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Measurement of spin polarization

Spin and Charge Transport

Abstract:

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 (2020) and here (2019) . More details about spin polarization of the electron gas is here
The method is based on calculations of the spin pumping in a magnetic field described here



Content

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(1).Spin polarization of conduction electrons and its basic properties

(measurement method ): Measurement of Hall angle vs magnetic field applied along magnetization

 

External magnetic field Hz is applied along magnetization M (shown as green ball) of nanomagnet. Since Hz is applied along M, the magnetization and corresponded Anomalous Hall effect (AHE) are not influenced by the magnetic field Hz. In contrast, the spins of conduction electrons are aligned along Hz . As a result, the number of spin- polarized electrons increases. It cause of a larger contribution of the Inverse Spin Hall effect (ISHE) and therefore an increase of the Hall angle vs. Hz.
(what is measured) The increase of the Hall angle vs. increasing magnetic field applied along magnetization
(which parameters are evaluated) AHE, ISHE, spin polarization of conduction electrons.
method has been developed in 2019-2020 by Zayets
Click on image to enlarge it

(2). Direct measurement methods of spin polarization (in short)

(2a) limitation for the measurement of spin polarization
(2b). method 1: Hall measurement
(2c) method 2: Magneto-optical measurement
(2d). method 3: measurement of the total magnetic moment of conduction electrons

(3). Indirect measurement methods of spin polarization (exotic)

(4).Measurement of spin polarization using Hall effect

(1). Explanation in short
(2). Experiment
(3).What is the spin polarization?
(5a). Spin-pumping 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

(7). Voltage-controlled magnetic anisotropy (VCMA) effect & spin polarization

(8). Spin-Orbit Torque (SOT) effect & spin polarization

(9). Oscillations of 2nd derivative

(10). Spin polarization vs film thickness & interface type

(11) Questions & Answers

.........


What is the spin polarization sp of the conduction electrons

The conduction electrons in a ferromagnetic metal can be divided into 3 groups (See here): (group 1) spin-polarized electrons, which spins are directed in the same direction; (group 2) spin-unpolarized electrons, which spins are equally distributed in all direction; (group 3) spin-inactive electrons, which energy is substantially bellow the Fermi energy and which do not participate in the spin transport. The spin-polarization is the ratio of the number of spin-polarized electron to the total number of the spin-polarized and spin-unpolarized electrons.

Methods to measure of spin polarization of conduction electrons

Method 1: Hall measurement

The Hall voltage or the Hall angle αHall can be measured using cross-bar structure (See here). αHall has 3 contributions: (contribution 1) Ordinary Hall effect αOHE, which is linearly proportional to external magnetic field H. (contribution 2) Anomalous Hall effect αAHE, which is linearly proportional to the total magnetic moment M (to total spin) of localized electrons . (contribution 3): Inverse Spin Hall l effect αISHE, which is linearly proportional to the total magnetic moment (to total spin) of conduction electrons and therefore to the spin polarization sp.

(dependence on H in a nanomagnet): AHE contribution is constant. (Reason): The spins of localized are firmly fixed along the easy axes and are not affected by magnetic field applied along the easy axis. ISHE contribution non-linearly increases with H. (Reason): The spins of spin-unpolarized conduction electrons are aligned along the magnetic field. As a result, the spin polarization increases.

Method 2: Magneto- optic measurement

The Faraday rotation angle θFaraday or the Kerr rotation angle can be measured using crossed polarizer and analyzer (See here). Measured θFaraday has 3 contributions: (contribution 1): Paramagnetic contribution θparam from a thick substrate, which is linearly proportional to H. (contribution 2): θd is polarization rotation by localized d-electrons, which is linearly proportional to the total local moment M (total spin) of localized electrons. (contribution 3): θcond is polarization rotation by conduction electrons, which is linearly proportional to the total magnetic moment (to total spin) of conduction electrons and therefore to the spin polarization sp.

Method 3: measurement magnetic moment (magnetization)

Magnetic moment μtotal can be measured by a magnetometer. μtotal has 3 contributions: (contribution 1): Paramagnetic contribution μparam from a thick substrate, which is linearly proportional to H. (contribution 2): Magnetic moment μd of localized d-electrons. (contribution 3): Magnetic moment μd of conduction electrons, which is linearly proportional to number of spin-polarized conduction electrons sp ⋅ ncond

Spin polarization sp of conduction electrons is evaluated from measured dependence of αHall or θFaraday or θKerr or μtotal on the external magnetic field H.

All these methods are direct methods to measure the spin polarization, because they measure the magnetization of the conduction electrons and therefore the number of the spin- polarized conduction electrons.

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 below-described method, the dependence of the Hall angle on an applied magnetic field is measured.

Similary, the measurements of the dependence of the magneto-optical (MO) constants or the tunnel resistance or the magnetization measured by a magnetometr 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 spin-polarized electrons from another metal.

In the below- described method, tthe spin polarization is modulated by applying an external magnetic field Hext along the spin direction of the spin polarized electrons.

The spin polarization increases when a magnetic field is applied along spin direction of the spin-polarized 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 aligment is described by the Gilbert damping parameter  in the LandauLifshitz 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 spin-polarized conduction electrons changes and therefore the total magnetic moment of the conduction electrons changes as well. Nearly-all magnetic properties depends on the the total magnetic moment of the conduction electrons. Specifically the Hall angle, magneto-optical constant, the total magnetic moment and all magneto-transport properties are changed when the spin polarization is changed.

Main condition & limitation for the measurement of spin polarization

Main 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 polarization

Why these methods are direct?

A. All below-described 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 spin-polarized 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 spin-polarized conduction electrons. Therefore, the measurement of the magnetization of the conduction electrons gives the most direct evaluation of the number of the spin-polarized 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 & symmetry

A 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 fully-saturated 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, Scond is the total spin of the conduction electrons, and Sd is the total spin of the localized d-electrons

Magnetic parameters, which reversed with reversal of external magnetic field: (1) magnetization M or the total magnetic moment of localized d-electrons Md, (2) the total magnetic moment of spin polarized electrons Mcond ; (3) the Hall voltage and the Hall angle αHall; (4) magneto-optical constants: Faraday rotation angle θFaraday , 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

 

method 1: Hall measurement

contributions

total (measured) αHall (sum of 3 contributions)

spin polarization sp
Hall angle αHall has 3 contributions: (1st linear contribution (blue line)) Ordinary Hall effect (OHE); (2d constant contribution (red line)) Anomalous Hall effect (AHE); (3d non- linear contribution (black line)) Inverse Spin Hall effect (ISHE);

Since ISHE contribution depends of the spin polarization sp of conduction electrons, fitting of measured αHall gives spin polarization sp of conduction electrons.

Click on image to enlarge it.

Measurement of spin polarization. (Method 1): Hall measurement

The Hall angle αHall is asymmetrical with respect to H reversal. Therefore, it defines as a difference of αHall measured at opposite H:

αHall can be linearly proportional to material parameters of the same symmetry

where Md is the total magnetic moment of localized d-electrons and Mcond is the total magnetic moment of conduction electrons. The total moment of all spin-unpolarized electrons is zero. Since all spins spin-polarized electrons are directed in one direction, Mcond =μcond ·nTIA , where nTIA is the number of spin-polarized conduction electrons, μcond is the magnetic moment of one conduction electron.

 

Therefore, Eq(3.2) is simplified as

nd is the number of spin-active localized d-electrons and μd is the magnetic moment of one localized d- electron.

By definition of the spin polarization sp (See here) is percentage of spin-polarized electron among conduction electrons:

where ncond is the total number of spin-polarized and unpolarized electrons.

The magnetization M of a ferromagnetic metal is defined as

Substitution of Eqs. (3.0) and (3.0a) gives

or

where βISHE is redefined as βISHE =βISHE· μcond ·ncond

The 1st term βOHE·H describes proportionality of αHall to external magnetic field. This effect is called the ordinary Hall effect.

The 2nd term βAHE·M describes proportionality of αHall to magnetization of the ferromagnetic metal. This effect is called the Anomalous Hall effect.

The 3d term βISHE·sp describes proportionality of αHall to the spin polarization of the conduction electrons. This effect is called the Inverse Spin Hall effect.

 

method 2: magneto-optical measurement

contributions

total (measured) αMCD (sum of 3 contributions)

spin polarization sp
(left) Total measured αMCD has has 3 contributions: (1st non-linear contribution, black line) contribution of spin-polarized conduction electrons αMCD,cond ;, which is proportional to the number of the conduction electrons; (2d constant contribution, red line) contribution of d-electron αMCD,d-, which is proportional to the number of the d- electrons; (3d linear contribution) paramagnetic contribution αMCD,param , which is due transition of paramagnetic electrons.

Since αMCD,cond depends of the number of spin-polarized electrons and therefore on the spin polarization sp of conduction electrons, fitting of measured αMCD gives spin polarization sp of conduction electrons.

Click on image to enlarge it.

Measurement of spin polarization.(Method 2): Magneto-optical measurement

A measurement of the Faraday rotation angle θFaraday or Kerr rotation angle θKerr 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 mesurement of Faraday rotation angle θFaraday or Kerr rotation angle θKerr.

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 spin-polarized 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 αMCD 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, αMCD can be measured as

 

 

 

 

 

Measurement of spacial distribution of spin polarization in GaAs. Optical spin injection

Spacial distribution of Kerr rotation angle in GaAs. Red circle shows the spot, at which focused circular polarized light excites spin-polarized electrons. The Kerr rotation angle is proportional to spin polarization sp of the conduction electrons. The spin polarization decays as the photo-excited electrons diffuse from the laser spot. (a) The electrical field is not applied. The spin polarization decays symmetrically in all directions. (b) Electrical field is applied ("+" at the right and "-" at the left). As the electrons are drifted along the electrical field, the spin polarization is drifted as well.

The fact that the Kerr rotation angle is proportional to the spin polarization of the conduction electrons is used in this measurement!!!

S. A. Crooker and D. L. Smith, PRL 94, 236601 (2005). Permission
Click on image to enlarge it.

Each spin-polarized electrons contribute to αMCD, therefor αMCD is proportional to the number of spin-polarized electrons. Since the symmetry and size of the localized d-electrons and conduction electrons are very different, therefore the interaction efficiency of these electrons with photons is very different and these spin-polarized electrons contribute differently to αMCD. Additionally, there is a paramagnetic contribution due to the Zeeman splitting.

Eq.(3.12) can be obtained also the symmetry fact (like that αMCD

Substitution of Eqs. (3.0) and (3.0a) gives

where we redefine

 

 

 

 

 

 

 

 

 

method 3: measurement of magnetic moment

contributions

total (measured) μtotal (sum of 3 contributions)

spin polarization sp
(left) Total measured magnetic moment μtotal has has 3 contributions: (1st non- linear contribution, black line) total magnetic moment of conduction electrons μcond.elec;, which is proportional to the number of the conduction electrons; (2d constant contribution, red line) total magnetic moment of d-electrons μd-.elec, which is proportional to the number of the d- electrons; (3d linear contribution) paramagnetic moment μd-.elec, which is contribution of a substrate and paramagnetic electrons.

Since μcond.elec depends of the number of spin-polarized electrons and therefore on the spin polarization sp of conduction electrons, fitting of measured μtotal gives spin polarization sp of conduction electrons.

Click on image to enlarge it.

Measurement of spin polarization.(Method 3): Measurement of Magnetic moment

The magnetic moment μtotal 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 μtotal is the sum of magnetic moments of d-electrons, magnetic moments of the spin-polarized 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 polarization

requirements 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 spin-polarized 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 spin-unpolarized 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 spin-polarized 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.

Non-magnetic metal:

change of spin polarization sp. (Method 3)Spin injection

The

 


Indirect (exotic) methods of measurement of spin polarization (sp)

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, spin-dependent photoluminescence, spin-dependent 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 sp-dependence 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 spin-dependent..

These methods are exotic, because the physics of the spin-dependent tunneling and the spin-dependent photoluminescence are complex and are not fully understood yet. It is very speculative when the spin polarization is estimated from a fitting of very-approximate 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 of spin polarization is important for spintronics
Click on image to enlarge it.

Measurement 1: From tunneling magneto-resistance (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: 70-90 %

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 superconductor-metal contact.

Estimated spin polarization for FeCoB: 30-50 %

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 superconductor-metal contact; (3) clear under estimation of the value of the spin polarization;

 

Measurement 3: using spin-dependent photoluminescence, spin-dependent electroluminescence and spin LED

C. Aku-Leh,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 circular-polarized light emitted from a semiconductor, in which spin-polarized current is injected

note: with some limitations, the the spin-dependent photoluminescence can be considered as a direct measurement of the spin-polarization

Estimated spin polarization for FeCoB: 60-95 %

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 spin-light interaction can cause a systematic error; (see here and here);

Measurement 4: from spin-injection and spin-detection experiment (non-local spin-detection)

In the non-local spin- detection experiment, the spin-polarized 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 non-magnetic and magnetic materials.

 

 


Method 1: Hall measurement

Increase of spin polarization vs H increase makes the loop non-linear

Spin polarization increases in a magnetic field due to precession damping  
 
The spin polarization is evaluated from non-linear dependence of Hall angle αHall on H. The green dash line shows the imaginary case if spin polarization would not depend on magnetic field. The red dot line shows the case when spin polarization of metal is close to 100 %.Click on image to enlarge it. Fig. 1(b)Spin precession and precession damping in a magnetic field Hext . During the precession the spin aligns itself along the direction of the magnetic field. Click on the image to enlarge it.  

 

 

Merits of this measurement method:

Merit 1: Simplicity of measurements

Merit 2: ability for measurement of spin-polarization 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:

Measurement of spin polarization

Measured spin polarization as a function of an external magnetic field

Volt 54B ud40 Ta(2.5)/FeB(1.1)/ MgO(6)/ nanowire width 1000 nm nanomagnet length 500 nm. Fit by Eq.(1.8) gives (region of nanomagnet):sp0=81.2% ; Hpump=0.425 kG ;

Click on image to enlarge it.

Spin polarization sp in a magnetic field is calculated as

where sp0 is the spin polarization in absence of an external magnetic field, Hpump is the pumping magnetic field (a material parameter)

The measured Hall angle αHall is the sum of the Hall angle αOHE of the ordinary Hall effect, the Hall angle αAHE of the anomalous Hall effect and the Hall angle αISHE of the inverse spin Hall effect. It can be calculated as

where αAHE is the Hall angle of Anomalous Hall effect, αISHE,0 is the Hall angle of the Inverse Hall effect in the absence of an external magnetic field, βOHE is the Hall coefficient of ordinary Hall effect; H is the magnetic field applied perpendicularly to the film.

 

 

Measurement of spin polarization

FeB
Hall angle αAHE 1st derivative 2nd derivative
sp0=81.7% ; Hpump=0.834 kG ; Ta (2.5 nm):FeB(1 nm):MgO; sample:
Hall rotation angle αHall as a function of applied magnetic field H ; 1st and 2nd derivatives of αHall normalized to its value at H=0. Black triangles show the measured data. The red line shows the fitting by Eq.(1.12)
Click on image to enlarge it.
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 non-linear 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 αOHE of ordinary Hall effect (OHE), the Hall angle αAHE of Anomalous Hall effect (AHE), the Hall angle αISHE of the Inverse Spin Hall effect (ISHE) ?

A. The αISHE is linearly proportional to spin polarization (sp) of electron gas, the magnetization and the strength of the spin-orbit (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 αAHE can be calculated as

 

where σxx, σxy are diagonal and off-diagonal components of the conductivity tensor; a is the proportionality constant;M is out-of-plane component of magnetization

 

Parameters, which makes Hall angle αAHE 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

Measurement of spin polarization

Fig.2. A Fe nanomagnet connected to the Hall probe

The Hall voltage increases under external magnetic field!! From fitting of the dependence of Hall voltage on magnetic field the spin polarization is evaluated

Click on image to enlarge it.

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 non-magnetic 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/mm2. The aHall in the ferromagnetic metal was evaluated as

where  sferro , snonMag  are conductivities of ferromagnetic and non-magnetic metals;  tferro , tnonMag are their thicknesses, VHall 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 aISHE, aAHE and aOHE 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 spin-polarized and spin-unpolarized electrons. The spin polarization sp of the electron gas is defined as a ratio of the number of spin-polarized electrons to the total number of the spin-polarized and spin-unpolarized electrons:

where nTIA and nTIS are the numbers of spin-polarized and spin-unpolarized electrons, respectively.

How to divide all conduction electrons into the group of spin-polarized and spin-unpolarized electrons?

Detailed explanation is here. Explanation in short:

In fact, all conduction electrons in a ferromagnetic metal are divided into 3 groups: of spin-polarized, spin-unpolarized electrons and spin-inactive electrons. In the group of the spin-polarized electrons,   the spins of all electrons are in the same direction. In the group of the spin-unpolarized electrons, the spins are distributed equally in all directions. Additionally, there are some electrons, which are "spin-inactive". A pair of these electrons with opposite spins occupies one quantum state. The occupation of quantum states by the electrons of both the spin-polarized and spin-unpolarized 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 time-inverse 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 "spin-inactive". 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 spin-polarized and spin-unpolarized electrons is distributed mainly nearly the Fermi energy. See details here.

 


 


 

Measurement of spin polarization using the Hall effect

Measurement of spin polarization. Comparison of different materials

FeB
Hall angle αAHE 1st derivative 2nd derivative
sp0=81.7% ; Hpump=0.834 kG ; Ta (2.5 nm):FeB(1 nm):MgO; sample:
FeCoB
Hall angle αAHE 1st derivative 2nd derivative
sp0=89.4 % ; Hpump=1.098 kG ;Ta (5 nm):Fe0.4Co0.4B0.2 (1 nm):MgO ; sample:
Hall rotation angle αHall as a function of applied magnetic field H ; 1st and 2nd derivatives of αHall normalized to its value at H=0. Black triangles show the measured data. The red line shows the fitting by Eq.(1.12)
Click on image to enlarge it.

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 spin-polarized electrons to the total number of the spin-polarized and spin-unpolarized 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 spin-unpolarized electrons into the group of the spin-polarized electrons. The spin relaxation is the conversion in the opposite direction.

Detailed explanation about spin polarization is here.

 

Spin-pumping rate:

The spin pumping describes the conversion of electrons from the group of the spin-polarized electrons into the group of spin-unpolarized electrons. The conversion rate of the spin-pumping is described as

where tpump is the spin pumping time, nTIA and nTIS are the numbers of spin-polarized and spin-unpolarized electrons, respectively.

Details about different mechanisms of spin pumping is here

Spin-relaxation rate:

The spin damping describes the conversion of electrons from the group of the spin-polarized electrons into the group of spin-unpolarized electrons. The conversion rate of spin-relaxation can be described as

where trelax is the spin relaxation time.

Details about different mechanisms of spin relaxation is here

Spin 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

Alignment of spins of conduction electrons along a magnetic field is the reason of increase of the spin polarization of the electron gas

Fig.3 Precession of electron spin and the spin precession damping in a magnetic field. The red arrow represents electron spin. The grey arrow shows the direction of magnetic field. The data was calculated by solving Landau-Lifshiz equation

Spin precession and precession damping in a magnetic field. During the precession the spin aligns itself along the direction of the magnetic field. Click on the image to enlarge it.
 

 

There are two reasons:

(1) major reason: increase of spin pumping in a magnetic field

It is due to alignment of spins of spin-unpolarized electrons along a magnetic field (see Fig. 3)

(2) minor reason: suppressing of spin relaxation by a magnetic field

 

Spin-pumping rate induced by a magnetic field:

In a magnetic field, the spins of spin-unpolarized 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 spin-unpolarized electrons (spins are equally distributed in all directions). (details See here) As a result, there are more spin-polarized electrons. The spin-pumping (See Eq.19 below)

where tH,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 sp0 is the spin polarization in absence of an external magnetic field (a material parameter) , which is calculated as

Hpump is the pumping magnetic field (a material parameter), which is calculated as

 


Factors, which may influence the measurements

In 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 spin-dependent scatterings: Even spin may be rotated after a spin-dependent 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 field

Can 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.

 

 

 

 

 

 


Spin polarization & VCMA effect

Dependence of spin polarization on gate voltage

 
  Fig. 14 Spin polarization of FeB nanomagnet as a function of the gate voltage

click on image to enlarge it

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 magnetization-switching mechanism can be used as a data recording method for low-power 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)

 

Figure 14 shows the gate-voltage dependence of the spin polarization in the FeB sample. The measured voltage-dependent change of the spin polarization is -0.245 %/V. The change is substantial and it can be reliably measured. For example, the observed voltage-dependence of 1st derivative of αAHE(Fig.3(c)) is about 5 %. For the FeCoB sample the voltage-dependence of sp was smaller ~ -0.0068%/V. For both samples, the sp depends linearly on the gate voltage and the polarity of the dependence is the same as the polarity of the voltage-dependence of anisotropy field Hanis, coercive field Hc , Hall angle αAHE, magnetization switching time tswitch , and retention time treten.

 

 

VCMA and TMR effects. It would be interesting to correlate this gate tuning of spin polarization with the bias-dependent TMR experiments which have been traditionally explained with barrier height modulation.

A. I guess there are several contributions to the bias-dependence of TMR. The voltage dependence may have some contribution. However, it should be a contribution, which is polarity dependent. The voltage-control 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.

 

 

 

 

 

 


Spin polarization & SOT effect

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

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

Fig.14 click on image to enlarge it

 

 

 

 

The effect of Spin-Orbit 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 3-terminal 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.

Figure 14 (a) shows the measured spin polarization sp of the FeB sample as a function of the current density. The sp decreases for both polarities of the current. The used current density is relatively large and the decrease of sp is assumed to be due to the heating of the nanomagnet. The increase of the nanowire resistance confirms the increase of the nanowire temperature. In order to exclude the influence of heating, the spin polarization was measured at the same current, but for two opposite current directions. Figure 14(center) shows the change of the spin polarization as the polarity is reversed.  The change of the sp linearly depends on the current. The measured slope is +0.00524 %/(mA/mm2) for the FeB sample and -0.018 %/(mA/mm2) for the FeCoB sample.

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 non-zero. 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.

 

 


 

Oscillations of 2nd derivative

Oscillations of measured 2nd derivative vs H

     
There are clear oscillations of measured 2nd derivative of αAHE vs H of an unknown origin.

 

 

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

nanowire with two pairs of Hall probes

Measurement setup

spin polarization

hysteresis loop

Backside Hall probe is connected to a nanomagnet. FeB is thicker in this region and top of FeB is covered by MgO. The Front side Hall probe is connected at side of a nanomagnet. FeB is thinner in this region and at its top covered by SiO2. The distance between two Hall pairs is 11 μm. Measured spin polarization vs applied perpendicular magnetic field; Fit by Eq.(1.8) gives (region of nanomagnet):sp0=81.2% ; Hpump=0.425 kG ; (region at side): sp0=63.5 % ; Hpump=0.88 kG ;  

Volt 54B ud40 Ta(2.5)/FeB(1.1)/ MgO(6)/ nanowire width 1000 nm nanomagnet length 500 nm. Sample is soft. Hc~0 Oe, Hani~1.7 kG,

click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Measurement of spin polarization. Comparison of different thicknesses of the same nanowire

nanomagnet (thicker FeB covered by MgO)
Hall angle αAHE 1st derivative 2nd derivative
sp0=81.2% ; Hpump=0.425 kG ;
side (thinner FeB covered by SiO2)
Hall angle αAHE 1st derivative 2nd derivative
sp0=63.5 % ; Hpump=0.88 kG ;
Volt 54B ud40 Ta(2.5)/FeB(1.1)/ MgO(6)/ nanowire width 1000 nm nanomagnet length 500 nm. Sample is soft. Hc~0 Oe, Hani~1.7 kG,

triangles show the measured data. The solid line shows the fitting by Eq.(1.12)

Click on image to enlarge it.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Ambiguility of Fitting

Ambiguity for the evaluation of the AHE and ISHE contributions, Ambiguity for the  evaluation of  the spin polarization

Ambiguility of fitting

Fig.5 Ambiguility of the data fitting. Two possible fittings: (solid lines) PS=60 %, αISHE = 827 mdeg αAHE= 400 mdeg  (dash lines) PS==20 %, αISHE= 137 mdeg αAHE= 1080 mdeg. HS=3.985 kGauss and  αOHE=0.2 mdeg/kGauss for both cases. Both fitting give the identical total Hall angle (pink line) and its 1st derivation.

click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


A Disturption of the measurement due to existance of static domain

Underisirable influence of static domains

Fig.7 Influence of static domain on the measurement. The of large size nanomagnet (3 mm x 3 mm). (a) Hysteresis loop (b) absolute value of the Hall angle αHall. (c)  Derivative dαHall/dHz. There are spark changes of the derivative in the regions of existence of static domains. In the regions of absence of static domains the change of derivative is monotonic. Different line color corresponds to different part of scan of the magnetic field

click on image to enlarge it

 

This measurements can be used only for a sufficiently small samples.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Questions & Answer

Can the spin polarization sp of the conduction electrons be created in a non-magnetic metal by an external magnetic field?

Yes.

Can you see a non-linear dependence of the Hall angle vs an external magnetic field in a non-magnetic 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 non-magnetic metal, the HS is substantially larger than in a ferromagnetic metal due to faster spin relaxation rate. As a result, 1st derivitive 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?

Decomposing of measured hysteresis loop of the hall angle into its 3 contributions

Fig.6 Different contribution to hysteresis loop. (a) total Hall angle αHall; (b) decomposing the total Hall angle into contributions; (c) AHE contribution; (d) ISHE contribution. Parameters used: αAHE=20 mdeg; αISHE=10 mdeg; α0HE=1 mdeg/kG;

click on image to enlarge it

(about another possible Hall effect)

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 Sconduct 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 Slocal  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 aHall

where H is external magnetic field, Slocal is the total spin of localized electrons and Sconduct 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 H3, H2∙ Sconduct, H2∙Slocal, 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.

 

 

 

 

 

Dependence of spin polarization PS on the equlibrium spin polarization PS,O and HS

Fig.4 Spin polarization PS and its 1st order derivative as a function of external magnetic field Hz.  (a),(b) the equilibrium spin polarization in absence of magnetic field (Hz=0) is varied at constant HS =7 kGauss. (c),(d) the HS is varied at constant equilibrium spin polarization of 25 %

click on image to enlarge it

 

 

 

 

 

 

 

 

The phenomenological model for the ISHE includes the spin-polarization 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 magneto-optical experiments ?

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 1st and 2nd order derivatives

The 1st and 2nd 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 1st and 2nd 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 spin-polarization of the electrons is only mildly affected As a function of H. ?

You are absolutely correct. 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 sp-d 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 HS. The fitting gives unambiguously the value of this magnetic field HS. The evaluated magnetic field HS is not related to intrinsic magnetic field. The HS describes the effectiveness of alignment of spins of the conduction electrons along the external magnetic field. The smaller HS 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 PS,0 does depend on the intrinsic magnetic field. However, it also depends on the strength of the sp-d 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 Hanis. We measure the as Hanis function of a perpendicularly- applied external magnetic field Hz. Figure shows the measured dependence. Since the Hanis increases a few times under Hz=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.

 

 

 

 

 

 


I am strongly against a fake and "highlight" research

 

 

 

 

I truly appreciate your comments, feedbacks and questions

I will try to answer your questions as soon as possible

 

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