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:"damplike" torque"fieldlike" torque; 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.
Origin of SpinOrbit TorqueThe creation of spinpolarization of conduction electrons by an electrical current is the origin of the SpinOrbit Torque. How an electrical current can spin polarize conduction of electrons?The spindependent scatterings spinpolarize conduction electrons (See here)
Two types of SpinOrbit TorqueInterfacetypeThe 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). BulktypeThe 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)
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 thermoactivated reversal. The modulation of the Δ changes the probability thermoactivated 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. Experiment
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.
Major 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.
Method to measure "damplike" torque and "fieldlike" torque This 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: "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: "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.

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