Anisotropic magnetoresistance (AMR), Planar Hall effect V.Zayets

Two features of one effect: Anisotropic magneto- resistance & Planar Hall effect

FeB nanomagnet on top of tantalum (Ta) nanowire. A Hall probe is aligned to the nanomagnet. The nanovoltmeter (at the left) measures the Hall voltage and resister meter (at the right) measures the total resistance. The FeB magnetization (green ball) is in-plane. Direction of external in-plane magnetic field H_\|\| (green arrow) is rotating. The magnetization follows the H_\|\| rotation. There is a electron current j_V flowing through the nanomagnet.
j_M is magnetically- generated current, which direction is along (or opposite) to magnetization direction and which magnitude is proportional magnetization component along current.
AMR effect due to j_M component \|\| current j_V. When magnetization \|\| to wire, is opposite to and wire resistance increases. When magnetization ⊥ to j_V , j_M =0 and wire resistance is smaller.
Planar Hall effect due to j_M component ⊥ current j_V. The perpendicular component of j_M and therefore Hall voltage is largest when magnetization is at 45 deg. with respect to j_V
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Anisotropic magnetoresistance (AMR)

wiki page is here and classic review paper on the AMR is T.R. Mcguire and R.I. Potter, IEEE Trans. Magn. (1975)

The effect of the anisotropy magnetoresistance describes the fact the resistance of a ferromagnetic metal becomes smaller, when the magnetization of the metal changes from parallel to perpendicular direction in the respect to the direction of the electrical current flowing in the metal. Spin- dependent scatterings originate this effect.

Planar Hall effect

The Planar Hall effect is a Hall voltage created when magnetization is in-plane (in one plane with current and Hall probe. . It is largest when at 45 degrees with respect to current direction and Hall probe.

AMR and the Planar Hall effect are two features of a single magneto- transport

(measurements): Both the AMR and the Planar Hall effect are measured by rotating magnetization in-plane. Usually, the resistance is highest when magnetization is parallel to the current and is lowest when magnetization is perpendicular to the current. The Hall voltage is largest when angle between current and magnetization is 45 degrees. There is no Hall voltage, when magnetization is either parallel or perpendicular to the current.

(another measurement) The magnetization also can be rotated from perpendicular to plane direction to parallel-to current direction, Similarly, the resistance is highest when magnetization is parallel to the current. There is no Planar Hall effect (there is no Hall voltage) in case of such rotation.

Mathematical description of AMR+ Planar Hall effect

Anisotropic Magneto Resistance (AMR) & Planar Hall effect (PHE)

AHE	PHE

Magneto-resistance as function of magnetization direction
AHE. Radar plot of resistance	PHE. Radar plot of Hall angle

Magneto-resistance (upper plot) and resistance (lower plot)as function of angle φ of in-plane magnetization direction with respect to current direction. The magnetization is in-plane.	Hall angle θ_PH as function of angle φ of in-plane magnetization direction with respect to current direction. The magnetization is in-plane.

AMR: Resistance is smaller when magnetization perpendicular to the current and larger when magnetization parallel to the current

Planar Hall effect: Hall angle θ_PH is largest positive at φ =-45^o and φ =180^o-45^o. Hall angle θ_PH is largest negative at φ =+45^o and φ =180^o+45^o. θ_PH =0 at φ =0^o; +90^o; -90^o; 180^o

Sample Volt 54a R13C (see here about sample details)

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AMR+ Planar Hall effect is an even magneto- transport effect (see here), which polarity is not reversed when direction of a substantially substantially large magnetic field is reversed.

Both the AMR and the Planar Hall effects are described by one equation, which is describes the magnetically- created current j_M as:

where is j_v the conventional electron current flowing through metallic wire from source to drain, α_AMR is AMR coefficient or AMR angle., M is the magnetization

The projection of j_M parallel to j_v describes the AMR effect:

The projection of j_M perpendicularly to j_v describes the AMR effect:

where φ is the angle between magnetization M and current j_v.

AMR effect:

The total flowing through the wire is a sum of and :

The current is largest and therefore the wire resistance is smallest when M perpendicular to the wire (φ =+90^o or -90^o). In this case, j_AMR=0 and j=j_M.

The current is smallest and therefore the wire resistance is largest when M parallel to the wire (φ =+90^o or -90^o). In this case

The wire resistance is

It is largest, when M parallel to the wire:

The AMR ratio is define as relative change of resistance when the magnetization is rotated from to be parallel to to be perpendicular with respect to current direction

The resistance vs. φ can be calculated as

Planar Hall effect

The Hall voltage V_Hall. for the Planar Hall effect is calculate as (see here)

The magnitude of Hall voltage is largest V_Hall, when magnetization M is at 45 degrees with respect to current j_v (φ =+45^o or -45^o or 180^o+45^oor 180^o-45^o). V_Hall is positive at -45^o and 180^o-45^o and negative at +45^o and 180^o+45^o. V_Hall=0 at φ =0^o; +90^o; -90^o; 180^o

The Hall angle θ_PH for the Planar Hall effect is calculate as

The maximum Hall angle at φ =+45^o; -45^o; 180^o+45^o;180^o-45^o is calculated as

The Hall angle θ_PH for the Planar Hall effect vs φ is calculate as

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AMR vs Planar Hall effect

Comparison of Eq.1.8 and Eq.1.11 gives

(note) Eq.1.12 is often used to measure the AMR in a nanomagnet.

A direction measurement from the resistance change often contains a systematic error due to the contact resistance and the nanowire resistance. In contrast, the benefit of the AMR measurement from the Planar Hall effect benefits of a locality of measurement and independence from contact and wire resistances .

Measurement of AMR

(bulk material): By a measurement of change of resistance when magnetization changed from perpendicular to parallel direction with respect to electrical current.

(a nanomagnet):From a measurement of the Planar Hall effect. The resistance of nanomagnet is substantialy smaller than a parastic resistance of nawire and contacts. It makes unreliable the evaluation of the AMR from the resistivity change.

AMR & Planar Hall effect in different materials

AMR in ferromagnetic metals

Ni_xFe_1-x	Ni_xCo_1-x

Anisotropic magnetoresistivity ratio in percent for Ni_xFe_1-x alloys at room temperature. Bulk sample.	Anisotropic magnetoresistivity ratio in percent for Ni_xCo_1-x alloys. Bulk sample.

data from T. R. Mcguire and R. I. Potter, “Anisotropic Magnetoresistance in Ferromagnetic 3D Alloys,” IEEE Trans. Magn., 1975.

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The AMR is know to be high in nickel containg materials and can reach cose to 10%. In Fe and Co, the AMR is not as large, depends on device geometry and typically is smaller than 0.1 %.

(about a peak of AMR) There is a peak for dependence of the AMR on Ni content, where AMR has a maximum. For a long time it has not been understood why this content of Ni (~0.9) is so special for the AMR.

Both the saturated magnetization and spin polarization of conduction electrons are small in nickel. Why Ni is so special for AMR and the AMR is substantially larger in Ni than e.g. in Fe and Co?

The AMR effect is origanated from special spin- dependent scatterings, which depends on mutul spin direction of locolized and conduction electrons. Therefore, the scattering probability for spin- polarized conduction electrons is larger, for example, toward the left than towards right. It occurs due to influence of the spin of localized d-electrons on the strength of the spin- orbit interaction of the conduction electrons (See below). Such influence occurs only for a spific orbital configuration. The orbital of Ni is well matched t that condition and therefore the AMR in Ni is large. The orbitals of Fe and Co do not promote this kind of the spin-orbit interaction. As a result, the AMR is weak in Fe and Co.

Why there is an optimum Ni concentration in NiFe and NiCo, when the AMR is largest?

The AMR linearly proportional to three paramenters: (parameter 1): total spin of localized d- electrons (magnetization); (parameter 2): the total spin of spin- polarized conduction electrons (spin polarization) ; (parameter 3): the probability of required spin- dependent scattering. In Ni, the probability of the spin- dependent scatterings is high, but both the spin polarization and magnetization are small. Adding some Fe or Co to Ni increases the spin polarization and the magnetization, which leads to an increase of the AMR. At the same time, the adding of Fe or Co reduces the probability of the required spin-dependent scatterings, which leads to a decrease of the AMR. There is an optimum Ni concentration, at which the spin polarization and the magnetization are large enough, but the the probability of the spin-dependent scatterings is not reduced much.

Origin of AMR & Planar Hall effect

(general notes):

(note 1) As a magento- transport effect, AMR/PHE is originated from spin- dependent scatterings of the conduction electrons.

(note 2) In a general case, the spin direction of spin-polarized conduction electrons and localized electrons are the same, which is the magnetization direction

(There are a few exceptions).

(note 3) The current j_AMR/PHE of the AMR/PHE effect flows along the magnetization direction.

There are 3 contributions to AMR/PHE effect:

(contribution 1): Kondo- type contribution ⇔

It is the main contribution. The contribution is due to a scattering of the spin-polarized conduction electron on the localized electrons and which is proportional to mutual direction of spins of the localized and spin-polarized conduction electrons. The current j_AMR/PHE of the AMR/PHE effect can be described as:

where j is the electron current, S_local is the total spin of the localized electrons, S_condis the total spin of the conduction electrons.

(contribution 2): 2nd-AHE-like contribution ⇔

The contribution is proportional to the spin of the localized electrons S_local, but is independent of the spin of the conduction electrons S_cond.

The symmetry of this contribution is the same as if the current experience twice the Anomalous Hall effect (AHE). In fact, the Hall current cannot experience AHE second time due to the geometrical restriction (see below). However, spin-dependent scatterings may have similar symmetry.

(contribution 3): 2nd- ISHE-like contribution ⇔

The contribution is proportional to the spin of the conduction electrons S_cond, but is independent of the spin of the localized electrons S_local.

The symmetry of this contribution is the same as if the current experiences twice the Inverse Spin Hall effect (ISHE). Simalary, the Hall current cannot experience ISHE second time due to the geometrical restriction (see below). However, spin-dependent scatterings may have similar symmetry.

Classical incorrect origin of AMR: Two consequent Hall rotations.

The AMR effect and the effect of Planar Hall resistance can be explained as two consequent Hall rotations as follows

AMR as two consequent Hall rotation. (Incorrect suggestion)

Incorrect classical suggestion	the real case	Charge accumulation & compensate current j_comp

The result of the Hall rotation for current j_V is perpendicularly- flowing current j_Hall,1 Hall . The current j_Hall,1 itself experience the Hall rotation. The result of this rotation is the current j_Hall,2, which flows perpendicularly to j_Hall,1 and opposite to j_V.	in the reality, an electron current can only flow from a source to a drain and therefore only along j. Any current flowing perpendicularly to j_V creates a charge accumulation and is balanced by exactly opposite current j_comp. As a result, in total there is no current perpendicularly to j_V and the current j_Hall,2cannot be created.	The Hall current j_Hall,1, which flows perpendicularly to to the wire, creates a charge accumulation at sides of the wire. The charger accumulation creates the compensated current, which balance . As a result, in total there is no electron current perpendicularly t the wire

When magnetic field H (or magnetization M) is perpendicular to bias current j_V, there is a Hall current j_Hall flowing perpendicularly to j_V

The Hall current j_Hall flows perpendicularly to H and therefore itself experience the Hall effect. The second Hall current j_Hall,2 can be calculated as

using the vector relation

The Eq. (2.3) is simplified as

The first term of Eq. (2.5) describes the AMR and Planar Hall effects:

The second terms of Eq. (2.5) describes the direction-independent magneto-transport effect

Eq 2.6 became same as

Why this classical interpretation of AMR is incorrect?

A. This explaination requires a flow of electron current perpendicularly to the wire. However, there is no such a current. The Hall current j_Hall is fully ballanced the opposite current, which is induced by the Hall voltage. As there is no perpendicular current, there is no secondary Hall effect.

(reason 1) Change accumulation on side of wire. Balancing of the Hall current

Any electron current, which flows perpendicularly to a metallic wire, creates a charge accumulation on sides of the wire. A balance current flows opposite to the Hall current. Therefore, in total there is no current, which flows perpendicularly to the wire, the balance current fully compensate the Hall current and the second Hall current cannot be created.

(reason 2) polarity of AMR

As can be seen from Eq.2.8, the experimentally observed AMR is opposite to that expected from two consequent Hall effects. When magnetization is perpendicular to the current, there is Hall effect, two 90 deg rotations makes the 2nd Hall current flowing opposite to the main current and therefore the resistance should increase. Experimentally observed dependence is opposite (See Fig.), the resistance is largest when the magnetization is parallel to the current.

(note) Even though the two consequent Hall rotations is not origin of the AMR effect, it give all correct description of all even magneto transport effect The first term in Eq.2.5 describes the AHE + Planar Hall effect. The second term describes the spin- dependent conductivity.

(note about wire capacitance) in the case of an AC current, the charge accumulation on the sides of wire are modulated and therefore the current j_Hall,1 is not fully balanced by j_balanc and j_Hall.2 exists. As a result, the two consequent Hall effects to MNR effect in the case of an AC current.

(note) The Hall effect changing the capacitance of the wire. As a result, the capacitance of the wire can be changed by external magnetic field.

(note) Another effect, which contributes to the capacitance of a metallic fire is the effect of the spin- detection of the spin- dependent conductivity.

(note) The dependence of capacitance of a metallic (e.g. ferromagnetic) wire on an external magnetic field is used in a high-sensitivity low- cost sensors of a magnetic field. For example, any smart phone is equipped by such sensor.

(note about Quantum Hall effect) in a very large magnetic field and a very low temperature, the Hall angle becomes 90 degrees, the electrons starts to move a circle paths and there is no j_balanc. This effect is called the Quantum Hall effect.

Family of AMR/Planar Hall effects

The AMR& Planar Hall effect is an even magneto- transport effect (See here). Any even magneto-transport effect should be described as a vector product of two vector sets: (H, S_local,S_conduct )*(H, S_local,S_conduct ), where H is external magnetic field, S_local, is the total spin of localized electrons and S_conduct is the total spin of conduction electrons. Different vector product of these vectors defines a different type of the AMR& Planar Hall effect

Kondo- type AMR/Planar Hall effect

(origin) Kondo- type AMR/Planar Hall effect is originated from interaction of spin of localized electrons S_local and spin of conduction electrons S_conduct. Kondo- type AMR/Planar Hall effect is linearly proportional to the spin polarization of conduction electrons.

(note) Prof. Kondo studied the interaction of spins localized and conduction electrons in order to explain the origin of Anomalous Hall effect (AHE). However, the AMR effect is a 1st-order magneto- transport effect, which is linearly proportional to M) and therefore such even interaction cannot contribute to the AMR. In contrast, the interaction of spins of localized and conduction electrons contributes to the 2nd AMR/Planar Hall effect, which is proportional to a vector of magnetization M and magnetization of M. The scattering probability of the Kondo interaction is proportional to a product of the spin of the conduction electron, which is proportional to M, and the spin of the localized electron, which is again proportional to M. It makes the Kondo interaction the 2nd order magneto-transport effect, which is proportional to a vector product of M and M.

The scattering of the spin-polarized conduction electrons on localized d-electrons produces an electron current j_M.

The component of j_M, which is along the wire and, therefore, parallel to the main current j_V, is originating the AMR effect. When j_M is parallel to j_V, for a fixed applied voltage the total current increases meaning the wire resistance decreases. When j_M is opposite to j_V, the total current decreases meaning the wire resistance increases. Therefore, the magnetization- dependent current makes the resistance to become dependent on the magnetization direction, which is the effect called Anomalous Magneto-Resistance (AMR).

The component of j_M, which is perpendicular to the wire and, therefore, perpendicular to the main current j_V, is originating the Planar Hall effect (PHE) effect. When a current flows perpendicular to a wire, a charge is accumulated at side edges of the wire. The voltage of this accumulated charge is called the Hall voltage.

The current j_M, which is due to a spin-dependent scattering of spin-polarized conduction electrons on the spins of the localized electrons, should be proportional to the total spin of localized electron S_local, to the total spin of conduction electrons S_conduct and to the electron current j_V. There are a few possible options when a vector jm can be represented as a linear product of 3 vectors.

Additionally, the scattering probability should symmetrically depend on the spins of localized and conduction electrons. The scattering only on mutual spin direction without any preferences for spin of the localized or conduction electrons. This condition reduces the possibility only to one option:

Since S_conduct is linearly proportional to spin polarization P_s and S_local is a constant, Eq. (3.1) is simplified as

where unit vectors M and m are unit vectors in spin directions of conduction and localized electrons.

AMR effect:

Assigning φ+Δ as angle between spin of localized electrons and current j_V and φ-Δ as angle between spin of spin- polarized conduction electrons and current j_V, the projection j_AMR of magnetic current j_Vf, which flows along the wire and, therefore, parallel to the main current j_V, and which defines the AMR effect, is calculated from Eq.(3.2) as

Therefore, the AMR current can be calculated as

Except that the the AMR constant is linearly proportional to the spin polarization , the Kondo- type AMR effect is nearly the same as classical AMR (see Eq.1.2)

Planar Hall effect

The Planar Hall effect is described by the perpendicular components of the magnetic current:

(note) Only the spin- polarized conduction electrons contribute to the AMR/Planar Hall effect. The spin- unpolarized conduction electrons do not contribute

(note) Kondo- type AMR/Planar Hall effect is linearly proportional to the spin polarization P_s of conduction electrons.

Origin 1of Kondo- type AMR/Planar Hall effect: Spin-orbit interaction

(fact ) The total spin S_conduct of spin-polarized conduction electrons is linearly proportional to the number of spin-polarized electrons and therefore to the spin polarization P_s

Spin- dependent conductivity. Spin- detection effect.

Another Kondo-type magneto- transport effect is the Spin- dependent conductivity. This effect describes the Spin detection effect (See here) and the in-plane giant magnetic resistance (GMR) effect.

This Kondo-type magneto- transport effect is proportional to the scalar product of total spins of localized and conduction electrons:

Since this magnetic current is always along the main current j_M || j_V. the effect can be describe as a change of the total current

as in this case

or conductivity

if φ is the angle between the spin of localized electrons (magnetization) and the spin of conduction electrons and P_s is the spin polarization of the conduction electrons. The spin- dependent conductivity can be described as

(fact 1 about spin- dependent conductivity) Spin-detection

The spin detection effect describes the creation of charge and voltage along diffusion path of a spin current. The origin of the spin- detection effect is the spin- dependent conductivity.

A spin diffusion current is a current of spin without a current of charge. It consists of two currents of spin-polarized and spin-unpolarized electrons, which flow in opposite directions. When

(fact 2 about spin- dependent conductivity) The effect of the spin- dependent conductivity does not contribute to the AMR or the Planar Hall effect

The spin

(fact 3about spin- dependent conductivity) In-plane GMR

The spin

(fact 4 about spin- dependent conductivity) spin- dependent conductivity is large at an interface

The spin

(fact 5 about spin- dependent conductivity) spin- dependent conductivity is large in a material with a low conductivity

The spin

AHE- like AMR/Planar Hall effect

(origin) Two subsequent scatterings of spin- unpolarized conduction electrons

AHE- like AMR/Planar Hall effect is proportional only to the total spin S_local of localized d- electron (but not to the spin S_cond of conduction electrons):

Since the total spin of localized d- electrons is along magnetization, the AHE- like contribution is calculated as

where M is an unit vector directed along magnetization.

(note) this contribution is independent of spin polarization of conduction electrons

(note) Both the spin- polarized and spin- unpolarized conduction electrons contribute to the AHE- like AMR/Planar Hall effect.

ISHE- like AMR/Planar Hall effect

(origin ) Two subsequent scatterings of spin- polarized conduction electrons

ISHE- like AMR/Planar Hall effect is proportional only to the total spin S_cond of spin-polarized conduction electrons (but not to the spin S_locallocalized d- electron):

Since S_cond is linearly proportional to the number spin- polarized electrons and therefore to the spin polarization P_s, the ISHE- like AMR/Planar Hall effect is described as:

where M is an unit vector directed along magnetization

(note) this contribution is proportional to a square of spin polarization P_s of conduction electrons

(note): Only the spin- polarized conduction electrons contribute to the ISHE- like AMR/Planar Hall effect. The spin- unpolarized conduction electrons do not contribute

OHE- like AMR/Planar Hall effect

(origin ) The Lorentz force during Two subsequent scatterings of a conduction electrons

In this case the direction of magnetic current is along the direction of external magnetic field H

MH- type AMR/Planar Hall effect

(origin 1) Simultaneously with a spin- dependent scattering an electron experiences the Lorentz force.

It is a prominent feature of a magnetically- hard material (e.g. SmCo), in which the magnetization is firmly fixed along its easy direction and is nearly independent on external magnetic field H.

For this type, there is component (~S_cond ), which is linearly proportional to the spin polarization and there is a component (~S_local), which is independent of spin polarization P_s.

Additionally MH- type AMR/Planar Hall effect is divided into two additional types:

(type 1) in which magnetically-created current is parallel to the magnetic field: j_M || H

(type 2) in which magnetically-created current is parallel to the magnetization: j_M || M

AMR of FeB film

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Incorrect assumption that the Lorentz force is the origin the AMR effect

The resistance of FeB wire as function the angle between applied magnetic field and the direction of electrical current in the wire. 0 and 180 deg correspond to the case when magnetic field is along the wire. -90 and 90 deg correspond to the case when magnetic field is perpendicular to the wire and in film plane.

AMR is about 0.09%.

Important! The resistance becomes larger when the magnetic field is applied along the wire.

This polarity of AMR is the same for the most of metal only with a few exceptions of some monocrystal metals.

Sample Hall 16. Measured on 2016/02/29

When an electron moves between scatterings in an magnetic field, it experiences the Lorentz force. The Lorentz force turns out the electron from a straight path. Therefore, the magnetic field slows down the movement of the electron and the electrical resistance should increase when a magnetic field is applied perpendicularly to the electrical current.

It is important: The metal resistance decreases when a magnetic field is applied perpendicularly to the wire (AMR effect) comparing to the case of no field or a field applied along the wire (See experimental measurements ).

It is clear evidence that not the Lorentz force, but spin-dependent scatterings is origin of the AMR effect.

Q. What is the difference between cases when magnetic field is applied along electron current or perpendicular to current?

Spin polarization of electron gas is along to the applied magnetic field. When magnetic field is perpendicular to the electron current, the scattering is spin-dependent and scattering probabilities towards left and right are different. When magnetic field is along the electron current, the scattering becomes spin-independent.

Q. Why the resistance of a metallic wire becomes larger when a magnetic field changes from perpendicular to parallel directions with respect to the direction of electrical current in the wire?

When magnetic field is perpendicular to electron movement, the scattering probability toward left and right are different. In contrast when magnetic field is along to electron movement the scattering probability toward left and right are same.

It is important that the total scattering probability are different for perpendicular and parallel directions of the magnetic field, because of the non-linear nature the Fermi-Dirac distribution. Usually

and the metal resistivity is larger when magnetic field is along the electron current.

However, in some rare cases when there is a substantial energy dependence of the density of state at the Fermi surfers, the resistivity may be larger for a perpendicular magnetic field.

Kondo Mechanism of AMR + Planar Hall effect: Spin - orbit interaction

Scatterings become spin-dependent, when a electron moves perpendicular to the magnetic field	Scatterings become spin-independent, when a electron moves along the magnetic field

Fig.20

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Q. The electrons in a metal move in all directions, why only electrons, which move along the electron current, contribute to the AMR effect?

A. All electrons contribute to the AMR effect. The contribution of the electrons, which move in the opposite directions, are of opposite signs. Along electron current the number of electrons, which moves along and opposite current, are different and their contributions do not compensate each other.

Q. Is any correlations between the AMR effect and Anomalous Hall effect, the Spin Hall effect and Inverse Spin Hall effect?

AMR effect due to orbital moment of conduction electrons

Kondo mechanism: Anisotropic magnetoresistance (AMR) due to orbital momentum of electron

case 1: orbital moment parallel to electron moving direction

Magnetization is along current	Magnetization is perpendicular to current

Magnetic moment (magnetization) of localized electrons is along y-direction. Orbital moment of electron is along its movement direction. Orbital moment of a conduction electron, which moves along y-axis, is along y-axis and therefore its magnetic energy is higher. is higher than o

case 2: orbital moment perpendicular to electron moving direction

Magnetization is along current	Magnetization is perpendicular to current

Magnetic

Sphere of directional distribution of conduction electrons. The length of a vector from axis origin to sphere is proportional to the number of electrons moving in the vector direction. In absence of electrical current J, the sphere is symmetrical and an equal amount of conduction electron moving in any direction. When electrical current flows along y-axis, sphere is shifted towards positive y- direction and more electrons flow along positive y- direction than in opposite direction.

Red/ blue bars show amount of magnetic energy, which a conduction electron experience in magnetic field H (green arrow) of spin of localized electrons (violet ball with arrow). Rotating arrows show orbital moment of conduction electron. When spin of localized electrons is along orbital moment, the magnetic energy is largest (bar is fully red) and the scattering probability in that direction is higher.

The size of arrows in lift- top corner proportional to the scattering probability in shown direction

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Dependence of AMR on external magnetic field

AMR vs external magnetic field

room temparature	T= 77° K	T= 4.2° K

Resistivity of Ni_0..9942Co_0.0058 as a function of applied magnetic field H at room temperature. The points A and B mark the field, at which magnetic domain are removed. The point B isat a higher applied field than A because of demagnetization.	Resistivity of Ni_0..9942Co_0.0058 at 77° K.	Resistivity of Ni_0..9942Co_0.0058 at 4.2° K

data from T. R. Mcguire and R. I. Potter, “Anisotropic Magnetoresistance in Ferromagnetic 3D Alloys,” IEEE Trans. Magn., 1975.

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Questions & Answers

(AMR/PHE vs. AHE) 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. At least, there is no a direct relation.

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 S_d of localized d- electrons (or the magnetization M) and (3) the total spin S_cond of the spin-polarized conduction electrons (or the spin polarization). A linear magneto-transport effect is linearly proportional either to H or S_d or S_cond. The ordinary Hall effect is proportional to H. The AHE is linearly proportional to S_d. The inverse spin Hall effect (ISHE) is linearly proportional to S_cond. A 2nd order magneto-transport effect is proportional to a product of a pair from H, S_d and S_cond. 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 (S_d) and is independent of spin polarization of the conduction electrons (S_cond). 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.

(simple explaination of the origin and symmetry of AMR/PHE) I am working in field of magnetic materials. I have tried every means to understand physics behind planar hall effect in ferromagnetic thin films. I am unable to capture it effectively. Please, help me to with the same.

The following explanation might be slightly over simplified. I hope it will help you to grasp the main idea beyond the Planar Hall effect .

(PHE/AMR symmetry) The Planar Hall effect (PHE) and the Anomalous Magneto-resistance (AMR) are two features of one effect called in short PHE/AMR. The PHE/AMR is the second order magneto-transport effect, which means it is a vector product of the magnetization, electric current and the magnetization again. It means the magnetization is included twice into the vector product. It is why it is the 2nd order magneto-transport effect. The in-plane GMR is also the 2nd - order magneto transport effect and is a very close cousin of the PHE/AMR effect. The PHE/AMR effect depends on both the total spin of the localized electrons and the total spin of the conduction electrons (the spin- polarized conduction electrons to be exact). In an equilibrium, the spins of spin-polarized conduction electrons and the spins of the localized electrons are always parallel to each other (except for some specific cases) and it is simpler to describe both their direction by one vector, the magnetization. Once again, even though the PHE/AMR is proportional to the vector product of spins of localized and localized electrons, the magnetization is used twice in the formula for the PHE/AMR.

(Physical origin of the PHE/AMR) The origin of the PHE/AMR is the spin- dependent scattering of conduction electrons, which probability depends on mutual spin directions of conduction and localized electrons. It means that when the spin of a conduction electron is parallel to the spin of a localized electron, the probability of a scattering to the left is, for example, larger than the probability to the right. In contrast, when the spin of a conduction electron is opposite to the spin of a localized electron, the probability of a scattering to the left direction is smaller than the scattering probability to the right direction. Since the conduction electrons in a ferromagnetic metal are spin-polarized, there are more conduction electrons, whose spin is parallel to the total spin of the localized electrons (to the magnetization). As a result, in the total for all conduction electrons the probability of a scattering of a conduction electron towards the left direction is larger than the scattering probability towards the right direction. It results that the electrons are accumulated at the left side of a metallic wire and are depleted at the right side. It means the negative charge is accumulated/ depleted at the opposite sides of the wire and there is a Hall voltage due to this charge, which could be measured by a voltmeter. If one could reverse the spin direction of spin- polarized electrons opposite to the spin of the localized electrons by a spin injection, for example (it is very hard to do), the Hall voltage would change its polarity.

(Physical origin of the AMR) The origin of the AMR is very similar. When the magnetization is perpendicular to the current, the scattering to the left means the scattering along the current. Therefore, in this case the scattering probability for a conduction electron along the current is larger than the scattering probability opposite to the current. As a result, there are more electrons moving in the direction of the electron current, the conductivity becomes larger and correspondingly the resistance becomes smaller. When the magnetization is turned along current, there is no such difference in the scattering probability and, therefore, the resistance increases.As before, it is the case when the spin of a conduction electron is parallel to the spin of a localized electron. If one could reverse the spin direction of spin- polarized electrons opposite to the spin of the localized electrons, the polarity of the resistance change would be reversed.

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.

(about AMR in a ferromagnettic wire) I've been trying to observe the AMR effect by measuring the resistance of a "permalloy" wire, unfortunately without success. I use a short (20mm) and thin (0.5mm) permalloy (81% Ni, 14.7% Fe, 4.5% Mo) straight wire. I measure its resistance using a highly sensitive ohmmeter. The resistance is about 0.5 Ohm and does not change one bit regardless of any externally applied magnetic field. Questions: - Is there anything wrong with the way I measure the resistance? - Is there anything wrong with the alloy in use? - Is AMR a "geometrical" effect in the sense that it requires a certain conductor geometry configuration to work, such as a planar film rather than a thin wire? If so, why? - Is there any setup required for the alloy before use, such as magnetization/demagnetization? If so, why?

A. I have never measured AMR in a ferromagnetic wire by myself and I can only suggest a possible problem. As far as I know, the measurement of the magneto-impedance is often used for a ferromagnetic wire. It is a very popular magnetic sensor. In fact, almost every cell phone has a ferromagnetic-wire sensor with a measurement of the magneto-impedance. The magneto- impedance is the complex resistance, which is measured at a RF frequency. I never heard that somebody has measured AMR in a ferromagnetic wire.

---------------- suggested problem

(problem 1: magnetic field is small) AMR is proportional to the magnetization direction, but not the applied magnetic field. Often a very large magnetic field is required to to turn the magnetization from its easy axis. E.g. a several- Tesla of magnetic field is required to turn magnetization of a thick Fe film from the in-plane to perpendicular-to-plane direction.

(problem 2: resistance is too small) For a reliable measurement of AMR, one should be able to reliably measure 0.01-0.1% change of resistance. For R=0.5 Ohm it is difficult. The contact resistance can be a problem. The heating can be a problem. The thermal resistance for your wire is not small. Reaching thermal stability can be difficult.

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( As for the material and geometry:) AMR in some materials is very small. There is a bulk contribution and there is an interface contribution to AMR. Therefore, there is a geometry dependence for AMR. You can check this classic review on AMR for the specific material T.R. Mcguire and R.I. Potter, IEEE Trans. Magn. (1975)

I use a source meter to measure resistance in nanomagnets. In FeB and FeCoB, AMR is small, I prefer to measure Planar Hall effect instead. AHE and PHE are two sides of one effect. In FeTbB nanomagnet, AMR is large.

Regarding problem 1: "AMR is proportional to the magnetization direction, but not the applied magnetic field." Keeping that in mind, how is an AMR magnetometer sensor able, after all, to measure the external magnetic field, including its amplitude? Is it the case that the permalloy film is magnetized with a vector M (pointing to the easy axis), then any external field B can be measured (including amplitude) just by sensing the cosine of angle M + B relative to the easy axis? If so, does it require prior magnetization of the permalloy film in the direction of the easy axis? Is that why AMR magnetometer sensors require a SET/RESET action prior to utilization? Keeping that in mind, does it mean that in order to observe AMR on a permalloy wire (or on film), the wire needs to be "permanently" magnetized first in the direction of the easy axis (using a very strong field, as you suggested)?

Regarding problem 2: That's probably not the case, we use high-end ohmmeter which is able to sense 0.01% changes quite reliably. Nothing changes when we introduce a strong external magnetic field. If AMR had worked, we would've expected to see *some* change in resistance, but it's absolutely static completely indifferently to any externally applied strong magnetic field.

A general question: do you feel that the AMR effect is "strong" or "reliable" enough to be measured and used on anything other than a perfectly calibrated nano-film on an IC (such as in off-the-shelf AMR magnetometer sensors)?

(about wire-base sensor of a magnetic field)

The magneto-impedance effect, but not AMR effect, is used for an application of a ferromagnetic wire as a sensor of magnetic field. You can check works of the group from Basque University (San Sebastian). They have been researching the wire-type sensor for over a decade.

In a classic AMR-based magnetic- field sensor, a ferromagnetic film, but not wire, is used. The magnetization direction is changing in-plane from being along current (and in-plane) to perpendicular to current (and in-plane) directions. A weak shape anisotropy is used, therefore, a small magnetic field can be detected. The detection of both the amplitude and direction is possible.

In a modern MTJ-based magnetic- field sensor, the magnetic field may be applied perpendicular to the plane. In this case, a thin film, which has the anisotropy field of one or a few kGauss, is used. The anisotropy field is small for a thin film, because of a strong spin-orbit interaction at the interface and a weak influence of the bulk of the film in comparison to the interface. The precision of the sensor is about 0.1 % of the anisotropy field.

(about structure of magnetic domain and the direction of the easy axis in a ferromagnetic nanowire)

I guess the direction of the equilibrium magnetization (the easy axis) in a ferromagnetic wire is along the wire. In each magnetic domain, the magnetization changes direction to opposite to that of the neighbor domain being always along wire. A weak magnetic field along the permalloy wire should remove all domains. Only in a hard magnetic material, it is necessary to SET/RESET in order to remove the magnetic domains.

I guess it is possible that some mechanical stress may make the magnetic easy axis to be different from the direction along the wire. In this case the situation becomes more complex.

(about strength of AMR)

The Ni- containing compounds are famous for a strong AMR, but it does not mean that your specific wire has strong AMR. You should be sure that the contributions of the contact resistance and all external resistance is much smaller than your wire resistance. Try to measure the thermal-resistance in your wire. Try to monitor resistance in time (measure each 0.5 sec) and increase the current, you should see a clear increase of the wire resistance.

(about reliability of a AMR sensor)

As a small footprint sensor of a magnetic field, I think a MTJ-based sensor is much better and more reliable than an AMR sensor. When a source of microwave is available (e.g. a smartphone), a magneto- impedance sensor is much better and more reliable than an AMR sensor.

(why I measure AMR in all my nanomagnets)

The measurement of AMR is important for my study of magneto-transport properties of the nanomagnets, because AMR is proportional to the product of the total spin of localized electrons and the total of the conduction electron. When the magnetization is tilted from the nanomagnet normal, there is a non-zero angle between the spin of conduction electrons and the spin of localized electrons. From AMR measurements it is possible to evaluate this angle. This is very important data for me.

(AMR/PHE in an an Antiferromagnet) (from Vinay) I want to know if AMR/PHE is possible in Antiferromagnets? Because AFMs generally have zero overall magnetization then how AMR/PHE will generate?

(about AMR/PHE is an Antiferromagnet) Yes, the AMR/PHE generally occurs in an antiferromagnet with some exceptions. Only the case, when there is no AMR/PHE, is the case when the electron gas of the conduction electrons is not spin polarized, which is a rare case. Even the zero- spin polarization of the electron gas is not a hard condition, because the spin polarization can be created by the electrical current due to the spin dependent scatterings (~= due to Spin Hall effect).

In contrast to the Anomalous Hall effect (AHE), AMR/PHE does not change its sign at the compensation point. Another critical point might exist where AMR/PHE becomes zero due to the zero spin polarization of the electron gas at this point.

(the reason why AMR/PHE exists in an Antiferromagnet) The origin of the AMR/PHE effect is the dependence of the scattering probability of the spin-polarized conduction electrons on the spin of the localized electrons. An antiferromagnet usually consists of two magnetic lattices with opposite spin direction. Even though two magnetic lattices have exactly the same spin (therefore the total spin is zero), the electron scattering probability is usually different for these two magnetic lattices, which results in a non-zero AMR/PHE. It is easier to understand this effect in the case a compensated ferromagnet FeTb, in which spins of Fe and Tb are antiferromagnetically coupled and the concentration of Tb is adjusted so that the total spin of Fe and Tb are the same and there is no net magnetization. The spatial symmetry of localized electrons of Fe is the d- type. The spatial symmetry of localized electrons of Tb is the f- type. The spatial symmetry of conduction electrons is mostly the p- type. The scattering probability is high only for electrons of a similar symmetry or closed symmetries. For this reason, the scattering probability is moderate between conduction electrons and localized electrons of Fe, because the p- and d- symmetries are close (nearly similar). However,scattering probability is very small between conduction electrons and localized electrons of Tb, because the p- and f- symmetries are very different. Even though overall magnetization of FeTb is zero, the conduction electrons are scattered only by Fe, whose magnetization is substantial, with almost no contribution from Tb. It results in a substantial AMR/PHE effect in the antiferromagnetic FeTb ( I have not checked it experimentally yet, but it just has to be).. Of course, for each type of an antiferromagnet, the specific might be different, but I hope you understand the main idea.

(the simplest way to understand all features of any magneto- transport effect in an Antiferromagnet) The best way is to understand the magneto- transport features in a compensated ferromagnet at the beginning. In this material all features of magneto- transport are clear and straightforward. A compensated ferromagnet is a compound of d- symmetry (Fe or Co) and f- symmetry (Tb or Gd) metals. The exchange interaction is strong only between electrons of the same symmetry. For this reason, two clear magnetic lattices (Fe-electron lattice and Tb-electron lattice) are formed even in an amorphous FeTb, in which Fe-Fe, Tb-Tb and Fe-Tb distances are the same. There is a strong PMA for each magnetic lattice. Therefore, the magnetization configuration is rather simple. Since there is only scattering between the conduction electrons and localized electrons of Fe, the spin direction of spin- polarized conduction electrons is the same as the spin direction of the Fe electrons.

Anomalous Hall effect (AHE), which is proportional to the spin of localized electrons, is proportional only to magnetization of Fe for the same reason: there is only scattering between the conduction electrons and localized electrons of Fe. The polarity of the Inverse Spin Hall effect (ISHE), which is proportional to the spin polarization of the conduction electrons, also follows the magnetization direction of Fe as the spin of conduction electrons does. . There is an intrinsic magnetic field, which is a difference between magnetic fields induced by Fe and Tb spins. When there are more Fe atoms, the intrinsic magnetic field is along Fe spins. When there are more Tb atoms, the direction of the intrinsic magnetic field is reversed, which results in the polarity reversion of AHE and ISHE effects at the compensation point . AMR/PHE is proportional to the product of the total spin of the localized electrons (magnetization) and the total spin of spin-polarized conduction electrons. At the compensation point both the Fe spin and spin of conduction electrons are reversed, but the polarity of their product does not change. For this reason, there is no change of AMR/PHE at the compensation point.

*(Polarity of AMR/PHE**) (from Vinay)* I do see some experimental observations where parallel resistance is lower than perpendicular. What could be the possible reasons of it?

(positive vs. negative polarity of AMR/PHE ) Even though the positive polarity is more common, some materials have the negative polarity of AMR/PHE. All nanomagnets, which I have measured so far, have only the positive polarity of the AMR/PHE. The classical review paper on AMR (T.R. Mcguire and R.I. Potter, IEEE Trans. Magn. (1975)) reports mostly the positive polarity, but still there are materials having the negative polarity (See Fig.10). At present, it is not 100 % clear what is the reason for the polarity change, even though there are some reasonable suggestions. Even more, the true origin of the AMR/PHE is not fully understood yet. It is clear that the AMR/PHE has several contributions, which makes this effect very interesting. For example, I measure AMR/PHE for nearly all nanomagnets, which I measure, but yet I have not analyzed the raw data, because both the current- and gate-dependencies of AMR/PHE are complex and unexpected. It is because of the complex origin of the AMR/PHE effect.

(the polarity of AMR/PHE as a feature of the polarity of spin-orbit interaction) The origin of AMR/PHE is due to the asymmetry of the spin-dependent scattering. Due to the spin of localized electrons, the scattering probability of conduction electrons to the left and to the right with respect to the current direction are different. The same spin-dependent scatterings are also the origin of the Anomalous Hall effect (AHE). AHE is independent whether the conduction electrons are spin-polarized or not. A ferromagnetic metal has both groups of the spin-polarized and spin-unpolarized electrons. The spin-dependent scatterings of the spin-polarized electrons also contribute to the AMR/PHE effect. There is a difference in the scattering probability, because the magnetic field of the spin-orbit interaction H_SO is different for the electron scattered to the left and to the right. When H_SO is along the spin direction of the spin-polarized electrons, the energy of a scattered electron becomes smaller and , therefore, the electron scattering probability becomes larger. In contrast, when H_SO is opposite to the spin direction of the spin-polarized electrons, the energy of a scattered electron becomes larger and , therefore, the electron scattering probability becomes smaller (see Fig. 10).

In short, the polarity is determined by whether the electron scattering probability is larger for the scattering to the left than the right or it is larger to the right than to the left. The different polarity of the magnetic field H_SO of spin- orbit interaction for the electron movement to the left and to the right makes the scattering probability different. Therefore, the difference of the Hso polarity defines the sign of the AMR/PHE effect.

(the polarity of AMR/PHE vs. electron- or hole- dominated conductivity of a metal) The scattering probability is proportional to the product of the Fermi function and the density of states of the metal at the Fermi level. When the density of states increases for a higher energy, the metal is electron- dominated. When the density of states decreases for a higher energy, the metal is hole- dominated. If the metal is strongly hole- dominated, the change of the density of state can overcome the change due to the Fermi function and, therefore, the scattering probability to state of a higher energy may become larger than the scattering probability to state of a lower energy. As a result, the polarity of the AMR/PHE effect changes. There are several possible contributions to the AMR/PHE effect. The above- described polarity change is only for the contribution shown in Fig. 10.

(about AMR and PHE relation)

(quastion 1) PHE voltage looks to be proportional to the first derivative of AMR by angle between current density and magnetization (this looks to be direct evidence from the formulas). (PHE ~ d(AMR)/d(thetaM)) Is there a possibility to show that PHE ~ d(AMR)/d(thetaH)) or PHE ~ d(AMR)/d(H)), where thetaH is the angle between current and field H? In experiments, it often looks like this, even in systems, where magnetization and the field angles are not the same.

(quastion 2) What is the physical origin of the similarity between the off-diagonal components of the resistivity tensor (PHE components) and to the derivatives of the diagonal ones (AMR components) by thetaM?

PHE and AMR are two sides of one single effect, which is often called the PHE/AMR effect.

In the case of the PHE/AMR effect, the magneto- current is neither parallel nor perpendicular to the main current , but its direction is somewhere between. The perpendicular- to wire component of the magneto-current creates a charge accumulation at the wire edges and, therefore, the Hall voltage. This component is responsible for the PHE effect. The parallel- to wire component of the magneto-current flows either parallel or opposite to the main current.At a fixed voltage, the total current flowing through the metallic wire either increases or decreases due to the magneto- current. It is absolutely similar as the resistance of the wire changes due to the magneto- current. As a result, the parallel- to wire component of the magneto-current is responsible for the AMR effect. In the simplest case, the proportionality between PHE and AMP effect is

PHE~ cos(φ)

AHE~sin(φ)

where φ is the angle of the magneto-current with respect to the wire direction.

If one wants to compare specifically the Hall voltage to the change of resistivity, the math is slightly more complex, but it is basically the same as above.

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I do not see the reason why the first derivative of AMR should be relevant. The cosine changes to sine, maybe that is why.

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The PHE/AMR is the 2nd- order magneto-transport effect. It means it is proportional to vector product of current and (ether H or S or s) and (ether H or S or s) , where H is the external magnetic field, S is the total spin of localized electrons and s is the total spin of the spin-polarized conduction electrons. The magnetization is S+s.

In the simplest cases when magnetization is in-plane or perpendicular- to- plane, H, S and s are parallel and the PHE/AMR is simple. In other cases, they are not and the PHE/AMR becomes very complex, but interesting.

I am strongly against a fake and "highlight" research

Dr. Vadym Zayets

v.zayets(at)gmail.com

more Chapters on this topic:

Spin and Charge Transport

Abstract:

The AMR and the Planar Hall effect describe one single odd magneto-transport effect, which describe magnetic generation of a current flowing along the magnetization direction.

Content

click on the chapter for the shortcut

(10) about AMR/PHE is an Antiferromagnet

(10.a) reason why AMR/PHE exists in an Antiferromagnet

(10.b) the simplest way to understand all features of any magnetotransport effect in an Antiferromagnet

(11) Polarity of AMR/PHE

(11.1) the polarity of AMR/PHE as a feature of the polarity of spin-orbit interaction

(11.2) the polarity of AMR/PHE vs. electron- or hole- dominated conductivity of a metal

(12) about AMR and PHE relation

Two features of one effect: Anisotropic magneto- resistance & Planar Hall effect