Abstract:

The family of the Hall effects and the magneto- resistance effect are described. The effect are distinguished by their symmetry with respect to a reversal of a substantially large magnetic field and direction in which magnetically created current flows. When it flows parallel to the main current, the resistivity of the wire is changed and the effect called the Anisotropic magneto- resistance (AMR). When the magnetically- created current flows perpendicularly to the wire, a charge is accumulated at sides of the wire and the voltage is created across wire, which is called the Hall voltage, and the effect is called the Hall effect. Additionally, magneto- transport effects are distinguished by whether their due to spin features of either localized d- electrons or conduction electrons or both.

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Content

 

1.Odd and even magneto- transport effects

2. 1st- order odd magneto- transport effects

2.1. Ordinary Hall Effect (OHE)

2.2 Anomalous hall Effect (AHE)

2.3 Inverse Spin Hall effect (ISHE)

2.4 Spin Hall effect

3. 2nd- order even magneto- transport effects

3.1 AMR and Planar Hall effect: two features of one single effect

3.2 AMR

3.3 Planar Hall effect

3.4 Kondo type AMR/ Planar Hall effect

3.5 Spin- dependent conductivity

3.5a In-plane GMR effect

3.6 AHE- type AMR/ Planar Hall effect

3.7 ISHE- type AMR/ Planar Hall effect

3.8 OHE- type AMR/ Planar Hall effect

3.9 MH- type AMR/Planar Hall effect

4. DC measurement of the Hall effect

5. RF measurement of magneto- transport effects

6 Influence of a Hall probe on a Hall measurement

7. Magneto- capacitance induced by the Hall effect

(10). Questions & Answers

(q1) Planar Hall/ AMR vs. AHE

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

 

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Distinguish features of a magneto- transport effect:

(Distinguish feature 1) a magnetically- generated current is linearly proportional to current j_bias

(Distinguish feature 2) a magnetically- generated current is reversed its polarity when to current j_bias is reversed

 

(Distinguish feature 3) a magnetically- generated current is proportional either an external magnetic field H or/and total spin of localized electrons S_local or/and total spin of conduction electrons S_cond .

(Distinguish feature 4) an electrical current should change distribution of at least one magnetic property of conduction electrons. It should change the distribution of spins or orbital moments

In the case of an ISHE- like effect, the electron current changes the distribution of the spins. In the case of an AHE- like effect, the electron current changes the distribution of the orbital moments.

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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

Anisotropic Magneto- resistance (AMR), Planar Hall effect, , spin detection effect

 

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1st- order odd magneto- transport effects

These effects are linearly proportional to either an external magnetic field H_ext , the total spin S_local of localized d- electrons or the total spin S_cond of conduction electrons

Three types to linear odd Hall effects

 

(Linear odd Hall effect 1) Ordinary Hall effect (OHE)

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):

a_OH is the rotation angle of the ordinary Hall effect (in mdeg/kG). H is external magnetic field. a_OH is positive for the hole- dominated conductivity. a_OH is negative for the electron- dominated conductivity. j_V 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. j_Hall is electrical current flowing perpendicular to wire; j_V is electrical current flowing along to wire due to applied voltage V; 

 

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

 

 

 

 

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(Linear odd Hall effect 2) Anomalous Hall effect (AHE)

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):

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

where M is an unit vector in direction of magnetization; j_Hall is electrical current flowing perpendicular to wire; j_V is electrical current flowing along to wire due to applied voltage V; 

 

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

 

 

 

 

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(Linear odd Hall effect 3) Inverse Spin Hall effect (ISHE)

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):

or the same

where  P_s is spin polarization. m is unity vector along spin- direction of spin-polarized conduction electrons; a_IIHE is the rotation angle of the Inverse Spin Hall effect (in mdeg); S_conduct is the total spin of conduction electrons; j_Hall is electrical current flowing perpendicular to wire; j_V is electrical current flowing along to wire due to applied voltage V; 

 

(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

 

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(Linear odd Hall effect 3a) Spin Hall effect (ISHE)

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):

 

a_AH is the rotation angle of the Anomalous  Hall effect (in mdeg).  S_local 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  P_s 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.

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2nd- order even magneto- transport effects

Three contributions to linear even Hall effect in a ferromagnetic nanomagnet

(contribution 1) Anomalous

 

 

 

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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

 

(generated current flows perpendicularly to bias current) Hall effect

(current

 

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In-plane Giant Magneto- resistance (GMR) effect

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.

 

 

 

 

 

 

 

 

 

 

 

 

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Influence of a Hall probe on a Hall measurement

 

 

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

where V is the voltage applied to the nanowire, α_Hall 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 w_p also includes the length of the probe. It may cause a systematic error in the measurement of the Hall angle α_Hall

 

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.

 

 

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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

RF measurement of Inverse-Spin Hall effect (ISHE) in a nonmagnetic material
Experimental setup 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 and I. Solomon, “Experimental evidence of the anomalous hall effect in a nonmagnetic semiconductor,” Phys. Rev. Lett., vol. 29, no. 25, pp. 1676–1679, 1972. 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.
click on image to enlarge it

 

 

 

 

 

 

 

 

(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

 

 

 

 

 

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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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Questions & Answers

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.

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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.

 

 

 

 

 

 

 

 

 

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I am strongly against a fake and "highlight" research