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

Measurement of Inverse Spin Hall effect (ISHE). Measurement of spin polarization
Spin and Charge TransportAbstract:An experimental method to evaluate the spin polarization of conduction electrons in a ferromagnetic nanomagnet is described. The method is based on a measurement of dependence of the Inverse Spin Hall effect on the spin polarization of conduction electrons. The spin polarization is modulated by applying an external magnetic field along spin direction of the spin polarized electrons. The spin polarization is increasing with the magnetic field, because the alignment of spins of the spin unpolarized electrons along the magnetic field.Papers on this topic is here PRB (2020) , PRB (2020) suppliment, arXiv (2020) and arXiv (2019) . More details about spin polarization of the electron gas is hereThe method is based on calculations of the spin pumping in a magnetic field described hereContentclick on the chapter for the shortcut(1).Spin polarization of conduction electrons and its basic properties
(2) (Main idea): Experimental method to measure spinpolarization: measurement of Inverse Spin Hall effect (ISHE) using Hall setup(2a) Hall angle in a ferromagnetic nanomagnet. AHE+OHE+ISHE contributions.(2b) Excellent fitting of measured data for FeB and FeCoB nanomagnets(2c) Experimental setup for measurement of ISHE and spin polarization(2d) Methods to check the absence of magnetic domains(2e) Dependence of ISHE and spin polarization on external magnetic field(2f). Ambiguity of Fitting(2h) Dependence of AHE and ISHE contributions on temperature and current. SpinTorque effect (SOT)(2g) Dependence of AHE and ISHE contributions on gate voltage. VCMA effect.(3). Overview of possible measurement methods of spin polarization. Direct methods.(3a) limitation for the measurement of spin polarization(3b). method 1: Hall measurement(3c) method 2: Magnetooptical measurement(3d) method 3: measurement of the total magnetic moment of conduction electrons(4). Overview of possible measurement methods of spin polarization. Indirect methods (exotic)(5).Measurement of spin polarization using Hall effect(1). Explanation in short(2). Experiment(3).What is the spin polarization?(5a). Spinpumping rate induced by a magnetic field(5b).Spin polarization of electron gas in a magnetic field(6). Other factors, which may influence the measurements(6a). Reduction of spin relaxation in a magnetic field(6b). Increase of magnetization in a magnetic field(6c). Oscillations of 2nd derivative(6d). Spin polarization vs film thickness & interface type(6e). Disruption of the measurement due to existence of static domains(6f). Independence of Inverse Spin Hall effect (ISHE) from Anomalous Hall effect (AHE)(12). Video presentations(14) Questions & Answers(a) about another possible Hall effects(b) about intrinsic magnetic field(c) about unambiguous evaluation of spin polarization(d) about a perfect alignment of magnetization in a nanomagnet(e) about a possibility of usage of the method for a ferromagnetic film and a disturbance due to a static domain6. Explaination video......... What is the spin polarization sp of the conduction electronsThe conduction electrons in a ferromagnetic metal can be divided into 3 groups (See here): (group 1) spinpolarized electrons, which spins are directed in the same direction; (group 2) spinunpolarized electrons, which spins are equally distributed in all direction; (group 3) spininactive electrons, which energy is substantially bellow the Fermi energy and which do not participate in the spin transport. The spinpolarization is the ratio of the number of spinpolarized electron to the total number of the spinpolarized and spinunpolarized electrons. (Main idea): Experimental method to measure spinpolarization: measurement of Inverse Spin Hall effect (ISHE) using Hall setup
Three contributions to Hall effect in a ferromagnetic nanomagnet(contribution 1) Anomalous Hall effect (AHE)More details see hereAHE is linearly proportional to the total spin of localized electrons M. (Dependence on external magnetic field) ~ constant The localized magnetic moments are firmly fixed in a ferromagnetic nanomagnet. They can only be switched between its two stable direction perpendicularly to film. (contribution 2) Ordinary Hall effect (OHE)More details see hereOHE is linearly proportional to the magnetic field (Dependence on external magnetic field) linear (~*H) It is induced by the Lorentz force and therefore it is linearly proportional to external magnetic field (contribution 3) Inverse Spin Hall effect (ISHE)More details see hereAISHE is proportional to the total spin of conduction electrons m. (Dependence on external magnetic field) nonlinear, the same as spin polarization The conduction electrons are two groups: spin polarized and spin unpolarized electrons. In a magnetic field along spin of spinpolarized electrons, the spins of spin unpolarized electrons are aligned along the magnetic field. As a results, the number of spin polarized electrons becomes larger, the total magnetic moment of conduction electrons increases and the ISHE contribution increases. Total Hall angle α_{HE}α_{HE} is sum of 3 contributionsTotal Hall angle α_{HE} of nanomagnet vs. external perpendicular magnetic field H: where α_{OHE} is angle of the OHE contribution, which is linear vs. H ; α_{AHE} is angle of the AHE contribution, which is independent of H ; α_{ISHE} is angle of the ISHE contribution, which is a nonlinear of H ; P_{S} (H) is the spin polarization of conduction electrons, which nonlinearly increases with H.
Excellent fitting of measured data for FeB and FeCoB nanomagnets
(fact 1) The Hall angle α_{Hall} nonlinearly increases under an external perpendicular magnetic field H for all measured FeB and FeCoB. As 2020.05, more than 100 nanomagnets have been measured (See here) (fact 2) All measured dependencies of α_{Hall} vs. H are perfectly fitted by Eq.(5) (a sum of OHE, AHE and ISHE contributions) (fact 3) The perfect fitting can be only achieved only for a smallsize nanomagnet with the perpendicular magnetic anisotropy (PMA), in which the magnetization is firmly fixed perpendicularly to the surface and does not change under the magnetic field H. (fact 4) The fitting is impossible for a large nanomagnets, in which static magnetic domains exist and the magnetization is smoothly changing under H due to movement of domain walls. (See below for details) (fact 5) The fitting is impossible for a continuos film, in which static magnetic domains exist and the magnetization is smoothly changing under H due to movement of domain walls. (fact 6) The fitting is impossible for a in an antiferromagnet or a compensated ferromagnet, in which the magnetization is changing due a re alignment of opposite spins of different magnetic super lattices. (fact 7) The Hall angle in a ferromagnet always has an ISHE contribution, which makes dependence of α_{Hall} vs. H to be nonlinear. (fact 8) The equilibrium spin polarization P_{S} in FeB and FeCoB is neither zero (P_{S} (H =0) ≠ 0) nor one (P_{S} (H =0) ≠ 1). Otherwise, P_{S} (H ) would linear. (fact 9) The measured dependence α_{Hall} vs. H is different for different materials, different samples, and different nanomagnets See (Fig.5 on the right) (fact 10) The AHE contribution α_{AHE} is slightly different for similar nanomagnets fabricated on same wafer. The ISHE contribution α_{ISHE} is sometimes different but sometimes the same for similar nanomagnets fabricated on same wafer. See below for details
Experimental setup for measurement of ISHE and spin polarization
more details see below(Measurement): The Hall angle α_{HE} was evaluated from the measured Hall voltage (See here). The Hall voltage is measured by the ”Hall bar” measurement setup.
(Main requirement): Condition of field independence of localized magnetic moments requires to use only a mono domain nanomagnet. Only a nanomagnet of small size is in the single domain state.
Methods to check the absence of magnetic domains (method 1): Smooth 1st and 2nd derivatives of vs H without any sparks or sharp steps. (See below for details) (method 2) Similarities of 1st and 2nd derivatives for different samples (See below for details) Nearly identical dependence of derivatives in different nanomagnets excludes the possibility of existence of any magnetic domains, because the movement of the domain wall should be individual and different for each nanomagnet due to different distributions of fabrication defects, edge irregularities, and a slight difference in shape of the nanomagnets
Dependence of ISHE and spin polarization on external magnetic field
E.g. the spin polarization of the conduction electrons can arise due to the influence of an external magnetic field, the mechanism considered by Landau and Lifshitz a long time agoSee details of calculation of dependence of spin polarization on magnetic field SpinPolarizationVsMagneticField.pdfExplanation about spin polarization is hereAbout the change of the spin polarization in external magnetic field is here.The spin polarization P_{S} of the electron gas is defined as a ratio of the number of spinpolarized electrons to the total number of the spinpolarized and spinunpolarized electrons: where n_{TIA} and n_{TIS }are the numbers of spinpolarized and spinunpolarized electrons, respectively. In an external perpendicular magnetic field H the spin polarization increases as where P_{S,0}_{} is the equilibrium spin polarization in absence of magnetic field (H=0); H_{S}_{} is the scaling magnetic field; From Eq.7, the dependence of the Hall angle α_{Hall} on the external perpendicular magnetic field H is described by the following nonlinear function in which α_{OHE}, α_{AHE}, α_{ISHE}, P_{S,0}_{} and H_{S}_{} are all independent of H. How Eq.7 is calculated from Eq.6?Eq.7 was calculated from a requirement of balancing of conversion rate of spinpolarized to spin unpolarized electrons and backward rate of of spinunpolarized to spin polarized electrons. See this pdf file, or below or full details here
What is the scaling magnetic fields H_{S}_{}?The scaling magnetic fields H_{S} can be calculated as (See this pdf file): where g is the gfactor, λ is a phenomenological damping parameter of LL equation, and μ_{B} is the Bohr magneton , τ_{relax} is the spin relaxation time (note about H_{S}) The nonlinearity is the key distinguish feature of the ISHE component and therefore the spin polarization is evaluated from measured nonlinearity of the Hall angle. A parameter, which defines the nonlinearity, is the scaling magnetic field H_{S}. When H_{S} becomes larger, the ISHE contribution becomes close to linear and it is difficult to distinguish it from the linear OHE contribution. For the case H_{S} ≥ 30kG the ISHE becomes nearly linearly (Fig. 4(b,d)) and this method stops working. H_{S} increases when (1) spin relaxation becomes faster and the the spin relaxation decreases; (2) spin polarization becomes close to 100% (P_{S,0}_{} ~1); (3) spin damping becomes weaker and the damping parameter λ decreases
(bad sample) In a sample with a larger number of defects and imperfections, the spin relaxation is larger and therefore H_{S} is larger
Ambiguity of Fitting
Ambiguity for the evaluation of the AHE and ISHE contributions, Ambiguity for the evaluation of the spin polarizationThere is an ambiguity for evaluation of spin polarization, ISHE and AHE contribution from of fitting of experimentally measured Hall angle α_{Hall}. Even though the fitting is perfect for 0, 1st and 2nd derivatives, the found fitting function can be identically represented by different set of parameters e.g. , . etc.
The dependence of Hall angle α_{Hall} on an external perpendicular magnetic field H in nanomagnet with PMA is described as where α_{OHE} is the angle of the OHE contribution, α_{AHE} is the angle of the AHE contribution and α_{ISHE} is the angle of the ISHE contribution, PS,0 is spin polarization at H=0 and HS is the scaling magnetic field. The ambiguity is originated from the fact that the functional dependence (S1.1) does not change when the set of three initial fitting parameters changes to a new set , which is related to the initial set as See ambiguility.pdf for details. Figure 5 shows the measured Hall angle α_{Hall} vs. external perpendicular magnetic field H and its 1st derivative . Two perfect fittings by Eq.S1.1 are shown by the solid and dashed lines. The solid lines show the case of a higher spin polarization, a higher ISHE contribution and a smaller AHE contribution. The dashed lines show the case of a smaller spin polarization, a smaller ISHE contribution and a higher AHE contribution. There is no AHE contribution for ∂α_{Hall}/∂H (Fig. S2(b)) and there is no ambiguity for ∂α_{ISHE}/∂H. It should be noted that the total Hall angle α_{Hall} and its derivatives are absolutely the same for both fittings (solid and dashed lines). The difference of between solid and dashed lines is a different representation of one identical function (Eq.S1.1) by different sets of parameters . There is no ambiguity for a fitting of experimental data by the function of Eq.S1.1 and there is only one function of Eq.S.11, which gives the best fit for each experimental data. However, this one function can represented by different sets of . The parameter H_{S} is obtained from fitting unambiguously. The 1/H_{S} describes the effectiveness of spin alignment along magnetic field. From LL equation, the alignment is faster when the Gilbert constant is larger. Also, the effectiveness is larger when the spin relaxation is smaller. The 1st and 2nd derivatives are larger when H_{S} is smaller.
(Example 1) FeB of Fig.5 (ac) Following possible fitting sets give an identical fitting function, which perfectly fits to the experimental data(fitting set 1) H_{S}=3.13 kG; P_{S,0}=0.4 ;α_{AHE} = 268 mdeg; α_{ISHE} = 886 mdeg; α_{OHE} = 0.2 mdeg/kG; (fitting set 2) H_{S}=3.13 kG;P_{S,0}=0.3 ;α_{AHE} = 395 mdeg; α_{ISHE} = 759 mdeg; α_{OHE} = 0.2 mdeg/kG; (fitting set 3) H_{S}=3.13 kG;P_{S,0}=0.2 ;α_{AHE} = 490 mdeg; α_{ISHE} = 664 mdeg; α_{OHE} = 0.2 mdeg/kG; (fitting set 4) H_{S}=3.13 kG;P_{S,0}=0.1 ;α_{AHE} = 563 mdeg; α_{ISHE} = 590 mdeg; α_{OHE} = 0.2 mdeg/kG; (Example 2) FeCoB of Fig.5 (eh) Following possible fitting sets give an identical fitting function, which perfectly fits to the experimental data(fitting set 1) H_{S}=6.18 kG;P_{S,0}=0.5 ;α_{AHE} = 657 mdeg; α_{ISHE} = 783 mdeg; α_{OHE} = 0.2 mdeg/kG; (fitting set 2) H_{S}=6.18 kG;P_{S,0}=0.4 ;α_{AHE} = 788 mdeg; α_{ISHE} = 653 mdeg; α_{OHE} = 0.2 mdeg/kG; (fitting set 3) H_{S}=6.18 kG; P_{S,0}=0.3 ;α_{AHE} = 851 mdeg; α_{ISHE} = 560 mdeg; α_{OHE} = 0.2 mdeg/kG; (fitting set 4) H_{S}=6.18 kG;P_{S,0}=0.2 ;α_{AHE} = 951 mdeg; α_{ISHE} = 490 mdeg; α_{OHE} = 0.2 mdeg/kG;
Measurement procedure
Since there is a fitting ambiguity, how it is possible to evaluate any parameters?Despite of existence of the ambiguity, an important information about spin polarization and its properties can be evaluated. (point 1) The scaling field H_{S} is evaluated unambiguously. A larger H_{S} means that either the spin polarization is larger or the spin relaxation(point 2) Even though the absolute values of ISHE and AHE contributions cannot be evaluated unambiguously, their small changes can be evaluated (e.g. when a current or gate voltage or temperature is changed)
Evaluation of small changes of α_{AHE} and α_{ISHE}. Example of Fig.23 It is assumed that for a small change of the Hall angle (1) there are no changes of H_{S} and α_{OHE}. (2) The change is due to change of α_{AHE} and α_{ISHE} (or P_{S,0})At left side of Fig.23, the measured Hall angle α_{Hall} and its first derivative are shown for a different current density. Both the 0 and 1st derivatives are different for a different current density. (fitting, step 1) Finding change of the ISHE contribution Since AHE contribution is a constant vs H, it does not contribute to the 1st derivative, the change of dα_{Hall}/dH is due a change of only the ISHE contribution. Also, all curves are clearly parallel to each other. Therefore, there is only a constant offset between them. The offset can be fitting curves to each other. In the case of Fig 23 (button raw) all curves are coincide when dα_{Hall}/dH (black line)= dα_{Hall}/dH (blue line) +0.13 mdeg/kG dα_{Hall}/dH (red line)= dα_{Hall}/dH (blue line) +0.06 mdeg/kG (fitting, step 2) Finding change of the AHE contribution When the dependence α_{Hall} vs H is corrected for the ISHE contribution, all curves of α_{Hall} (upper raw) become parallel to each other. Therefore, there is only a constant offset between them. The offset can be fitting curves to each other. In the case of Fig 23 (upper raw) all curves are coincide when α_{Hall}(black line)=α_{Hall}(blue line) +(0.7 mdeg +0.13 mdeg/kG *H) α_{Hall}(red line)=α_{Hall}(blue line) +(1.9 mdeg +0.13 mdeg/kG *H) (result of fitting) The change of the ISHE contribution dα_{ISHE}/dH (43 mA/μm^{2}) dα_{ISHE}/dH (43 mA/μm^{2}) =0.13 mdeg/kG dα_{ISHE}/dH (21 mA/μm^{2})= dα_{ISHE}/dH (43 mA/μm^{2}) =0.06 mdeg/kG The change of the AHE contribution α_{AHE}(43 mA/μm^{2})α_{AHE}(43 mA/μm^{2}) =0.7 mdeg α_{AHE} (21 mA/μm^{2})=α_{AHE}(43 mA/μm^{2}) =1.9 mdeg
(note about AHE)There is almost always a detectable change of the AHE contribution when (1) current is changed or (2) gate voltage changed or (3) between different nanomagnet of one sample (4) between different samples. (note about ISHE)For some cases the ISHE change is detectable (See Fig.23), but for some cases the ISHE change is not detectable (even when the AHE change is clearly detectable). See Fig.7 and Fig.8
Dependence of AHE and ISHE contributions on temperature and current. SpinTorque effect (SOT)
The effect of SpinOrbit torque (SOT effect) describes the fact that magnetic properties of ferromagnetic nanowire may depend on the magnitude and polarity of an electrical current flowing through the nanowire. For example, under a sufficiently large current the magnetization of the nanowire may be reversed. The direction of the magnetization reversal depends on the polarity of the current. The effect may be used as a recording mechanism for 3terminal MRAM. The origin of the SOT effect is the spin Hall effect, which describes the fact that an electrical current may create a spin accumulation.
Why are the slopes of Figs. 14 (b) and 14(c) different?A. The polarity of the spin polarization generated by the Spin Hall effect are different at opposite sides of nanomagnet (see here). In the case of a symmetrical nanomagnet, the generated spin polarization is the same at opposite sides of the nanomagnet, the total generated spin polarization is zero and there is no SOT effect. Our studied samples are asymmetric. The ferromagnetic metal is contacting the MgO at one side and the Ta at another side. As a result, the total spin polarization generated by the Spin Hall effect is nonzero. However, in this case the contributions from each interface are nearly equal. It can explain the observed substantial change of the slope for different nanomagnets on the same wafer and the different slope polarities for the FeB and FeCoB samples. The enlargement of structure asymmetry and optimizing interfaces may increase the current induced change of sp.
Dependence of AHE and ISHE contributions on gate voltage. VCMA effect.
The VCMA effect describes the fact that in a capacitor, in which one of the electrodes is made of a thin ferromagnetic metal, the magnetic properties of the ferromagnetic metal are changed, when a voltage is applied to the capacitor. For example, under an applied voltage the magnetization direction of the ferromagnetic metal may be changed or even reversed. This magnetizationswitching mechanism can be used as a data recording method for lowpower magnetic random access memory and all metal transistor. Until now the physical origin of the VCMA effect has not been clarified. However, several possible physical mechanisms have been discussed. (See here for more details)
VCMA and Tunnel magnetic Resistance (TMR) effects. It would be interesting to correlate this gate tuning of spin polarization with the biasdependent TMR experiments which have been traditionally explained with barrier height modulation.A. I guess there are several contributions to the biasdependence of TMR. The voltage dependence may have some contribution. However, it should be a contribution, which is polarity dependent. The voltagecontrol change of the spin polarization always changes its sign, when the voltage polarity is reversed. I have checked many samples already. The spin polarization always linearly increases under a negative gate voltage and it always linearly decrease under a positive gate voltage.
Overview of possible measurement methods of spin polarization. Direct methods.
How to measure the spin polarization sp of the conduction electrons?It requires to measure a material property, which depends on the spin polarization (i.e. Hall angle, a MO constant, sample magnetization, tunnel resistance etc). In order to evaluate the spin polarization, the spin polarization is changed in controllable way and the dependence of the material property on the change of the spin polarization is measured. From a fitting of measured data to the known dependence of the material property on the spin polarization , the absolute value of the spin polarization is evaluated. In the belowdescribed method, the dependence of the Hall angle on an applied magnetic field is measured. Similarly, the measurements of the dependence of the magnetooptical (MO) constants or the tunnel resistance or the magnetization measured by a magnetometer on modulated spin polarization are also the effective methods to measure the spin polarization of the conduction electrons. How to change the spin polarization?The spin polarization can be changed by applying an external magnetic field, by applying the elastic stress, by illuminating the sample by circular polarized light, by injecting spinpolarized electrons from another metal. In the below described method, the spin polarization is modulated by applying an external magnetic field H_{ext} along the spin direction of the spin polarized electrons. The spin polarization increases when a magnetic field is applied along spin direction of the spinpolarized electrons.(The reason why it increases): The spins of spin unpolarized electrons are aligned along the magnetic field. As a result, the number of spin polarized electrons becomes larger. The process of spin alignment is described by the Gilbert damping parameter in the Landau Lifshitz equations (See here)
Why magnetic properties depend on the spin polarization of the conduction electrons? Which magnetic properties specifically depend on the spin polarization?When the spin polarization sp of the conduction electrons changes, the amount of the spinpolarized conduction electrons changes and therefore the total magnetic moment of the conduction electrons changes as well. Nearlyall magnetic properties depends on the the total magnetic moment of the conduction electrons. Specifically the Hall angle, magnetooptical constant, the total magnetic moment and all magnetotransport properties are changed when the spin polarization is changed. Main condition & limitation for the measurement of spin polarizationMain condition is that under applying of an external magnetic field only the spin polarization of the conduction electrons changes, but all other parameters remain unchanged. It means that (condition 1) The magnetic field is applied along magnetization and the spin direction of the spin polarized electrons. The magnetization direction remains along magnetic field during the whole measurement. (condition 2) There are no any magnetic domains. Over whole sample, the magnetization is directed along the external magnetic field for any used values of the magnetic field. (condition 3) The temperature, bias current etc. remain unchanged during the whole measurement In any measurement of the spin polarization, the change of a material parameter is traced under a change of spin polarization of conduction electrons. The simplest way to change the spin polarization of the conduction electrons is to apply an external magnetic field. However, it is possible to evaluate the spin polarization only if one parameter, the spin polarization changes under the external magnetic field and all other parameters remains unchanged. E.g. in the case when there is a movement of magnetic domain under an increasing magnetic field, the evaluation of the spin polarization is impossible. Direct measurement methods of spin polarizationWhy these methods are direct?A. All belowdescribed measurement methods are based on a measurement of the magnetization of the conduction electrons. The magnetization of the conduction electrons is the sum of the magnetization of each spinpolarized electrons and therefore it is linearly proportional to the number of the spin polarized conduction electrons. The magnetization of the conduction electrons is the main feature, which characterizes and distinguish the spinpolarized conduction electrons. Therefore, the measurement of the magnetization of the conduction electrons gives the most direct evaluation of the number of the spinpolarized electrons.Additional merit of a direct measurement of sp is that the magnetization of the conduction electrons has distinguished symmetry properties, which can be verified and a possible systematic error can be avoided.
Spin polarization & symmetryA general form of dependency of material parameters on different magnetic properties can be found from symmetry rules. The most of important magnetic properties can be found from the TCP symmetry. In the following more simplified symmetry rules are used A measurement of the spin polarization required that the sample is always fullysaturated without any magnetic domains. It means that after the external magnetic field His reversed, the magnetization, the spins of d electrons and spin of conduction electrons are fully reversing following H. From the symmetry, any material parameter, which reverses its sign, when the external magnetic field is reversed can be calculated as where a,b,c are constants, which sign is not reversed under reversal of H, S_{cond} is the total spin of the conduction electrons, and S_{d} is the total spin of the localized delectrons Magnetic parameters, which reversed with reversal of external magnetic field: (1) magnetization M or the total magnetic moment of localized delectrons M_{d}, (2) the total magnetic moment of spin polarized electrons M_{cond }; (3) the Hall voltage and the Hall angle α_{Hall}; (4) magnetooptical constants: Faraday rotation angle θ_{Faraday} , the Kerr rotation angle, constants of magnetic circular dichroism (MCD). Only magnetic parameters, which reversed their sign with reversal of external magnetic field, can be used for evaluation of the spin polarization
Measurement of spin polarization. (Method 1): Hall measurementThe Hall angle α_{Hall} is asymmetrical with respect to H reversal. Therefore, it defines as a difference of α_{Hall} measured at opposite H: α_{Hall} can be linearly proportional to material parameters of the same symmetry where M_{d} is the total magnetic moment of localized delectrons and M_{cond} is the total magnetic moment of conduction electrons. The total moment of all spinunpolarized electrons is zero. Since all spins spinpolarized electrons are directed in one direction, M_{cond} =μ_{cond} ·n_{TIA} , where n_{TIA} is the number of spinpolarized conduction electrons, μ_{cond} is the magnetic moment of one conduction electron.
Therefore, Eq(3.2) is simplified as n_{d} is the number of spinactive localized delectrons and μ_{d} is the magnetic moment of one localized d electron. By definition of the spin polarization sp (See here) is percentage of spinpolarized electron among conduction electrons: where n_{cond} is the total number of spinpolarized and unpolarized electrons. The magnetization M of a ferromagnetic metal is defined as Substitution of Eqs. (3.0) and (3.0a) gives or where β_{ISHE} is redefined as β_{ISHE} =β_{ISHE}· μ_{cond} ·n_{cond} The 1st term β_{OHE}·H describes proportionality of α_{Hall} to external magnetic field. This effect is called the ordinary Hall effect. The 2nd term β_{AHE}·M describes proportionality of α_{Hall} to magnetization of the ferromagnetic metal. This effect is called the Anomalous Hall effect. The 3d term β_{ISHE}·sp describes proportionality of α_{Hall} to the spin polarization of the conduction electrons. This effect is called the Inverse Spin Hall effect.
Measurement of spin polarization.(Method 2): Magnetooptical measurementA measurement of the Faraday rotation angle θ_{Faraday }or Kerr rotation angle θ_{Kerr} or difference in absorption between left and right circular polarized light (MCD effect) can be used to evaluate the spin polarization of conduction electrons. The following explains how the spin polarization can be evaluate from a measurement the MCD absorption. Similar evaluation can be done from a measurement of Faraday rotation angle θ_{Faraday} or Kerr rotation angle θ_{Kerr}. A circular polarized photon has spin equals 1. Therefore, a spin polarized electron can be excited only, for example, by left, but not by right circular light as it required by the Selection rules for electronic transition. In the case when there are spinpolarized electrons, the absorption of left and right circular polarized becomes different. This effect is called the magnetic Circular Dichroism.(See here and here and here) The difference in absorption coefficient α_{MCD} between left and right circular polarized light changes its sign, when spin direction of the spin polarized electron is reversed. When spins are reversed by external magnetic field H, α_{MCD} can be measured as
Each spinpolarized electrons contribute to α_{MCD}, therefor α_{MCD} is proportional to the number of spinpolarized electrons. Since the symmetry and size of the localized delectrons and conduction electrons are very different, therefore the interaction efficiency of these electrons with photons is very different and these spinpolarized electrons contribute differently to α_{MCD}. Additionally, there is a paramagnetic contribution due to the Zeeman splitting. Eq.(3.12) can be obtained also the symmetry fact (like that α_{MCD} Substitution of Eqs. (3.0) and (3.0a) gives where we redefine
Measurement of spin polarization.(Method 3): Measurement of Magnetic momentThe magnetic moment μ_{total } of a sample can be measured by magnetometer (SQUID or VCM) Similarly, the magnetic moment should be reversed, when the external magnetic field is reversed The total measured magnetic moment μ_{total }is the sum of magnetic moments of delectrons, magnetic moments of the spinpolarized conduction electrons and induced magnetic moment (paramagnetic type) Substitution of Eqs. (3.0) and (3.0a) gives Direct Measurement of sp. Step 2. Change of spin polarizationrequirements of the measurement of sp: (requirement 1): The measured material parameter should be dependent on spin polarization sp of conduction electrons (magnetization of conduction electrons). (requirement 2): The spin polarization sp should be changed in controllable way. Therefore, the dependence of the material parameter on the spin polarization can be measured and the spin polarization sp can be found from a fitting. Controllable change of spin polarization sp:change of spin polarization sp. (Method 1)Applying a magnetic field along the spin direction of spinpolarized electrons. Details of only this method are described below.It is the most simple method and the most favorable method. An external magnetic field aligns the spin of the spinunpolarized electrons. As a result, the spin polarization of the conduction electrons increases (See details here) change of spin polarization sp. (Method 2)Applying a magnetic field perpendicularly to the spin direction of spinpolarized electrons. (Hanle effect) The perpendicular magnetic field reduces the spin polarization. Additionally, the direction of the spin polarization slightly tilts towards the magnetic field. Ferromagnetic metal: It is difficult to use this method for a ferromagnetic metal because (1) There is a substantial intrinsic magnetic field inside a ferromagnetic metal. The external magnetic field should be comparable with this field in order to produce any changes. (2) The magnetization and therefore the spin polarization turns towards the magnetic field. Nonmagnetic metal: change of spin polarization sp. (Method 3)Spin injection The
Overview of possible measurement methods of spin polarization. Indirect (exotic) methods.Why these methods are indirect and exotic?A. All these methods are based on a measurement of complex dependencies of a material property on the spin polarization of the conduction electrons, like the tunneling resistance, spindependent photoluminescence, spindependent electroluminescence. These methods are indirect, because the measurement material properties depends not only on sp, but on many other parameters. It is very hard to distinguish the material property dependence on sp from similar dependencies on another material properties. E.g. additionally spdependence the tunneling magneto resistance depends on shape, size and symmetry of electron wavefunction in the tunnel barrier and in electrode near interface. All these these parameters may or may not be spindependent.. These methods are exotic, because the physics of the spindependent tunneling and the spindependent photoluminescence are complex and are not fully understood yet. It is very speculative when the spin polarization is estimated from a fitting of veryapproximate and suggested dependencies of the complex properties on the spin polarization. Note: Using an indirect measurement of the spin polarization, it is always possible to obtain some number of spin polarization. However, the data should verified and calibrated using a more reliable direct measurement method.
Measurement 1: From tunneling magnetoresistance (TMR) using Julliere formula M. Julliere, Phys. Lett. A 54, 225 (1975).The TMR ratio of a magnetic tunnel junction (MTJ) is proportional to the spin polarization of its electrodes. In the most simplified case, the TMR ratio Estimated spin polarization for FeCoB: 7090 % merits: Simplicity of measurements demerits: systematic error due to substantial limitations and approximations of Julliere formula (e.g. )
Measurement 2: from Andreev reflection R. Meservey and P.M. Tedrow, Phys. Rep. 238, (1994). R.J. Soulen Jr RJ et al, Science 282(1998).The spin polarization is evaluated from the tunneling properties of a superconductormetal contact. Estimated spin polarization for FeCoB: 3050 % merits: ??? demerits: (1) limitation of only a low temperature measurement; (2) a systematic error due to simplifications and approximations for calculations of the transport through a superconductormetal contact; (3) clear under estimation of the value of the spin polarization;
Measurement 3: using spindependent photoluminescence, spindependent electroluminescence and spin LED C. AkuLeh,et al .Phys. Rev. B 76, 155416 (2007). B.T. Jonker, Proc. IEEE 91, 727 (2003).The spin polarization is evaluated from the amount of circularpolarized light emitted from a semiconductor, in which spinpolarized current is injected note: with some limitations, the the spindependent photoluminescence can be considered as a direct measurement of the spinpolarizationEstimated spin polarization for FeCoB: 6095 % merits: (1) spatial distribution of spin polarization can be checked. demerits: (1) incorrect description of spin injection can cause a systematic error; (2) incorrect description of complex features of spinlight interaction can cause a systematic error; (see here and here); Measurement 4: from spininjection and spindetection experiment (nonlocal spindetection) In the nonlocal spin detection experiment, the spinpolarized electrons are injected by a pair of ferromagnetic electrodes and detected by another pair of ferromagnetic electrodes. From dependence of the spin detection voltage of different parameters of the measurement and device structure, a very rough estimation of the injected spin polarization in the paramagnetic metal and the spin polarization of ferromagnetic electrodes can be obtained. merits: (1) simplicity of experiment demerits: (1) all features of the spin injection and the spin detection have not been understood yet; (2) both the spin injection and the spin detection substantially depend on the quality and chemistry of interface between nonmagnetic and magnetic materials.
Method 1: Hall measurement
Merits of this measurement method:Merit 1: Simplicity of measurements Merit 2: ability for measurement of spinpolarization even in a nano sized object;
Main idea:The spin polarization is evaluated from the measured dependency of the Hall angle on an applied perpendicular magnetic field
Experimental Fact 1: Hall rotation angle increases under applied external magnetic field (See Fig.3 below)Formulas:
Spin polarization sp in a magnetic field is calculated as where sp_{0 }is the spin polarization in absence of an external magnetic field, H_{pump} is the pumping magnetic field (a material parameter) The measured Hall angle α_{Hall} is the sum of the Hall angle α_{OHE} of the ordinary Hall effect, the Hall angle α_{AHE} of the anomalous Hall effect and the Hall angle α_{ISHE} of the inverse spin Hall effect. It can be calculated as where α_{AHE} is the Hall angle of Anomalous Hall effect, α_{ISHE,0} is the Hall angle of the Inverse Hall effect in the absence of an external magnetic field, β_{OHE} is the Hall coefficient of ordinary Hall effect; H_{⊥} is the magnetic field applied perpendicularly to the film.
Can experimentally observed increases of the Hall angle be due to the ordinary Hall effect ?A. The contribution of the ordinary Hall effect (OHE) depends linearly on the magnetic field . The experimentally measured dependence is nonlinear and it has three contributions the linear contribution due to OHE, constant contribution due to AHE and non linear contribution due to ISHE with respect to magnitude of external magnetic field H. Which parameters influence the Hall angle α_{OHE} of ordinary Hall effect (OHE), the Hall angle α_{AHE} of Anomalous Hall effect (AHE), the Hall angle α_{ISHE} of the Inverse Spin Hall effect (ISHE) ?A. The α_{ISHE} is linearly proportional to spin polarization (sp) of electron gas, the magnetization and the strength of the spinorbit (SO) interaction (See details here). The strength of the SO interaction mainly depends on the ratio of the holes and electrons in a metal. Therefore, it can be assumed that it does not change in an external magnetic field. In this case the Hall angle α_{AHE} can be calculated as
where σ_{xx}, σ_{xy} are diagonal and offdiagonal components of the conductivity tensor; a is the proportionality constant;M_{⊥} is outofplane component of magnetization
Parameters, which makes Hall angle α_{AHE} to be dependent on the magnetic field:From Eq.(1.11), three contributions can be identified.(1) major contribution: an increase of spin polarization due to increase of spin pumping The spin polarization increases in a magnetic field due alignment of spins of conduction electrons along the magnetic field (See Fig.1b).(2) minor contribution: an increase of spin polarization due to decrease of spin relaxation An external magnetic field may suppress some some spin relaxation mechanisms (See below for details). The reduction of the spin relaxation enlarges the spin polarization (See here for details)(2) minor & major contribution: a change of magnetization M_{⊥} An external magnetic field may suppress some some spin relaxation mechanisms (See below for details).Experiment
The spin polarization of a nanomagnet was evaluated using a Hall measurement The FeB, FeCoB and FeTbB films were grown on a Si/SiO2 substrate by sputtering. A Ta layer was used as nonmagnetic adhesion layer. A nanowire of different width between 100 and 3000 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 3000 nm were fabricated. When it is not mentioned, the Hall angle is measured at current density of 5 mA/mm^{2}. The a_{Hall} in the ferromagnetic metal was evaluated as where s_{ferro} , s_{nonMag} are conductivities of ferromagnetic and nonmagnetic metals; t_{ferro} , t_{nonMag} are their thicknesses, V_{Hall} is the measured Hall voltage, I is the bias current and R,L,w are the resistance, length and width of the nanowire, correspondingly. Each of a_{ISHE}, a_{AHE} and a_{OHE} reverse its sign, when M and H are reversed. In order to avoid a systematic error due to a possible misalignment of the Hall probe, the Hall angle was measured as What is the spin polarization?All conduction electrons in a ferromagnetic metal can be divided into groups of spinpolarized and spinunpolarized electrons. The spin polarization sp of the electron gas is defined as a ratio of the number of spinpolarized electrons to the total number of the spinpolarized and spinunpolarized electrons: where n_{TIA } and n_{TIS} are the numbers of spinpolarized and spinunpolarized electrons, respectively. How to divide all conduction electrons into the group of spinpolarized and spinunpolarized electrons?Detailed explanation is here. Explanation in short: In fact, all conduction electrons in a ferromagnetic metal are divided into 3 groups: of spinpolarized, spinunpolarized electrons and spininactive electrons. In the group of the spinpolarized electrons, the spins of all electrons are in the same direction. In the group of the spinunpolarized electrons, the spins are distributed equally in all directions. Additionally, there are some electrons, which are "spininactive". A pair of these electrons with opposite spins occupies one quantum state. The occupation of quantum states by the electrons of both the spinpolarized and spinunpolarized groups is one electron per a state. As a result, the spin of each state is 1/2 and the spin direction for each quantum state is defined. The spin direction represents the direction of the local breaking of the timeinverse symmetry for the state. When a quantum state is occupied by two conduction electrons of opposite spins, the spin of such quantum state is zero. As a result, the spin direction of this state cannot be defined and the electrons occupying this state are "spininactive". The electrons, which energy is substantially below the Fermi energy, mainly belong to this group. For example, nearly all of the “deep level” electrons belong to this group. In contrast, the energy of electrons of the groups of spinpolarized and spinunpolarized electrons is distributed mainly nearly the Fermi energy. See details here.
Spin polarization in an external magnetic field
Main idea: The spin polarization is evaluated from the measured dependency of the Hall angle on an applied perpendicular magnetic field
Why in one metal the spin polarization is smaller and in another metal is larger? What does influence the spin polarization?The spin polarization sp of the electron gas is defined as a ratio of the number of spinpolarized electrons to the total number of the spinpolarized and spinunpolarized electrons. The amount of electrons in each group is determined by a balance between the spin pumping and the spin relaxation. The spin pumping is the conversion of electrons from groups of spinunpolarized electrons into the group of the spinpolarized electrons. The spin relaxation is the conversion in the opposite direction. Detailed explanation about spin polarization is here.
Spinpumping rate: The spin pumping describes the conversion of electrons from the group of the spinpolarized electrons into the group of spinunpolarized electrons. The conversion rate of the spinpumping is described as where t_{pump} is the spin pumping time, n_{TIA } and n_{TIS} are the numbers of spinpolarized and spinunpolarized electrons, respectively. Details about different mechanisms of spin pumping is hereSpinrelaxation rate: The spin damping describes the conversion of electrons from the group of the spinpolarized electrons into the group of spinunpolarized electrons. The conversion rate of spinrelaxation can be described as where t_{relax} is the spin relaxation time. Details about different mechanisms of spin relaxation is hereSpin polarization: The spin polarization sp of electron gas can be found from the condition that in an equilibrium there is a balance between the spin pumping and the spin relaxation, which is described by the condition: How to increase the spin polarization?(1) Increase of spin pumping rate(2) Decrease of spin relaxation rate
Why does the spin polarization become larger in a magnetic field
There are two reasons: (1) major reason: increase of spin pumping in a magnetic fieldIt is due to alignment of spins of spinunpolarized electrons along a magnetic field (see Fig. 3) (2) minor reason: suppressing of spin relaxation by a magnetic field
Spinpumping rate induced by a magnetic field: In a magnetic field, the spins of spinunpolarized electrons aligns along the magnetic field due to the precession damping (See here). However, scatterings quickly re aligns spins of electrons into two groups of spin polarized (all spins are one direction) and spinunpolarized electrons (spins are equally distributed in all directions). (details See here) As a result, there are more spinpolarized electrons. The spinpumping (See Eq.19 below) where t_{H,pump} is the spin pumping time in a magnetic field. The spin pumping time in a magnetic field can be calculated as spin polarization of electron gas in a magnetic field The spin polarization sp of electron gas can be found from the condition that in an equilibrium there is a balance between the spin pumping and the spin relaxation, which is described from Eqs (1.2),(1.5) (1.6) by the condition: Substitution Eqs.(1.2) (1.6) Eq(1.7) into Eq.(1.1) gives the spin polarization sp of electron gas in a magnetic field (see more details here) as where sp_{0} is the spin polarization in absence of an external magnetic field (a material parameter) , which is calculated as H_{pump} is the pumping magnetic field (a material parameter), which is calculated as
Factors, which may influence the measurementsIn described measurements it was assumed that the increase of the Hall angle in a magnetic field is only due to the increase of spin polarization, which is induced by the spin pumping induced by the magnetic field. However,there are additional effect, which may also contribute to the increase of the spin polarization: The effects, which may cause the increase of the Hall angle in a magnetic field:(1) Reduction of spin relaxation; (2) Increase of magnetization Ignoring of these effect may cause a systematical error.Reduction of spin relaxation in a magnetic field
Is it possible that the spin relaxation is reduced in an external magnetic field? Does it influence the spin polarization?Absolutely. An external magnetic field reduces the spin relaxation. This reduction should be included into an evaluation of the spin polarization. An external magnetic field reduces all mechanism of the spin relaxation: reduction of mechanism 1 : the spindependent scatterings: Even spin may be rotated after a spindependent scattering, in a magnetic field it quickly rotates back to be along the magnetic field. Therefore, the spin dependent scattering does not lead to the increase of the spin relaxation. reduction of mechanism 2 : incoherent spin precession in a spatially inhomogeneous magnetic field: A large magnetic field levels out and fully compensates any possible inhomogeneities of internal magnetic field in a metal. A magnetic field reduces or even this type of the spin relaxation.
Increase of magnetization in a magnetic fieldCan direction or magnitude of the magnetization change in an external magnetic field.? How does it influence the spin polarization measurement?A. It does influence very much. (1). Magnetization inclination. Keeping the magnetization in the same direction is very important for these measurements. The sample geometry and the scan range of magnetic field should chosen to avoid any (even slight) magnetization inclination or domain movement. (2) Change of magnetization magnitude. In a magnetic field the spin polarization sp increases. The increase of sp may cause the increase of the magnetization as well. The amount of increase of the magnetization is difficult to measure. The correct measurement of such increase is still a challenging task.
Oscillations of 2nd derivative
The physical origin of the oscillations is not yet understood.
The oscillations exists for all measured samples. The period and amplitude of the oscillations are different from a sample to sample
Spin polarization vs film thickness & interface type
Disruption of the measurement due to existence of static domains
The important requirement for our measurements is the mono domain nature of the specimen which ensures stability of the localized moment in the external magnetic field. This stability is very important to exclude any possible contribution to the Hall effect due to a realignment of localized moments. A small size of the nanomagnet ensures the absence of static magnetic domain and the stability of the localized moments. In our measurements the existence of static domains can be clearly identified. Figures 6 (a,b,c) shows measurement of the Hall angle ant its 1st derivative for a relatively large (3µm x 3µm) FeCoB nanomagnet. At magnetic field less than 1.5 kG, there is a sparklike change of the 1st derivative indicating existence of static magnetic domains in this range of the magnetic field H. The spark like change is due the fast realignment of localized magnetic moments during a nucleation of static domain and movement of a domain wall across the nanomagnet (See more about static magnetic domains here). Additionally, there is a dependence of the 1st derivative on the scan direction of the magnetic field. The yellow and green lines are different from the red and blue lines for H <1.5 kG. It is an indication of an existence of a thermally activated mechanism, which is probably a mechanism to overcome obstacles during movement of a domainwall across the nanomagnet. Figures 6 (d,e,f) shows the same data for a smaller nanomagnet without static domains. All dependencies are smooth without sparks. All 4 lines perfectly coincide with each other.
See next part for more proofs.
Independence of Inverse Spin Hall effect (ISHE) from Anomalous Hall effect (AHE)
(proof 1:) Measurement of nearly identical nanomagnets
Figure 7 shows the data measured for three different nanomagnets fabricated at different parts of the same wafer. The 1st and 2nd derivatives (Fig.7 (b,c)) are nearly the same for these nanomagnets indicating that the ISHE contribution and consequently the spin polarization are the same in these nanomagnets. However, the α_{HE} (Fig.S3 (a)) and consequently the α_{AHE} are different for each nanomagnet. It shows that there is a parameter, which influences the AHE contribution, but does not influence the ISHE contribution and the spin polarization, and this parameter is different for each of these three nanomagnets. Additionally, the perfect matching of derivatives for different nanomagnets excludes the possibility of existence of any magnetic domains, because the movement of the domain wall should be individual and different for each nanomagnet due to different distributions of fabrication defects, edge irregularities, and a slight difference in shape of the nanomagnets.
(note) In general, there are nanomagnets with different α_{ISHE} and consequently of different spin polarization on the same wafer. That depends on the quality of growth and nanofabrication. See details for each sample here. Fig.7 shows a general case for a relativelygood sample (a low number of defects and shape irregularities).
(proof 2:) Measurement at different current density
Figure 8 shows the data measured at a different current density. The used current density is relatively high and the heating of the nanomagnet is expected. The heating was confirmed from the measured reduction of conductivity of the nanomagnet. The heating is not sufficient to change the ISHE contribution and therefore the spin polarization (Fig.8(b,c)). However, the AHE contribution is clearly reduced with the heating (Fig.8(a)). It again confirms the existence of a parameter, which influences the AHE contribution, but does not influence the ISHE contribution and the spin polarization.
(note) In general, there are nanomagnets, in which t α_{ISHE} depends on the current density. See details for each sample here. Fig.8 shows just one typical case.
See the influence of Hall angle by the Spin Orbit torque (SOT) here. The SOT describes the dependence on current polarity
Video presentationsMy presentation on this topic no MMM 2020 conference
Questions & AnswerCan the spin polarization sp of the conduction electrons be created in a nonmagnetic metal by an external magnetic field?Yes. Can you see a nonlinear dependence of the Hall angle vs an external magnetic field in a nonmagnetic metal?Not yet. For calibration of my magneto transport probe, I am measuring the Hall effect in a thick (30 nm) Ta or Ru nanowire. The dependence of the Hall angle vs H is very linear. It seems to be the major contribution is from the Ordinary Hall effect. In nonmagnetic metal, the H_{S} is substantially larger than in a ferromagnetic metal due to faster spin relaxation rate. As a result, 1st derivative of ISHE is much smaller than the OHE contribution (See Fig.4d) and therefore this makes it difficult to distinguish between the ISHE and OHE contributions. Do you explore any other possible explanations for the Hall angle observed in their experiments? You state explicitly that the ISHE is “one possible candidate for such a mechanism”, i.e. a mechanism contributing to the Hall angle beyond the simple model of “constant + linear dependence”, you never even mention other possible candidates, which must exist in the large family of Hall effects. If there are other possible mechanisms which could lead to similar experimental signatures in your measurements?
(about another possible Hall effects) The symmetry of the Hall effects limits possible contributions to a few possible contributions. Except AHE, OHE, ISHE, any other contribution should be at least 2 orders smaller. This fact is explained from the symmetry of the Hall effect. Any possible contribution to the measured Hall angle should obey two symmetry requirements: (1st symmetry requirement) : The Hall angle is linearly proportional to the electron current and therefore to the number of the conduction electron. The Hall angle should depend linearly at least on some property of conduction electron. (2d symmetry requirement) The Hall angle reverses its polarity when all the external magnetic field, the total spin of localized electrons and the total spin of conduction electrons are reversed. There is no other object or field in nanomagnet, which has the similar time reverse symmetry. The Hall angle should depends linearly on each of theses objects or a 3d order of their product. Since the Hall angle is small (< one degree), a 3d order product is very small. There are only three possible contributions, which satisfies these symmetry requirements: (1st contribution) Ordinary Hall effect, which is linearly proportional to external magnetic field H and wave vector of a conduction electron. The origin of this contribution is the Lorentz force. (2nd contribution) Inverse Spin Hall effect, which is linearly proportional to the total spin of the conduction electrons S_{conduct}_{} and therefore to the number of the spin polarized conduction electrons. The origin of this contribution is the spin dependent scatterings of spin polarized electrons such as a screw scattering, side jump scatterings and a scattering across an interface. The probability of these scatterings depends on the spin direction of conduction electrons, but not on the spin of the localized electrons. (3d contribution) Anomalous Hall effect, which is linearly proportional to the total spin of localized electrons S_{local}_{} and the orbital moment of conduction electrons. Since spins of localized electrons is already aligned along the easy axis, the contribution is independent of the external magnetic field applied along the easy axis. The symmetry restrictions, which limit the possible contributions to the Hall effect, can be understood as follows. A reversal of a sufficiently large external magnetic field causes the reversal of spins of localized, reversal of spins of conduction electrons and , as a result, the reversal of sign of the Hall angle α_{Hall} where H is external magnetic field,S_{local}_{} is the total spin of localized electrons and S_{conduct}_{} is the total spin of conduction electrons From Eq.(8.1), the Hall angle should be the linear sum of each time reversal parameter, which is describes as Next order contribution can only be of 3rd order like H^{3}_{} H3, H^{2} · S_{conduct}_{} , H^{2}·S_{local}_{} etc., which are much smaller and can be ignored.Fig.6 shows how a hysteresis loop of measured Hall angles is decomposed into the sum of different contributions.
There are many types of Hall effects. Why do you use only 3 mechanisms to explain your measurements?A nanomagnet with a strong PMA is a unique object, in which individual contribution of only 3 types of the Hall effects can be separated. Nowadays many new names of the similar Hall mechanism have appeared and this is a confusing issue. In fact, there are not so many "the Hall effects", which may exist. The symmetry does not allow it. The different mechanisms should be distinguished and separated only based only on some experimentally distinguishable property or effect. There are not so many of those in a nanomagnet. The first distinguish feature of Hall effect is the free parameter, on which the Hall effect depends. The free parameter should be magnetic. In case of our nanomagnet, there are only 3 such parameters: (parameter 1): external magnetic field H; (parameter 2): the total spin S_{local} of localized electrons; (parameter 3) the total magnetic moment S_{conduc} of conduction electrons. Only these three magnetic parameters can be changed independently. Any external distortion of the magnetic system of nanomagnet can always be described as a sum of changes of these 3 free parameters. According to these 3 free magnetic parameters, the 3 mechanisms of the Hall effect can be distinguished: OHE, AHE and ISHE. The second distinguish feature of Hall effect is the even or odd symmetry with respect to a reversal of magnetic field when (H, S_{local }, S_{conduc}) > (H,  S_{local},  S_{conduc}). In our standard DC measurement, we measure only the odd contribution: a_{OHE}(H)=  a_{OHE}(H) a_{AHE}(S_{local})=  a_{AHE}(v) a_{ISHE}(S_{conduc})=  a_{ISHE}(S_{conduc}) Since are independent free parameters, the total even Hall angle can be divided into a sum of three contributions: a_{Hall}(H, S_{local }, S_{conduc})= a_{OHE}(H)+ a_{AHE}(S_{local}) + a_{ISHE}(S_{conduc}) Next order contribution can only be of 3rd order like H^{3}, H^{2}∙ S_{conduct}, H^{2}∙S_{local}, etc., which are much smaller and can be ignored. The independence of parameters means that we can change one parameter without affecting another parameter. For example, we can inject externally the spin polarized conduction electron without affecting the spins of localized electrons. Or we can apply an external magnetic field plus the spin injection of the opposite spins. As a result, the spin polarization is not changed. Even types of Hall effect. (not measured in our standard DC measurement setup) For even type Hall effects: a_{Hall}(H, S_{local }, S_{conduc})= a_{Hall}(H, S_{local }, S_{conduc}) Obviously, the even type Hall effects is proportional to H^{2}, H∙ S_{conduct}, H∙S_{local} etc. An example of the even Hall effect is socalled the Planar Hall effect. Since we do not measure the even contribution, we do not discuss it. In absence of other independent free magnetic parameters, the OHE, AHE and ISHE are only possible odd mechanisms of the Hall effect. For example, let us consider the Hall effect in a nonmagnetic metal covered by a ferromagnetic isolator. The magnetization of the isolator may be considered as a free parameter and a new mechanism of the Hall effect can be assigned. However, it is incorrect to do it. The magnetization in the isolator aligns the spins of localized and conduction electrons in the nonmagnetic metal and therefore induces the AHE and ISHE. Even one give a new name of this Hall effect, it is still AHE + ISHE.
The phenomenological model for the ISHE includes the spinpolarization of the electrons as a crucial parameter (as it should for a model describing the ISHE). Thus, it could be helpful, if you could provide information regarding expected values for the spin polarization in their samples, ideally as a function of magnetic field, e.g. based on magnetooptical experiments ?(about unambiguous evaluation of spin polarization) At the moment, I do not want to speculate about the absolute value of spin polarization. However, it might be possible to trace whether the spin polarization increases or decreases from a sample to a sample or under a gate voltage (VCMA effect) or when current polarity is changed (SOT effect). The change of the spin polarization can be traced from the change of the 1^{st} and 2^{nd} order derivatives The 1^{st} and 2^{nd} order derivatives should be larger when the equilibrium spin polarization is smaller. E.g. if the equilibrium spin polarization is 95 %, it can increase under a magnetic field maximum to 100 % and therefore the change is small. In contrast, if the equilibrium spin polarization is 5 %, the change to 100 % is large and . The 1^{st} and 2^{nd} order derivatives should be larger. The analysis is still no as simple, because derivatives also depend on the value of the spin relaxation. (See Fig.4 )
Another question regards the relative strength of external magnetic field and internal effective field caused by the magnetization of the sample. As the change in the Hall angle over the entire field range is significantly smaller than the value of the Hall angle at zero field, I would assume that the internal magnetic field is much stronger than the external field and thus the spinpolarization of the electrons is only mildly affected As a function of H. ?(about intrinsic magnetic field) There is an intrinsic magnetic field in a nanomagnet and it is influence the spin polarization of nanomagnet. However, it is difficult to estimate the value of the intrinsic magnetic field by this method, because additionally to the intrinsic magnetic field, the equilibrium spin polarization depends on the rate of the spin relaxation, the strength of the spd exchange interaction between conduction and localized electrons and the scattering rate between conduction and localized electrons. In fact, one fitting parameter of experimental data is some sort of a magnetic field, which we assigned as H_{S}. The fitting gives unambiguously the value of this magnetic field H_{S}. The evaluated magnetic field H_{S} is not related to intrinsic magnetic field. The H_{S} describes the effectiveness of alignment of spins of the conduction electrons along the external magnetic field. The smaller H_{S} is, the more effective alignment is. The spin alignment is less effective when the spin relaxation is faster. The spin alignment is more effective when spin precession damping is faster and therefore the Gilbert damping constant is larger. The equilibrium spin polarization P_{S,0} does depend on the intrinsic magnetic field. However, it also depends on the strength of the spd exchange interaction between conduction and localized electrons and the scattering rate between conduction and localized electrons. Therefore, it is difficult to distinguish the sole contribution of the intrinsic field. There are more direct methods of the intrinsic magnetic field than the described method. For example, we have developed another measurement method, in which it is possible to evaluate the strength of the intrinsic magnetic field. The method is based on a measurement of the strength of the perpendicular magnetic anisotropy (PMA). See . The idea of the measurement can be explained in short as follows. In our nanomagnet the PMA has an interfacial origin. The PMA is due to the spin orbit interaction (SO), which is enhanced at the interface. The SO by itself cannot break the time inverse (TI) symmetry and therefore the SO requires an external magnetic field to break the TI symmetry. Only in the presence of an external magnetic field, the SO can manifest itself. As a result, the strength of the SO and therefore PMA is proportional to the total magnetic field, which electrons at interface experience. The total magnetic field is the sum of the intrinsic and external magnetic fields. From a measurement of the PMA strength as a function of the external magnetic field, the intrinsic magnetic field can be estimated. The intrinsic magnetic field includes the demagnetization field. The strength of the PMA is characterized by the anisotropy field H_{anis}. We measure the as H_{anis} function of a perpendicularly applied external magnetic field H_{z}. Figure shows the measured dependence. Since the H_{anis} increases a few times under H_{z}=7 kG, we conclude that the applied external magnetic of about 7 kG is comparable to the intrinsic magnetic field. The intrinsic magnetic field is different from a nanomagnet to a nanomagnet, but it is within range of a few kGauss. Are you sure that all magnetic moments in a nanomagnet are aligned in one direction?
(about a perfect alignment of magnetization in a nanomagnet) There are several proofs, but I would like to mention 2 (experimental proof 1: linear dependence of M_{} vs. H_{}): In this measurement (details here and here) an external magnetic field H_{} is applied in plane (along the magnetic hard axis). The magnetization turns towards the inplane field. The dependence of the inplane component of the magnetization M_{} vs H is a perfect line, which is only possible when the nanomagnet is in a single domain state. (proof of perfect alignment): In the case if some region has magnetization different from the perpendicular, the inplane magnetic field H_{} expands this region in order to minimize the magnetic energy and the dependence is substantially different from linear.
(experimental proof 2: existence of a nucleation domain in process of magnetization reversal ): The experimental measurements of thermally activated magnetization switching in the same nanomagnets confirms that the magnetization switching mechanism of the studied nanomagnets is the nucleation domain type. A nucleation domain is an unstable magnetic domain, which exists for a very short time (~a few milliseconds) during magnetization switching. The very existence of the nucleation domain confirms that the sample is mono domain because such nucleation domain can exist only in a single domain nanomagnet in absence of a static domain. Additionally, the size of nucleation domain is measured to be between 40 nm and 90 nm for our FeB and FeCoB nanomagnets (details are here and here). Such measurement is impossible in case when there are static domains. (proof of perfect alignment): the nucleation domain can exists and therefore be measured in a single domain nanomagnet. Otherwise, the pre existed region of different magnetization just expands instead of creation of a nucleation domain.
(experimental proof 3: The stable static domain is at least a few times larger than the unstable nucleation domain. We have exactly similar result (even the measured signal is noisier) for a very small nanomagnets (d<50 nm), which size is even smaller than size of nucleation domain. The existence of a much larger static domain in such a small nanomagnet is absolutely impossible. (proof of perfect alignment): there is a minimum size of static domain. When the size of nanomagnet is smaller that this size, there is no static domains
Hysteresis loop of a ferrimagnetic film is also not rectangular shape. The magnetization monotonically increases with H. Does it means that your method can be used for a planar thin film or for a ferromagnet?(about a possibility of usage of the method for a ferromagnetic film and a disturbance due to a static domain). Existence of static domains in in a planar film or bulk ferromagnet makes very difficult to use this method. (magnetic domains of a ferromagnetic film) In a ferromagnetic film or in a bulk ferromagnet,, the magnetization monotonically increases with H due to a motion of domain walls. In a magnetic field the magnetic domains are smoothly rearranged in an increasing magnetic field making larger area of domains of magnetization along magnetic field. As a result, the magnetization is monotonically increases in a magnetic field. A modern Kerr microscope has an option, which allows to see magnetic domain distribution and redistribution while measuring a hysteresis loop of Kerr angle. This fact can be easily verified.
(magnetization of a singledomain nanomagnet) When the size of nanomagnet becomes smaller than the size of a magnetic domain, all spins of localized electrons are aligned in one perpendicular direction and the state of the nanomagnet becomes the singledomain state. The reason why all spins of localized electrons are perfectly aligned in a nanomagnet in one direction can be understood as follows. For existence of a static magnetic domain, the positive exchange energy of a domain wall should be balanced by a negative magneto static energy between dipoles of opposite magnetizations. When shape of nanomagnet is a circle with radius R and domain wall passes through its center, the magneto static energy is proportional to domain area (~R^{2}) and domain wall energy is proportional to its length (~R). When the size of nanomagnet decreases (decrease of R), the magneto static energy decreases faster, at some size it becomes unable to balance the domain wall energy and nanomagnet state becomes a single domain state.
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