more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11

SpinOrbit Torque (SOT) Spin and Charge TransportAbstract: An electrical current generates the spinpolarized electrons due to the Spin Hall effect. These spinpolarized changes the magnetic properties of the nanowire. This effect is called the effect of spinorbit 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:"magnetic field, which is induced by the spin accumulation; Anisotropy field H_{anis}; Energy of perpendicular magnetic anisotropy (PMA) E_{anis};Coercive field H_{c};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.Contentclick on the chapter for the shortcut(1). 3terminal MTJ(2). Origin of SpinOrbit Torque in short() Wrong path: Incorrect introduction "damplike" torque and nonexistent "fieldlike" torque(3). Magnetic parameters affected by SOT effect(video:) Measurement of coefficient of spin orbit interaction, anisotropy field in a nanomagnet and magnetic field created by a spin accumulation.(4). Measurement of SOT(measurement 1) "damplike" torque(measurement 2) "fieldlike" torque(measurement 3) SOT modulation of anisotropy field H_{anis} and PMA energy(measurement 4) SOT modulation of coercive field H_{c}(measurement 5) SOT modulation of delta Δ and retention time(measurement 6) SOT modulation of of effective magnetization M_{eff} and effective size of nucleation domain(measurement 7) SOT modulation of Hall angle(measurement 8) SOT modulation of spin polarization(5). What is the "damplike" torque?(6). Comparison between "damplike" torque and "fieldlike" torque(7). Influence of interface on SOT effect(8). Measurements of the SO torque using 2d harmonic lockin technique(9). Currentinduced magnetization reversal in FeBTbQuestions & Answers(q1) about systematic errors of 2nd harmonic measurements(q2) torque & spin dynamic & Quantum mechanic(q3) about FieldLike torque
6. Explanation video(video1) Measurement of coefficient of spin orbit interaction, anisotropy field in a nanomagnet and magnetic field created by a spin accumulation.(video2) Parametric magnetization reversal, Intermag 2022
.........Two method of spinreversal by spin polarized electrons(source 1 of spin polarized electrons) Spintransfer torqueIn this case a current of the spin polarized electrons flows from one material to another material. As a result, the spinpolarized electrons from one material are moved to another material and accumulated there. The spin accumulated electrons force the magnetization of the 2nd material to reverse and to be the direction of the spin polarization. (source 2 of spin polarized electrons) Spinorbit torque (SOT)In this case, the spinpolarized electrons are generated (created) inside of the ferromagnetic material ( Specifically, at the boundary of the material) by an electrical current. The spin accumulated electrons force the magnetization of the material to rotate or even reverse. The spin direction of the spin accumulated is different for the opposite directions of the current. Therefore, an electrical current of opposite directions switches the magnetization of a nanomagnet between its two stable states.
3 mechanisms of the magnetization reversal(mechanism 1) Direct spin injectionIn this case, the spin direction of the injected electrons is opposite to the spin direction of already existed electrons. When the number of injected localized electrons of spin opposite to the magnetization becomes larger than the number of initiallyexisted localized electrons of the spin along the magnetization, the magnetization of the nanomagnet is irreversibly reversed. The spin injection can be due to both the spintransfer or spinorbit torque (mechanism 2) Parametric reversalIn this case, the injected spinpolarized electrons (or their magnetic field) slightly change the magnetization direction in the resonance with the magnetization precession causing an increase of the precession angle. The magnetization of the nanomagnet is irreversibly reversed, when the precession angle becomes larger than 90 degrees. The parametric injection can be due to both the spintransfer or spinorbit torque (mechanism 3) thermallyactivated reversalIn this case, due to a random interaction with a nonzerospin particle, like a magnon or a circularly polarized photon, the number of localized electrons of spin opposite to to the magnetization may become larger than the number of initiallyexisted localized electrons of the spin along the magnetization. As a result , the magnetization of the nanomagnet is irreversibly reversed.
3terminal MTJ memory
The SOT effect is used as a writing mechanism for the 3terminal MTJ memory
merit: Reading circuit and writing circuits are separated.It improves: (1) memory durability; (2) operational speed; Data storage: By means of two opposite magnetization of the "free" layer.reading circuit: The reading voltage is applied between "free" and "pinned" layers The resistivity of the MTJ is lower, when the magnetizations of "free" and "pinned" layers are parallel. The resistivity of the MTJ is higher, when the magnetizations of "free" and "pinned" layers are anti parallel. writing circuit: The writing voltage is applied between sides of the nonmagnetic metal The spin current is generated at freelayer/ nonmagneticmetal interface , which induces the torque on the "free" layer due to the SOT effect. The SOT torque is opposite for the opposite polarities of the writing current and it reverses the magnetization into two opposite directions.
2teminals MTJ memory is described here
Origin of SpinOrbit Torque in shortThe creation (origin) of the spinorbit torque can be divided into two steps. At the first step, the spinpolarized electrons are created by an electrical current. At the second step, the created spinpolarized electrons affects the magnetic properties of the nanomagnet (step 1) Creation of of the spinpolarized electronsThe creation of the spinpolarized conduction electrons by an electrical current is called the Spin Hall effect (See here for details). The major mechanism of creation of spinpolarized electrons is the spindependent scatterings (See Spin Hall effect for more details) How an electrical current can create the spin polarized conduction electrons?When there are spindependent scatterings, the spinpolarized electrons are accumulated at side edges of an electrical wire (Spin Hall effect). For example, if initially the conduction electrons are not spinpolarized, the probability of a scattering of a spinup electron is larger to the left with respect to current direction and the probability of a scattering of a spindown electron is larger to the right, then there are more spinup electrons at the left side of wire and more spindown electrons at the right side of the wire.
Two sources of of generation of spinpolarized electrons:InterfacesourceThe spinpolarization is created due to the spindependent scatterings across an interface. Typically spindependent scattering occurs at an interface between a nonmagnetic heavy metal (like Pt, Ta, W) and ferromagnetic metal (like Fe,Co, FeCoB). BulksourceThe spinpolarization is created due to the spindependent scatterings in the bulk of ferromagnetic metal. Typically the spindependent scattering occurs in a ferromagnetic metal containing a heavy metal (like FeBTb)
The SOT effect is usually observed in a ferromagnetic metal, where there are two groups of conduction electrons: (group 1) spinunpolarized electrons and (group 2) spinpolarized electrons (see here). Correspondingly, there are two origins for creations of new spinpolarized electrons. Two origins of generation of spinpolarized electrons:from spinunpolarized electronsDue to spindependent scattering, some spinunpolarized electrons becomes spinpolarized. The spinpolarization of these created spinpolarized electrons are different on opposite sides of the wire. from spinpolarized electronsA spindependent scattering of alreadyexisted spinpolarized electrons creates the spinpolarized electrons of different spin direction. As a result, there are two groups of spinpolarized electrons of different spin directions: (group 1) large group of "alreadyexisted" spinpolarized electrons and (group 2) tiny group of "newlycreated" of spinpolarized electrons. These two groups quickly interact with each other (See Spin Torque)
(step 2) Influence of created spinpolarized electrons on magnetic properties on a nanomagnet(influence 1) Spin torque See details on the spin torque here.It is the case when the spin direction of "newlycreated" spinpolarized electrons is different from the spin direction of "alreadyexisted" spinpolarized electrons. In this case the Spin Torque is created. As a result, the spin direction of a large number of "alreadyexisted" spinpolarized electrons rotates toward the spindirection of a tiny number of "newlycreated" spinpolarized electrons. This effect is called the Spin Torque. Depending on the spin direction of "newlycreated" spinpolarized electrons and the corresponded direction the Spin Torque., two torque torque can be distinguished: "damplike" torque and "fieldlike" torque. (influence 2) Change of size of nucleation domain for the magnetization switching See details on thermally activated magnetization switching here.The electrical current induces the spintransfer torque (it is the mechanism of the current induced magnetization reversal in a MTJ). Under influence of the spintransfer torque, the domain wall of the nucleation domain for magnetization switching may may. As a result, (influence 3) Change of the spin polarization See details on spin polarization here and here.The electrical current creates the spin polarized electrons, which added to "alreadyexisted" spinpolarized electrons. Depending on the polarity of the electrical current, the spin direction of "newlycreated" spinpolarized electrons is either along or opposite to the spin direction of the "alreadyexisted" spinpolarized electrons. As a result, the total spin polarization either decrease or increases for two opposite directions of the electrical current. This influence makes currentdependent all magnetic properties, which depend on the spin polarization. (influence 4) Change of the PMA energy See details on thermally activated magnetization switching here.For a reason, which has not been understood yet, the PMA energy E_{PMA} is changed by the electrical current. It leads to currentdependency of anisotropy field H_{anis}, coercive field H_{c }and delta Δ.
Magnetic parameters affected by SOT effectcreation of damping like torqueThis SOT effect is similar to the effect produced by an usual magnetic filed H_{DL} , which is applied perpendicularly to the electrical current and perpendicularly to the magnetization. The direction of the magnetic field H_{FL} depends on the magnetization direction. When magnetization rotates along the zaxis. The magnetic field H_{DL} rotates as well.
creation of fieldlike torqueThis SOT effect is similar to the effect produced by an usual magnetic filed H_{FL} , which is applied along the electrical current. The magnetic field H_{FL} does not depend on the magnetization direction.
modulation of the anisotropy field H_{anis} and the energy of perpendicular magnetic anisotropy E_{PMA}The bias current generates a spinpolarized electrons. The spinpolarized electrons at may affect the magnetization near film interface and consequently the the strength of the perpendicular magnetic anisotropy (PMA)
modulation of coercive field
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 spinup to down state became different from switching field from spindown to up state modulation of delta Δ and retention timeThe Δ and retention time characterize stability of the magnetization against a thermally activated reversal. The modulation of the Δ changes the probability thermallyactivated magnetization switching modulation of effective magnetization M_{eff} and effective size of nucleation domainThe M_{eff} is magnetization of first magnetic domain (nucleation domain), which triggers the magnetization reversal. The bias current may move domain wall due to the spintransfer torque. As a result the size of the nucleation domain becomes smaller or larger. Consequently, the M_{eff} becomes smaller or larger. modulation of the Hall angleThe Hall angle or the Hall resistance depends on the magnetization of the ferromagnetic metal, spinpolarization of the conduction electrons and the strength of the spinorbit (SO) interaction. The bias current generates a spinpolarized electrons. As a result, the spin polarization of electron gas and its distribution across film changes. It causes the change of the Hall angle.
Measurement of SOT
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 nonmagnetic 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.
A obstacle of a SOT measurement is the heatingThe SOT becomes substantial at the current of about 10100 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 ~I^{2}, at a relativelysmall 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.
Wrong path: Incorrect introduction "damplike" torque and nonexistent "fieldlike" torque(fact) In order to explain rather complex measurement data of the 2nd harmonic method and the symmetrical asymmetrical contributions to FMR resonance, the "damplike" torque and the "fieldlike" torque were incorrectly introduced. Why have the "damplike" torque and the "fieldlike" torque been introduced? The currentinduced spin dynamics has many contributions (see above) and, therefore, is complex. It was incorrectly believed that the usage of two independent subjects as two independent torques could resolve all that complexity. (The torque as a subject of the Classical Physics to resolve problem of Quantum Mechanic): vs. The torque is a subject of the Classical Mechanics describing how a force changes rotation of an object. The spin does not describe any rotation (See here). In the Quantum Mechanic, the orbital moment describes the object rotation (See here). The spin describes the properties of the broken time inverse symmetry, according to which the spin can either precess or align along or opposite to an external magnetic field. Any introduction of the classical torque should fit to the fundamental properties of the time inverse symmetry
Method to measure the currentinduced in plane magnetic fieldThis method is similar to the method of measurement of anisotropy field H_{anis} (See here). In this measurement the inplane component of the magnetization is measured as a function of an external magnetic field H_{ext}. The dependence is linear (See here). Therefore, it can be measured with a high precision. The anisotropy field H_{anis} is defined as the inplane magnetic field, at which initiallyperpendicular magnetization turns completely into the inplane direction. Under a sufficient bias current, additionally to H_{ext}., there are two more additional magnetic fields: 1) effective field of "fieldlike" torque H_{FL} in direction of current and 2) effective field of "damplike" torque H_{DL} in direction perpendicularly to the current. Therefore, the magnetization experiences two magnetic field: Inplane magnetic field along bias current (the xaxis) H_{total,x}=H_{ext,x}+H_{FL} , Inplane magnetic field perpendicularly to bias current (the yaxis) H_{total,y}=H_{ext,y}+H_{DL} ,
As a result, the dependence M vs H_{ext} is shifted on H_{FL} (H_{ext} is applied along xaxis) or H_{DL} (H_{ext} is applied along yaxis). From the shift the H_{FL} are H_{DL} 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 2ndharmonic measurement using a lockin amplifier (See here) Both measurements give the same values of H_{FL} and H_{DL} 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 H_{FL} and H_{DL} on bias current can be evaluated. From measured dependence of M vs H, the contributions of the "fieldlike" and "damplike" torque can be separated in the case when they are in the same direction. In contrast to the 2ndharmonic measurement, in the direct measurement the undesired contribution due to sample heating can be removed. I use AHE measurement to evaluate the inplane component of the magnetization. The tunnel magnetoresistance (TMR) of magnetic tunnel junction (MTJ) can be used as well (See here).1st type of SOT effect: currentinduced in plane magnetic field incorrectly associated with "Damp like" torque
This SOT effect is similar to the effect produced by an usual magnetic filed H_{DL} , which is applied perpendicularly to the electrical current and perpendicularly to the magnetization. The "dampinglike" torque is described as where the effective magnetic field of the "dampinglike" torque is defined as (See Landau–Lifshitz equation)
How to measure it?
It can be measured by the same measurement, which is used to measure the anisotropy field (See here). The inplane component of magnetization is measured as a function of inplane magnetic field. The inplane magnetic field, which is applied perpendicularly to the electrical current. The H_{DL} gives the field offset for such measurement (See right Fig). From a linear fitting of measured dependence, the H_{DL} is evaluated. In the case of the "damplike" torque the dependence M vs H is not linear. Even in the case the fitting gives a high precision.
What is the "damplike" torque?
Difference between "damplike" torque and "fieldlike" torque
Properties of effective magnetic field of "damplike" 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 H_{ext} and the magnitude is the smallest (equals to 0), when the spin is parallel to H_{ext}.
What is the direction of the "damplike" torque? 3 components of the "damplike" torque can be distinguished. They are labeled as H_{damp,x} , H_{damp,y} and H_{damp,z}. Since the direction and magnitude of the effective magnetic field of "damplike" torque changes when the magnetization direction is changed, the following definition is used: H_{damp,x }aligns magnetization along the xaxis (along bias current) H_{damp,y} aligns magnetization along the yaxis. (inplane and perpendicularly to bias current) H_{damp,z} aligns magnetization along the zaxis. (perpendicularly to plane)
most probable direction of the "damplike" torque is H_{damp,x} It is because of the following reason: The bias current breaks the timereversal symmetry along the xaxis. Similarly, the timereversal symmetry breaks in this direction, when a magnetic field is applied along the xaxis. Then, damplike" torque is H_{damp,x} aligns the magnetization along this field Note: The existence of H_{damp,y} and H_{damp,z} is also allowed by the symmetry.
How H_{damp,x} , H_{damp,y} and H_{damp,z} change their magnitude and direction when magnetization is rotated in the yzplane and the xzplane It is important because from measurements of such rotation both is "fieldlike" torque and "damplike" torque are evaluated.
2nd type of SOT effect: currentinduced in plane magnetic field incorrectly associated with "Fieldlike" torque
This SOT effect is similar to the effect produced by an usual magnetic filed H_{FL} , which is applied along the electrical current. The "fieldlike" torque is described as where the effective magnetic field of the "fieldlike" torque is defined as
How to measure it?
It can be measured by the same measurement, which is used to measure the anisotropy field (See here). The inplane component of magnetization is measured as a function of inplane magnetic field. The inplane magnetic field is applied along the electrical current. The H_{FL} gives the field offset for such measurement (See right Fig). From a linear fitting of measured dependence, the H_{FL} is evaluated. The Hall measurement is used to evaluate the inplane component of the magnetization. The TMR can be used as well (See here).
The dependence of H_{FL} on the current is rather linear. All FeB and FeCoB samples, which I have measured by Nov. 2018, shows the same sign (positive) of H_{FL}.
3d type of SOT effect: SOT modulation of anisotropy field H_{anis} and PMA energy
4th type of SOT effect:SOT modulation of the coercive field
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 spindown to up and switching field between spinup 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.
5th type of SOT effect: modulation of delta Δ and retention time
6th type of SOT effect: modulation of effective magnetization M_{eff} and effective size of nucleation domain
7th type of SOT effect: SOT modulation of the Hall angle
8th type of SOT effect: The SOT modulation of spin polarization
"damplike" torque and "fieldlike" torqueThe "damplike" torque and "fieldlike" torque can be measured by following techniques: 1) from measurement of anisotropy field; 2) by 2flockin technique; 3) from STFMR measurements
Comparison between "damplike" torque and "fieldlike" torque
There are two substantially different types of the SpinOrbit 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 fieldlike torque. The second The second type does depend on the magnetization direction of the ferromagnetic metal. Such torque is called the antidamping torque. Even without any current, there are spinpolarized conduction electrons in a ferromagnetic metal. All conduction electrons in a ferromagnetic metal can be divided into two groups: group of spinpolarized electrons and group of spinunpolarized electrons (See here for details). As was mentioned above, the spindependent scatterings originate the SpinOrbit Torque. Since the properties of the spinunpolarized electrons does not depend on the magnetization direction, the scattering of these electrons creates the antidamping torque.. Spin direction of spinpolarized conduction electrons is parallel to the magnetization (See here for details). Therefore, the scattering of the spinpolarized creates the fieldlike torque The fieldlike torque 1.It does not depend on the magnetization direction of the ferromagnetic metal. 2. It is created due to scattering of the spinunpolarized electrons The antidamping torque 1.It does depend on the magnetization direction of the ferromagnetic metal. 2. It is created due to scattering of the spinpolarized electrons
Influence of interface on SOT effect
Measurements of the SO torque using 2d harmonic lockin technique
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 spinpolarization 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 lockin technique?1. "Fieldlike" torque The direct measurements see hereIt can be evaluated from the asymmetric (even) component of dependence of the 2dharmonic voltage vs the inplane magnetic field, when the magnetic field is applied along the current. 2. "Damplike" torque The direct measurements see hereIt can be evaluated from the asymmetric (even) component of dependence of the 2dharmonic voltage vs the inplane magnetic field, when the magnetic field is applied perpendicularly the current. 3. Currentmodulation of anisotropy field The direct measurements see hereIt can be evaluated from the symmetric (odd) component of dependence of the 2dharmonic voltage vs the inplane magnetic field. The result is the same whether the magnetic field applied along or perpendicularly to the current.
The fields of the the spinorbit torque can be calculated from the following dependance of the 2dharmonic voltage V_{Hall,2ω} vs applied inplane magnetic field H_{x} : where ΔH_{anis,ω} is the current induced change of the anisotropy field H_{anis}; ΔH_{off,ω} is the effective magnetic field H_{FL,ω} of the "fieldlike" torque, when H_{x} is applied along electrical current; and ΔH_{off,ω} is the effective magnetic field H_{DL,ω} of the "damplike" torque, when H_{x} is perpendicularly to the electrical current; the odd and even components can be calculated as R_{wire} is the is the ohmic resistance of the wire; R_{Hall,0} is the is the Hall resistance, when a inplane magnetic field is not applied ; H_{anisot} is the anisotropy field, which can be measured directly (See here) with a high precision or from 1st harmonic with a moderate precision. The H_{anisot} can be evaluated from the following dependance of the 1dharmonic voltage V_{Hall,ω} vs applied inplane magnetic field H_{x} :
for details description of 2d harmonic lockin technique, click there to expand
The 2d harmonic lockin technique measures the current modulation of the effective magnetic field H_{DL} of "damp like" torque, the effective magnetic field H_{FL} of "field like" torque and the anisotropy field ΔH_{anis} Without electrical current, the inplane component of the magnetization M_{x} depends on the applied external inplane magnetic field H_{x} as (see here) where H_{anis} is the anisotropy field As was demonstrated above, the spinorbit torque (SOT) produces the offset magnetic field ΔH_{off} and changes the anisotropy field H_{anis} on ΔH_{anis}. As a result, the Eq.(4.1) is modified as where ΔH_{off} equals to H_{FL }when the inplane magnetic field is applied along current and ΔH_{off} equals to H_{DL }when the inplane magnetic field is applied inplane 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 M_{z} 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 M_{z} is the perpendiculartofilm component of magnetization, R_{Hall,0} is the Hall resistance when the magnetization is perpendicular to the film (M_{z} =M). When the current is modulated with frequency ω , both the ΔH_{off} and ΔH_{anis} 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 V_{Hall,2ω} of the 2d harmonic (the coefficient at cos(2ωt)) as In a lockin measurement it is convenient to use the reference voltage V_{ω} rather than reference current I_{ω} where R_{wire} 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 ΔH_{off} and is an odd function in the respect to H_{x}. The first component is proportional to ΔH_{anis} and is an even function in the respect to H_{x}. Therefore, the voltage of the second harmonic can be calculated as Eq (4.17) can be written in a symmetrical form as where
Measurement of anisotropy field from the 1st harmonic (not recommended)
When current is small, the ΔH_{off} and ΔH_{anis} can be ignored. Than, the Hall voltage V_{Hall,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 V_{Hall,ω} of 1st harmonic as
The ratio of voltage of 1st harmonic to the voltage of 1st harmonic can be calculated as
Currentinduced magnetization reversal in FeBTbElectrical current can induce spin torque or reduce the exchange interaction between localized electrons. This can change the direction of magnetization of a material.
Two currentinduced effects, which can lead to the currentinduced magnetization reversal: 1) Currentinduced Spin torque2) Currentinduced reduction of the exchange interaction between localized electrons.
Both effects occur because of transfer of delocalized (conduction) spinpolarized electrons from a point to point, which alters an equilibrium spin polarization in a material. The spin torque occurs when the delocalized spinpolarized electrons are transferred from one material to another by a drift or a diffusion current. When spinpolarized 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 currentinduced spin torque: 1)The spintransfer torque. It occurs because of transfer of spinpolarized 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 spintransfer torque depends on mutual magnetization directions of the electrons. 2) The spinorbit (SO) torque. It occurs in magnetic or nonmagnetic metals in which there are substantial spindependent scatterings. Due to spindependent scatterings a spinpolarized 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.
Questions & Answers(about systematic errors of 2nd harmonic measurements) Regarding 2nd harmonic method, I have to disagree with you. The technic is reliable if you manipulate it correctly, and like any experiment there is always the risk of an “artifact” effect not taken into account. I believe that we have reached today a conclusion on how to perform an analysis using 2nd harmonic and to take into account spurious effects.. The problem of the 2nd harmonic measurement is that it has too many independent contributions, such as 1. magnetization precession due to spin injection 2. magnetic field Hoff, which is induced by the spin accumulation 3. Current dependency of anisotropy field 4. PHE/AMR effect. Three of them can be used for magnetization reversal by an electrical current. The fact is that the 2nd harmonic measurement does not have enough data to describe its own measured data, because of a large number of different independent contributions. The new method, which I have developed, measures each contribution individually and independently of other contributions. Each contribution has a rich and interesting Physics, which can be individually optimized for an efficient magnetization reversal.  The 2nd harmonic measurement has the similar tendency as the current dependency of the magnetic field Hoff, which is induced by the spin accumulation. Therefore, it is OK to use data of the 2nd harmonic measurement in a publication, in which different tendencies are studied and discussed, and in which some systematic error is not a big issue. However, for a technology optimization, the use of a direct and more reliable measurement is better.
(about field like torque) (from Sreyas Satheesh) I had some serious doubts regarding the field like torque terms. You had mentioned the fieldlike field to be independent of the magnetization direction and to be directed along the direction of the current. However, in some of the works, I had found it to be orthogonal to the direction of the current. Ref: 1)Garello, K. et al. Symmetry and magnitude of spinorbit torques in ferromagnetic heterostructures. Nature Nanotech. 8, 587–593 (2013). 2)Miron, I. M. et al. Perpendicular switching of a single ferromagnetic layer induced by inplane current injection. Nature 476, 189–193 (2011). (about torque & spin dynamic & Quantum mechanic) There is only one torque, which is damping (or anti damping torque) of the LandauLifshitz equation. The introduction of any possible torque of different types or a different direction violates the rules of the Quantum Mechanics. The spin is a pure quantum mechanical object and the torque is the object of classical physics. Therefore, strictlyspeaking it is incorrect to use the torque for a description of the spin dynamics. However, it is still possible to use the torque for the spin dynamics, when the torque closely mimics and well approximates all features of the quantummechanical dynamics of the spin. The reason for the use of the torque is to simplify the description and understanding of the spin dynamics. However, in contrast to the classic mechanic, in which the torque may have any direction and magnitude, the quantum mechanical rules limit the torque to only one possible direction and make the torque strength dependent on the spin precession angle. The spin dynamics, as any quantum mechanical process, is described by a transition between quantum levels. In the case of the spin, the lowerenergy level corresponds to spin direction along the magnetic field (spinup) and the lowerenergy level corresponds to spin direction opposite to the magnetic field (spindown) . Only possible other quantum states of the spin are the states, whose energy is between the spinup and spindown levels and which corresponds to the spin precession at a different spin precession angle. For example, in an equilibrium the spin is in the spunup state and there is no spin precession. When there is an injection of spindown electrons, both the spinup and spindown quantum states are partially filled, which corresponds to the spin precession. The spin precession is larger when there are more spindown electrons. This quantum mechanical process can be described rather well and reasonably correctly by the damping torque (or antidamping torque) of the LandauLifshitz (LL) equation. Except for the transition between the spinup and spindown quantum levels, which is described by the damping torque of LL Eqs, I do not see any other options for a possible quantum spin dynamic and, therefore, any possibility for introduction of the other torque. For example, another possible mechanism of the spin reversal, the parametric magnetization reversal, when the magnetization direction is modulated in the resonance with spin precession, is also described by the transition between the spinup and spindown quantum levels and, therefore, the same antidamping torque of LL Eqs. You can find more explanations about this in this video (click here) There is no such thing as the fieldlike torque. However, there is a magnetic field, which is induced by spinaccumulated electrons. Since the spin accumulation is created by the current, this magnetic field can be modulated by current and can be used for the parametric magnetization reversal. One of the inplane components of this magnetic field is incorrectly associated with the damplike torque and another inplane component is incorrectly associated with the fieldlike torque. The reason for that is the symmetry of the 2nd harmonic measurement with respect to the magnetization reversal. More details about this magnetic field and its measurement you can find in this video (click here). Two papers, which you have mentioned, are two important papers, in which the fieldlike torque was introduced based on the 2nd harmonic measurement. However, the problem of the 2nd harmonic measurement is that it is influenced by too many parameters and the data of measurement of the 2nd harmonic alone is not sufficient to explain the changes of all those parameters. Now the direct measurements have clarified the situation. You can find the explanation about it in the mentioned video. In the two mentioned papers, the puzzling data of the 2nd harmonic measurement was explained by the introduction of the fieldlike torque. It is a great, but incorrect idea. It is a very natural way of the development of science. Some great, but incorrect models may exist until new data clarifies the situation.
Video
(video): Measurement of coefficient of spin orbit interaction, anisotropy field in a nanomagnet and magnetic field created by a spin accumulation.
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