linear.
Data was measured using method described here, which gives measurement precision of coercive field better than 0.1 Oe.
Sample Ta(5)/FeB(0.9)/ MgO(6)/ Ta(1)/Ru(5) (Volt55 free44). Measurements date is 06. 2018
Click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

When current increases, two effects occur:

1.Heating

Even though the measurements of the coercive field are done in pulse mode, it is difficult completely avoid heating.

Due to the heating coercive field decreases. However, the decrease of the switching field between spin-down to up and switching field between spin-up to down states are absolutely identical and symmetrical (See here)

2. SOT effect

 

The coercive field was measured using method described here, which gives measurement precision of coercive field better than 0.1 Oe.

Change of coercive field due to the SOT effect (case 2: weaker heating)

Coercive field for the magnetization switching from spin-down to spin-up and spin-up to spin-down. Due to heating coercive field decreases. Due to SOT effect, the coercive field increases for spin-down to spin-up at a negative current and for spin-down to spin-up at a positive current. Since SOT effect is linearly proportional to current, but heating ~I2, only at a small current the Hc increases, but at a higher current effect of the heating dominates.

The difference of the coercivity field when polarity of the current reversed. The dependence is opposite for the magnetization switching from spin-down to spin-up and spin-up to spin-down.

Difference of switching fields between spin-down to up and spin-up to down states. The dependence is linear.
Sample ud30 Volt53B Ta(2.5):FeCoB(1):MgO Nanowire width: 1000 nm, length 200 nm.
Click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sample distribution of ΔHc

Measured sample distribution of the current- modulation of the coercive field ΔHc in FeB and FeCoB samples.
Click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 


5th type of SOT effect: modulation of delta Δ and retention time

modulation of delta Δ

modulation of retention time

modulation of delta Δ is nearly the same for magnetization switching from spin-down to up and from spin-up to down modulation of retention time is nearly the same for magnetization switching from spin-down to up and from spin-up to down
Sample ud30 Volt53B Ta(2.5):FeCoB(1):MgO Nanowire width: 1000 nm, length 200 nm
Click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


6th type of SOT effect: modulation of effective magnetization Meff and effective size of nucleation domain

modulation of effective size of nucleation domain

Sample ud30 Volt53B Ta(2.5):FeCoB(1):MgO Nanowire width: 1000 nm, length 200 nm Sample: ud49 Volt53, Ta(2.5) FeCoB(1):MgO, wire width 400 nm, nanomagnet length 500 nm
Dependence of size of nucleation domain on the electrical current. The black line shows for the case of magnetization switching from spin-down to spin-up state. The red line show for the case of switching from spin-up to spin-down state.
Click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 


7th type of SOT effect: SOT modulation of the Hall angle

SOT modulation of Hall angle

Difference of Hall for two opposite polarities of the bias current vs absolute value of current. The proximity of MgO modifies significantly
Sample: Ta(5):FeCoB ( 1 nm, x=0.3):MgO(7) Volt58A (L58B); nanowire width is 3000 nm, nanowire length is 25 um, length of um etched section is 3 um. For measurements of different magnetic properties of this sample click here
Click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 


8th type of SOT effect: The SOT modulation of spin polarization

Dependence of spin polarization on polarity of bias current (SOT effect)

Sample: FeB. Spin polarization as function of current density. At a higher current, the spin polarization decreases due to the sample heating. However, the decrease is different for opposite polarities of the current Sample: FeB. Change of the spin polarization under reversal of current direction Sample: FeCoB. Change of the spin polarization under reversal of current direction

click on image to enlarge it

 

 

 

 

 

 

 

 


"damp-like" torque and "field-like" torque

The "damp-like" torque and "field-like" torque can be measured by following techniques:

1) from measurement of anisotropy field; 2) by 2f-lockin technique; 3) from ST-FMR measurements


 

Comparison between "damp-like" torque and "field-like" torque

Difference between "Damp-like" torque and "Field-like" torque

Evaluation of type of SOT effective field "Field-like" torque "Damp-like" torque
In-plane component of magnetization as a function of applied in-plane magnetic field. The green line shows the case when there is no the spin-orbit torque. The blue line shows the case when the torque is "field-like" type. The red line shows the case when the torque is "damp-like" type. The effective magnetic field HFL of the "field-like" torque does not depend on the magnetization direction. Independently on the magnetization direction, it is always directed along the current.

The effective magnetic field HDL of the "damp-like" torque. It is always perpendicular to the magnetization direction. When the magnetization turns, the he effective magnetic field HDL turns as well.

 

Click on image to enlarge it

 

There are two substantially different types of the Spin-Orbit Torque.

One type does not depend on the magnetization direction of the ferromagnetic metal. It is only depend on the direction of the current. Such torque is called the field-like torque. The second

The second type does depend on the magnetization direction of the ferromagnetic metal. Such torque is called the anti-damping torque.

Even without any current, there are spin-polarized conduction electrons in a ferromagnetic metal. All conduction electrons in a ferromagnetic metal can be divided into two groups: group of spin-polarized electrons and group of spin-unpolarized electrons (See here for details). As was mentioned above, the spin-dependent scatterings originate the Spin-Orbit Torque. Since the properties of the spin-unpolarized electrons does not depend on the magnetization direction, the scattering of these electrons creates the anti-damping torque.. Spin direction of spin-polarized conduction electrons is parallel to the magnetization (See here for details). Therefore, the scattering of the spin-polarized creates the field-like torque

The field-like torque

1.It does not depend on the magnetization direction of the ferromagnetic metal.

2. It is created due to scattering of the spin-unpolarized electrons

The anti-damping torque

1.It does depend on the magnetization direction of the ferromagnetic metal.

2. It is created due to scattering of the spin-polarized electrons

 

 


Influence of interface on SOT effect

Dependence of SOT on the interface material

not-etched part etched part
 
Ta:FeB:MgO nanowire. Front half of the nanowire: the MgO and some FeB were etched and SiO2 was deposited instead. Back part of nanowire: FeB:MgO remains.  
Sample: Ta(5):FeCoB ( 1 nm, x=0.3):MgO(7) Volt58A (L58B); nanowire width is 3000 nm, nanowire length is 25 um, length of um etched section is 3 um. For measurements of different magnetic properties of this sample click here
Click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Measurements of the SO torque using 2d harmonic lock-in technique

Measurement of the Spin-orbit torque by 2d harmonic lock-in technique

asymmetric (even) component

symmetric (odd) component

it measures :(1) "field-like" torque, when the in-plane magnetic field is applied along the wire ( along electrical current); (2) "damp-like" torque, when the in-plane magnetic field is applied perpendicularly to the wire ( to electrical current); It measure
In this method the 2d harmonic of Hall voltage is measured as a function of an in-plane magnetic field
Click on image to enlarge it

 

The Hall voltage is proportional to the current, the perpendicular components of magnetization and the spin polarization of conduction electrons. When current is modulated with frequency ω, the magnetization direction, the magnetization magnitude and spin-polarization of the conduction electrons may be modulated due to the effect of the SO torque. As a result, the Hall voltage is modulated with frequency 2ω. Measuring this 2d harmonic the amplitude of the SO torque can be estimated.

 

What can be measured by the 2d harmonic lock-in technique?

1. "Field-like" torque

The direct measurements see here

It can be evaluated from the asymmetric (even) component of dependence of the 2d-harmonic voltage vs the in-plane magnetic field, when the magnetic field is applied along the current.

2. "Damp-like" torque

The direct measurements see here

It can be evaluated from the asymmetric (even) component of dependence of the 2d-harmonic voltage vs the in-plane magnetic field, when the magnetic field is applied perpendicularly the current.

3. Current-modulation of anisotropy field

The direct measurements see here

It can be evaluated from the symmetric (odd) component of dependence of the 2d-harmonic voltage vs the in-plane magnetic field. The result is the same whether the magnetic field applied along or perpendicularly to the current.

 

The fields of the the spin-orbit torque can be calculated from the following dependance of the 2d-harmonic voltage VHall,2ω vs applied in-plane magnetic field Hx :

where ΔHanis,ω is the current induced change of the anisotropy field Hanis; ΔHoff,ω is the effective magnetic field HFL,ω of the "field-like" torque, when Hx is applied along electrical current; and ΔHoff,ω is the effective magnetic field HDL,ω of the "damp-like" torque, when Hx is perpendicularly to the electrical current;

the odd and even components can be calculated as

Rwire is the is the ohmic resistance of the wire; RHall,0 is the is the Hall resistance, when a in-plane magnetic field is not applied ;

Hanisot is the anisotropy field, which can be measured directly (See here) with a high precision or from 1st harmonic with a moderate precision.

The Hanisot can be evaluated from the following dependance of the 1d-harmonic voltage VHall,ω vs applied in-plane magnetic field Hx :

 

for details description of 2d harmonic lock-in technique, click there to expand

 

The 2d harmonic lock-in technique measures the current- modulation of the effective magnetic field HDL of "damp- like" torque, the effective magnetic field HFL of "field- like" torque and the anisotropy field ΔHanis

Without electrical current, the in-plane component of the magnetization Mx depends on the applied external in-plane magnetic field Hx as (see here)

where Hanis is the anisotropy field

As was demonstrated above, the spin-orbit torque (SOT) produces the offset magnetic field ΔHoff and changes the anisotropy field Hanis on ΔHanis. As a result, the Eq.(4.1) is modified as

where ΔHoff equals to HFL when the in-plane magnetic field is applied along current and ΔHoff equals to HDL when the in-plane magnetic field is applied in-plane and perpendicularly to the current

In the case when

Eq.(4.2) can be simplified as

or

where

from (4.2) we have .

The z- component of the magnetization Mz can be calculated as

or

 

The Hall voltage is calculated as

when magnetization is not perpendicular to plane, the Hall voltage is calculated as

where Mz is the perpendicular-to-film component of magnetization, RHall,0 is the Hall resistance when the magnetization is perpendicular to the film (Mz =M).

When the current is modulated with frequency ω ,

both the ΔHoff and ΔHanis are modulated as well:

Using a trigonometric relation

and substituting Eqs (4.7) (4.10),(4.11) into Eqs. (4.7) gives the Hall voltage VHall,2ω of the 2d harmonic (the coefficient at cos(2ωt)) as

In a lock-in measurement it is convenient to use the reference voltage Vω rather than reference current Iω

2d harmonic

as a function of applied in-plane magnetic field.
Click on image to enlarge it

where Rwire is the resistance of metallic nanowire.

Substituting Eqs. (4.2) and (4.14) into Eq. (4.13) gives

or

The voltage of the second harmonic has two component. The first component is proportional to ΔHoff and is an odd function in the respect to Hx. The first component is proportional to ΔHanis and is an even function in the respect to Hx. Therefore, the voltage of the second harmonic can be calculated as

2d harmonic

as a function of applied in-plane magnetic field.
Click on image to enlarge it

Eq (4.17) can be written in a symmetrical form as

where

 


Measurement of anisotropy field from the 1st harmonic (not recommended)

1st harmonic

Perpendicular-to-plane component of magnetization as a function of applied in-plane magnetic field.
Click on image to enlarge it

 

When current is small, the ΔHoff and ΔHanis can be ignored. Than, the Hall voltage VHall,ω of 1st harmonic can be calculated from Eq.(4.9) as

substitution of Eq(4.1) into Eq.(4.20) gives the Hall voltage VHall,ω of 1st harmonic as

 

The ratio of voltage of 1st harmonic to the voltage of 1st harmonic can be calculated as

 

 

 

 

 

 

 

 

Measurement of anisotropy field Hanis

The arrow shows the direction and magnitude of the applied in-plane magnetic field. The ball shows the magnetization direction. Without magnetic field the magnetization is perpendicularly-to-plane. Under magnetic field, the magnetization turns toward magnetic field. The field, at which the magnetization turns completely in-plane, is called the anisotropy field. The dots of the right graph shows experimental data. Measurement date: May 2018.
Click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 


 

 

 

Current-induced magnetization reversal in FeBTb

Electrical current can induce spin torque or reduce the exchange interaction between localized electrons. This can change the direction of magnetization of a material.

Current-induced magnetization reversal in FeBTb film

 

Dependence on applied magnetic field

Coercive field significantly decreases when current density increases

click here or on image to enlarge it

Dependence on current density

magnetization is reversed by electrical current

click to enlarge

 

Two current-induced effects, which can lead to the current-induced magnetization reversal:

1) Current-induced Spin torque

2) Current-induced reduction of the exchange interaction between localized electrons.

 

Both effects occur because of transfer of delocalized (conduction) spin-polarized electrons from a point to point, which alters  an equilibrium spin polarization in a material.

The spin torque occurs when the delocalized spin-polarized electrons are transferred from one material to another by a drift or a diffusion current. When spin-polarized delocalized electrons are injected, it is not only change magnitude of spin accumulation, but  also it changes spin direction of spin accumulated electrons. As result, the spin direction of localized and delocalized electrons becomes different. This induces the torque,  which may turn or reverse the spin direction of the localized electrons.

Note: At one place an electron gas may have only one spin direction of its spin accumulation. In the case when the electrons with a different spin direction is injected, the spins quickly relax and the spin accumulation of only one spin direction remains. The final spin direction is different from initial spin direction and from the injected spin direction. Details see here and here

 

 

 

The spin torque may change magnetization direction in a material because of the exchange interaction between localized and delocalized electrons.

 

 

 

There are several effects which can cause the current-induced spin torque:

1)The spin-transfer torque.

It occurs  because of transfer of spin-polarized electron from material to material by a drift or diffusive spin current. Example: the spin transfer between electrodes  in a MTJ or GMR junction. The polarity of the spin-transfer torque depends on mutual magnetization directions of the electrons.


2) The spin-orbit (SO) torque.

It occurs in magnetic or non-magnetic metals in which there are substantial spin-dependent scatterings. Due to spin-dependent scatterings a spin-polarized current flows perpendicularly to the flow of spin- unpolarized drift current. The spin is accumulated at one side of a metallic wire and the spin is depleted at another side. The spin accumulation(depletion) may cause the spin torque at sides of the wire, which magnitude and direction is proportional to the drift current. The accumulated spin may  have different spin direction than the spin direction of the equilibrium spin polarization.


 

 

Current-induced magnetization reversal in FeBTb film

 

Negative bias

Coercive field significantly decreases when current density increases

click here or on image to enlarge it

Positive bias

magnetization is reversed by electrical current

click to enlarge

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Current-induced magnetization reversal in FeBTb film

 

delocalized (conduction p- or s-) electrons

The spin-polarized delocalized electrons are accumulated at the left side of the wire and they are depleted at the right side. Since the delocalized spin-polarized electrons mediates the exchange interaction between localized electrons, the exchange interaction becomes weaker at right side and stronger at left side.

click here or on image to enlarge it

localized (d- or f-) electrons

When a weak magnetic field is applied opposite to the magnetization of the wire, it is not sufficient to reverse magnetization of the wire. When current flows through the wire, delocalized electrons are depleted at right right side of wire and the exchange interaction between localized electrons is reduced in this area. Because of reduced exchange interaction, the weak external magnetic field becomes sufficient to reverse magnetization of localized electrons at the right side. This reversal triggers the magnetization reversal in whole wire.

click here or on image to enlarge it

click here to see both side reversal

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I truly appreciate your comments, feedbacks and questions

I will try to answer your questions as soon as possible

 

Comment Box is loading comments...