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### Reticle 11

Magneto transport. Family of Hall effects and AMR effects.

### Content

#### 7. Magneto- capacitance induced by the Hall effect

##### (q2) How to distinguish Planar Hall effect from Anomalous Hall effect experimentally?

.........

Distinguish features of a magneto- transport effect:

### Odd and even magneto- transport effects

All magneto- transport effects can be distinguished by their symmetry against reversal of substantially large magnetic field, which fully reverses spins of localized and conduction electrons.

#### (Odd magneto transport effects) (), which polarity is not reversed when the magnetic field and the spins are reversed

Polarity of effect is Reversed, when when the magnetic field + spin are reversed

Ordinary Hall effect (OHE), Anomalous Hall effect, Inverse Spin Hall effect

#### (Even magnetotransport effects) , which polarity is not reversed when the magnetic field and the spins are reversed

Polarity of effect is NOT Reversed, when when the magnetic field + spin are reversed

# 1st- order odd magneto- transport effects

### Ordinary Hall effect (OHE)

Ordinary Hall effect origin of OHE: Lorentz force OHE in Ru
(definition) The OHE describes the fact that charge is accumulated at sides of metallic wire, when an external magnetic field H is applied perpendicularly to the wire. The origin of the OHE is the Lorentz force The Lorentz force turns the electron (green bowl) from a straight movement. The electron, which moves in a magnetic field H, experience an electrical field perpendicularly to its movement (relativistic effect). This electric field interacts with the electron charge and turns the electron movement direction. It creates an electron current (Hall current) perpendicularly to the bias current. Hall angle vs an applied magnetic field H measured in ruthenium Ru (a non-magnetic metal). The Ru thickness is 25 nm. The dependence is nearly perfectly linear.
The movement of the conduction electrons (green balls) turns from a straight path due the Lorentz force induced by the magnetic field H. As a result, the electrons are accumulated at the side of the wire.   There are a very weak deviation from linear dependence (See here)
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##### More details see here

(origin) Origin of OHE is the Lorentz force. An electron experience an relativistic electrical field due to electron movement perpendicularly to the magnetic field. The The relativistic electrical field interacts with the electron charge (not spin) forcing the electron to move in its direction.

(interact with) Electron Charge of conduction electrons

OHE is linearly proportional to an external magnetic field H

(formula):

aOH is the rotation angle of the ordinary Hall effect (in mdeg/kG). H is external magnetic field. aOH is positive for the hole- dominated conductivity. aOH is negative for the electron- dominated conductivity. jV is the bias current along metallic wire (from electrical source to electrical drain). The hole- dominated conductivity in a material, in which density of states decreases at the Fermi level. The electron- dominated conductivity in a material, in which density of states decreases at the Fermi level

(note) The OHE is independent of the spins of localized and conduction electrons. It depends on the charge of carrier and its transport properties.

### Anomalous Hall effect (AHE)

(definition) The AHE describes the fact that charge is accumulated at sides of metallic wire, when the spins of localized d- electrons are in one direction (e.g. ferrimagnetic state) perpendicularly to the wire and an electron current flows through the wire
The conduction electrons (green balls) interacts with the aligned spin of localized d- electrons(blue ball). Due to such interaction the scattering probability of conduction electrons to the right becomes larger than to the left and they are accumulated at the right side of the wire.
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##### More details see here

(origin) Dependence of scattering of conduction electrons on the spin of localized electrons

(interact with) Rotational (Orbital) Moment of conduction electrons

AHE is linearly proportional to the total spin of localized electrons.

(formula):

aAH is the rotation angle of the Anomalous  Hall effect (in mdeg). Slocal is the total spin of localized electrons.  Since in the most of realistic cases, only the direction , but not magnitude of  Slocal  changes, Eq. can be simplified as

where M is an unit vector in direction of magnetization.

(note) The AHE depends on the total spin of localized d- electrons, but they are independent of the total spin of conduction electrons.

### Inverse Spin Hall effect (ISHE)

(definition) The ISHE describes the fact that charge is accumulated at sides of metallic wire, when the conduction electrons are spin- polarized and an electron current flows through the wire
The conduction electrons (green balls) are scattered on a charged defect (blue ball). The conduction electrons are spin- polarized (spin- up). Due to the spin-orbit interaction, the scattering probability for spin-up electrons is higher for a scattering to the right than to the left. As a result, the electrons (the charge) is accumulated at the right side of wire
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##### More details see here

(origin) Dependence of scattering of conduction electrons on the spin of localized electrons

(interact with) Rotational (Orbital) Moment of conduction electrons

ISHE is proportional to the total spin of conduction electrons

(formula):

aAH is the rotation angle of the Anomalous  Hall effect (in mdeg).  Slocal is the total spin of localized electrons. Since the the total spin of the spin-polarized electrons is linearly proportional to the number of spin polarized electrons, the Eq. can be simplified as

where  Ps is spin polarization. m is unity vector along spin- direction of spin-polarized conduction electrons

#### (origin of the Inverse Spin Hall effect) Spin- dependent scatterings

(explanation in short) The spin dependent scatterings means that the scattering probability of spin- up electron is higher to the left and the scattering probability of spin- down electron is higher to the right . For example, if the spin direction of the spin polarized conduction electrons is up, there are more electrons scattered to the left and as a result there is a charge current flowing to the left

### Spin Hall effect (SHE)

(definition) The SHE describes the fact that spin is accumulated at sides of metallic wire, when an electron current flows through the wire
The conduction electrons (green balls) are scattered on a charged defect (blue ball). The conduction electrons are spin- unpolarized (spins are distributed equally in all directions)). Due to the spin-orbit interaction, the scattering probability for spin-up electrons is higher for a scattering to the right than to the left and in contrast the scattering probability for spin-down electrons is lower for a scattering to the right than to the left. As a result, the spin-up electrons is accumulated at the right side of wire and the spin-down electrons is accumulated at the left side of the wire
(note) The SHE redistributes the spin- unpolarized electrons (all spin directions) into two separated places in which the number of either spin-up or spin-down electrons are larger. In total, the the number of electron with spin of a specific direction remains the same. It does not re align electron spin (as for example magnetic field(See here and here)
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##### More details see here

(origin) Dependence of scattering of conduction electrons on the spin of localized electrons

(interact with) Rotational (Orbital) Moment of conduction electrons

SHE is proportional to the total spin of conduction electrons

(formula):

aAH is the rotation angle of the Anomalous  Hall effect (in mdeg).  Slocal is the total spin of localized electrons. Since the the total spin of the spin-polarized electrons is linearly proportional to the number of spin polarized electrons, the Eq. can be simplified as

where  Ps is spin polarization. m is unity vector along spin- direction of spin-polarized conduction electrons

(note) Spin Hall effect (SHE) and Inverse Spin Hall effect (ISHE) are fully complementary effect. They have identical origins and in a material they have the same magnitude

(note) Both the SHE and ISHE depends on the total spin of conduction electrons, but they are independent of the total spin of localized electrons.

(note) The ISHE is linearly proportional to the number of spin- polarized conduction electrons. The SHE is linearly proportional to the number of spin- unpolarized conduction electrons.

# 2nd- order even magneto- transport effects

### Two parts of the same effect: the Hall effects and Anisotropic Magneto- Resistance (AMR) effect.

The Hall effect and AMR effect are in the same family of effects. The have similar properties, similar origins and similar symmetry.

(a good example) the classic AMR and the Planar Hall effect describes one single effect, which is the magnetic generation of a current parallel to the magnetization direction. The component of this current, which is parallel to the bias current, describes the AMR effect. The component of this current, which is perpendicular to the bias current, describes the Planar Hall effect.

### Two twins effects: Hall effect and Anisotropic magneto resistance

The Hall effect is defined

(current

## Magnetization directions are opposite. Resistivity is larger

F1,F2 are ferromagnetic metals, N is a non-magnetic metal. There is no exchange interaction between ferromagnetic layers. Therefore, their magnetization (shown by arrows) can be changed independently. The resistivity of the wire dependence on mutual directions of the magnetization in each layer.

Origin of in-plane GMR effect
Magnetization directions are parallel Magnetization directions are opposite
Spin directions of conduction electrons (balls) are the same in both layers and they are parallel to the spins of localized electrons (arrows). As a result, the resistance of each layer is smallest.. Spin direction are opposite in each layer. Since there are more spin-polarized electrons in the left layer, the spin-polarized electrons diffuse from the left into the right layer and the spin polarization in the right layer becomes the same as in the left layer and opposite to spin direction of localized electrons of the right layer. As a result, the resistance of the right layer becomes larger.
Arrows shows the spin direction of localized electrons (magnetization) in ferromagnetic layer. Balls shows the spins of spin- polarized conduction electrons. Color of balls indicates in which layer the conduction electron was made spin- polarized .
Spin-polarization of conduction electrons in the left layer is larger (Fro example, due to a weaker spin relaxation)
(note): Spins of conduction electrons are aligned along spins of localized electrons due to sp-d exchange interaction and sp-d scatterings. (See here). In an equilibrium in a single- material ferromagnetic metal, the spin directions of spin-polarized conduction electrons and localized d- electrons are parallel.
(note) The resistance of a material is smallest when spin directions of spin-polarized conduction electrons and localized d- electrons are the same due to the effect of the spin-dependent conductivity.
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#### The in-plane GMR effect describes the fact that resistance of a metallic wire, which consists of two ferromagnetic layers separated by a non-magnetic layer, depends on mutual magnetization directions of two ferromagnetic layers. It is the smallest, when magnetization directions are parallel and it is the largest, when the magnetization directions is opposite.

##### For experimental discovery of the in-plane GMR effect, Prof. Fert and Prof. Grünberg were awarded the Nobel Price in Physics in 2007.

(origin 1 of in-plane GMR effect) Spin proximity effect

The spin proximity effect (See details here) describes the fact that the spin polarized conduction electrons diffuses from the first ferromagnetic layer to the second ferromagnetic layer, change the spin polarization in the second layer and as a results the resistivity of the second ferromagnetic layer increases due to the effect of the spin- dependent conductivity.

(origin 2 of in-plane GMR effect) spin- dependent conductivity.

The conductivity of a ferromagnetic metal depends on mutual directions of spins of localized electrons and spins of conduction electrons. When spin-polarized conduction electrons diffuses from one ferromagnetic metal to the second ferromagnetic layer of a different magnetization directions, they make different the in the spin directions of localized and conduction electrons in the second layer and as a result the resistivity of the second layer becomes larger.

### (Origin of in-plane GMR effect):

When the magnetization directions in ferromagnetic layers are parallel, the spin directions of the spin-polarized conduction electrons are also the same and parallel to the magnetization (the spins of localized electrons). In this cases, the resistance of each layer is smallest. When the magnetization directions are opposite, the spin directions of conduction electrons are also opposite. In the case when in the first ferromagnetic layer the number of the spin polarized electrons is substantially larger than in the second ferromagnetic layer, a significant amount of the spin -polarized electrons from first layer diffuses into the second ferromagnetic layer and the spin direction in there become the same as in the first layer and opposite to the magnetization of localized electrons. As a result, the resistivity of the second layer becomes larger. The resistivity of a material is largest when the spin direction conduction electrons is opposite to the spin direction of the localized electrons due to the effect of the spin- dependent conductivity.

(note)When the total thickness of wire becomes smaller than the electron mean-free path, the electron gas becomes common through both ferromagnetic layers and the spin polarization is always the same in both layers. When magnetizations directions are parallel, the common spin polarization is the largest and parallel to each magnetization and therefore the resistance of each layer is smallest. When magnetizations directions are opposite, the common spin polarization is small (close to zero).As a result, the resistance becomes larger in each layer.

### Hall probe with a narrow contact

Fig.10a A metallic nanowire with a pair of Hall probe. An electron current flows from down to up and a Hall voltage is created perpendicularly to the wire due to the hall effect. Two metal contacts contact the opposite sides of nanowire to measure the Hall voltage. The width of nanowire w is smaller then the real width wp at measurement point, which includes the width of the Hall probe. Since the Hall voltage is proportional to the width of the wire or the width points of opposite charge accumulation, the uncertainty of the point, at which the charge is accumulated, may give a systematic measurement error. fig. 10b The probe a thin at contact point. As a result, the position of the charge accumulation is at about sides of the nanowire and therefore the correct Hall voltage is measured

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The Hall voltage VHall is linearly proportional to the width w of nanowire ( See HallAMRbasic.pdf)

where V is the voltage applied to the nanowire, αHall is the Hall angle, which is a material parameter, w is the width and L is the length of the nanowire.

Fig.10a shows a conventional structure for a Hall measurement in a metallic nanowire. Two metallic contact (probes) contact the opposite sides of the nanowire to measure Hall voltage. However, the effective width at the measurement point is wider than the nanowire width w. The effective width wp also includes the length of the probe. It may cause a systematic error in the measurement of the Hall angle αHall

Fig.10b shows an optimized design with a Hall probes, which are narrowing near contact. In this case, the charge is accumulated at the contacts and a possible systematic error is minimized.

##### To test all- metal transistor I have fabricated a hall probe as narrow as 30 nm.

Usually, the width of Hall probe about 1-2 mm is still OK but critical. The maximum- allowed width strongly depends on the sample structure.

#### (calculate it) For a reliable Hall measurement it is always better to simulate numerically the structure using e.g. Comsol.

There are several designs of Hall bars, which nearly fully exclude the undesired influence of the Hall probe.

### RF measurements of magneto-transport effects (Hall effects).

(main idea): The sample is illuminated by microwave at frequency of the Ferromagnetic resonance (FMR). The microwave excites the spin precession and additionally the microwave excites the electrical current. Since the Hall voltage proportional to both the spin direction and the current, which are both modulated by microwave, there are frequency beating of these two contribution. As a result, there is a DC component of the Hall voltage which is measured.

(merit of the method): It is possible to separate a studied Hall contribution from other contribution and measure a really weak Hall effect. For example, it is possible to measure a very weak ISHE effect in a paramagnetic metal.

### (Method 1). RF measurement of 1st order magneto-transport effects: AHE and ISHE

(what is modulated by RF):spin direction; electrical current

## Measurement of DC Hall voltage

 In an equilibrium, conduction electrons in InAs is not spin-polarized. When an external magnetic field is applied, the spins of conduction electrons are aligned along the magnetic field and the conduction electrons becomes spin- polarized. Under illumination by microwave radiation, the spins starts to precess around the magnetic field, which makes the Hall current. The microwave radiation excites the electrical current jz along z-direction. Additionally, the The microwave radiation excites the spin precession for the conduction electrons. Due to the spin precession the x component Sx of the spin at RF frequency. The Hall current flows along y- direction perpendicularly to Sx and jz. The frequency beating between Sx and jz creates a DC Hall current, which is detected by the DC nanovoltmeter.

# InAs

## Ge

DC Hall voltage VDC vs applied magnetic field measured in bulk InAs. Temperature1.3o K. Microwave frequency is 9.2 GHz DC Hall voltage VDC (dispersion derivative) vs applied magnetic field measured in bulk Ge. Temperature 2.2o K. Microwave frequency is 9.3 GHz
##### J. N. Chazalviel, “Spin-dependent Hall effect in semiconductors,” Phys. Rev. B, vol. 11, no. 10, pp. 3918–3934, 1975.

(experimental setup) A nonmagnetic InAs is illuminated by RF microwave radiation (blue/ green wave).The microwave radiation is produced by the RF generator (left-upper box) and the microwave antenna. The microwave RF radiation produces a Hall voltage, which is measured by the nanovoltmeter. Two metallic contact on the bulk InAs are used to measure DC Hall voltage.

### The FMR resonance is clearly detected in non-magnetic materials. It can be only the case when an external magnetic field makes the conduction electrons spin- polarized and the microwave radiation creates a spin precession.

(note) Sample are usually measured inside a microwave resonator. The FMR frequency depends on material parameters (e.g. the g- factor) and the external magnetic field. In a common FMR measurements, the microwave frequency is fixed and the magnetic field is scanned until the resonance condition is found.

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### (Method 2). RF measurement of 2nd order magneto-transport effects: Planar Hall effect/ AMR and spin- dependent conductivity

(what is modulated by RF):spin direction of localized d- electrons; spin direction of conduction electrons

## types of the Hall effect

Ordinary Hall effect (OHE)

The Charge of conduction electrons + magnetic field

Inverse Spin Hall effect (ISHE)

The Spin of conduction electrons + the Charge of localized electrons (or defects)

## Hall angle vs external magnetic field for each contribution

The Lorentz force turns the electron (green bowl) from a straight movement. The electron, which moves in a magnetic field H, experience an electrical field perpendicularly to its movement (relativistic effect). This electric field interacts with the electron charge and turns the electron movement direction. It creates an electron current (Hall current) perpendicularly to the bias current. The electrical field of a localized electron (or defect) creates a magnetic field of spin- orbit interaction HSO. The HSO depends on movent direction and position of a conduction electron. Since the electrical field (shown as red lines) is opposite at the left and right side of localized electron, the HSO is opposite at the left and right sides. When the spin of a conduction electron is along HSO , the electron energy is smaller and the probability to scattered into such state is higher. When the spin of a conduction electron is opposite to HSO , the electron energy is higher and the probability to scattered into such state is smaller. The difference in the scattering probabilities creates an electron current (Hall current) perpendicularly to the bias current. (OHE yellow line): Dependence is linear, because the Lorentz force is linearly proportional to external magnetic field Hz. (ISHE green line): Dependence is non-linear and αISHE reversed when magnetization M is reversed because the number of spin- polarized electrons (or spin polarization sp ) non-linearly increases with Hz. Polarity is following sp and therefore M. (AHE blue line): Dependence is a constant and αAHE is reversed when M is reversed, because αAHE is proportional only to M. (spM- type Hall red line): Dependence is non-linear and αspM , αspM polarity is not reversed when magnetization M is reversed because αspM is a product of M and sp. Both reverse their polarities. As a result, the polarity of does not change
Origin of OHE: Lorentz force. See details about OHE here Origin of ISHE: Dependence of HSO on movement direction and position of a conduction electron, which originates the spin- and direction- dependent scatterings. See details about ISHE here The dependencies of αHall on Hz are substantially different for each contribution. It allows to separate each contribution from an experimental measurement of vs Hz (See here)

Anomalous Hall effect (AHE)

The Charge (or orbital moment) of conduction electrons + the Spin of localized electrons

Kondo- type AMR/ Planar Hall effect

The Spin of conduction electrons + the Spin of localized electrons

## Total Hall angle as the sum of all contributions

A conduction electron has a orbital moment, which depends on a movement direction and position of the electron. Magnetic field, which created by the spin of localized electrons, interacts with the orbital moment of a conduction electron. The energy is higher when the spin M of localized electron is along the orbital moment of conduction electron and the energy is smaller when M is opposite to the orbital moment. The energy difference makes the scattering probability dependent on M and scattering direction, because of the directional dependence of the orbital moment. The difference in the scattering probabilities creates an electron current (Hall current) perpendicularly to the bias current. There is an exchange interaction between a conduction and localized electron, which depends on their mutual directions. Additionally, the exchange energy depends on the overlap of wavefunctions of conduction and localized electrons. Therefore, the exchange energy depends on the position and movement direction of the conduction electrons. This directional dependence of the exchange energy makes the scattering of the conduction electron spin- dependent and direction dependent. The difference in the scattering probabilities creates an electron current (Hall current) perpendicularly to the bias current. the dependence of Hall angle αHall vs Hz is no-linear, has a hysteresis loop and non-symmetric vs Hz polarity. The complex symmetry of the hysteresis loop is due to different symmetries of different contributions.
Origin of AHE: Dependence of electron scattering probability on spins of localized electron. See details about AHE is here Origin of spM Hall effect: Dependence of electron scattering probability of a conduction electron on mutual spin directions of conduction and localized electrons . See details about spM Hall effect is here αAHE=20 mdeg; αISHE=4 mdeg; αHall,spM=2 mdeg; αOHE=1 mdeg/KG; sp0=70% ;Hpump= 1kG;
shows a conduction electron. shows a localized d- electron. The arrow shows the electron spin, when the interaction with the spin is important.
click on image to enlarge it. Zayets 2020.03

## types of the Hall effect

Inverse Spin Hall effect (ferromagnetic metal)

The charge is accumulated, when a spin-polarized drift current flows

Spin Hall effect

The spin is accumulated, when a spin-unpolarized drift current flows

Inverse Spin Hall effect (non-magnetic metal)

The charge (and spin) is accumulated, when a spin diffusion current flows.

Under an applied voltage the electrical current (drift current) flows in the metal wire. When the metal is ferromagnetic, the drift current is spin-polarized. Therefore, there are more electrons with spin directed up. It causes more electrons be scattered into the left than into into the right. This is the reason for the charge accumulation at the right side of the wire. Under an applied voltage the drift current flows in the non-magnetic metal wire. The drift current is spin-unpolarized and the electrons have spin in any direction with an equal probability. Since the probability to be scattered to the left is higher for electrons with spin directed up and the probability to be scattered to the right is higher for There is a region of spin accumulation at backside of the wire. The diffusive spin current flows from the region of a higher spin accumulation to the region of a lower spin accumulation. This means that spin polarized electrons flow from back to front of the wire. In the opposite direction the spin-unpolarized electrons flow. The scattering probability of spin-up electrons into the right is higher. This is the reason for the charge accumulation at the left side of the wire.
This effect is used to measure the spin polarization in a ferromagnetic metal. See details here   This effect is called the ISHE- type spin detection. See details here
Figure shows the side- jump scatterings in the electrical field of a defect as an example. Any mechanism of spin-dependent scatterings contributes to these effects: mechanism 1, Skew scatterings (mechanism 2), side- jump scatterings (mechanism 4, mechanism 5)
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#### Is there any clear or evident relation between AMR/PHE and AHE? In my experience materials with strong AMR/PHE not necessarily exhibit strong AHE and vice-versa. But since they are both "magnetization angle dependent phenomena" I was just wondering if from a fundamental point of view they could be related, or if some reasonable theory relates them.

There is no relation between AMR/PHE and AHE.

From an experiment, this fact is known for a while. See, for example, T.R. Mcguire and R.I. Potter, IEEE Trans. Magn. (1975).

From theory point of view, the AMR/PHE and AHE have different symmetries and different physical origins. Therefore, they are two very different effects.

Difference 1: difference in the symmetry

The AHE is a linear magneto-transport effect. The AMR/PHE is a second- order magneto-transport effect. In a nanomagnet there are 3 independent variables, which time-inverse symmetry is broken: (1) externally-applied magnetic field H; (2) the total spin Sd of localized d- electrons (or the magnetization M) and (3) the total spin Scond of the spin-polarized conduction electrons (or the spin polarization). A linear magneto-transport effect is linearly proportional either to H or Sd or Scond. The ordinary Hall effect is proportional to H. The AHE is linearly proportional to Sd. The inverse spin Hall effect (ISHE) is linearly proportional to Scond. A 2nd order magneto-transport effect is proportional to a product of a pair from H, Sd and Scond. Additionally to the AMR/PHE, the in-plane GMR is also a 2nd order magneto-transport effect.

Difference 2: difference in the physical origin.

The AHE is dependent only on the magnetization (Sd) and is independent of spin polarization of the conduction electrons (Scond). In contrast, the AMR/PHE depends on both the magnetization and the spin polarization. The origin of the AHE is the spin-dependent scatterings of conduction electrons, which depend on the spin of a d- electron, but is irrelevant to the spin of a conduction electrons. The origin of the AMR/PHE is also spin-dependent scatterings of conduction electrons, but of different type, which depend on the angle between spin of a d- electron and the spin of a conduction electron.

#### How to distinguish Planar Hall effect from Anomalous Hall effect experimentally??

A 1st order magneto-transport effect (Anomalous Hall effect, Inverse Spin Hall effect & Ordinary Hall effect) can be easily distinguished experimentally from a 2nd order magento-transport effect (AMR/PHE, in-plane GMR etc.). Since the 1st order magneto-transport effect is linearly proportional to magnetization M + external magnetic field H, it reverses its polarity when H+M are reversed. In contrast, the 2nd order magento-transport effect is proportional to a square/product of magnetization M + external magnetic field H, it does not reverse its polarity when H+M are reversed.

It is a common rule for any magneto-transport measurement that two measurements are always done with the magnetization in the forward and reversed direction. Next, the symmetric and antisymmetric contributions of measurements are calculated. The anisymmerical contribution is associated with the 1st order magneto-transport effects and the symmerical contribution is associated with the 2st order magneto-transport effects. In this way any unwanted contribution of 1st order magneto-transport effects to a measurement of a 2st order magneto-transport effect can be avoided and vice versa.

As an example see my AMR/PHE measurement for nanomagnets here.

I am strongly against a fake and "highlight" research