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Anomalous Hall effect (AHE). Anisotropic magnetoresistance (AMR)

Spin and Charge Transport

Abstract:

The Anomalous Hall effect is the Hall effect, which is originated from the from the spin-dependent scatterings. The Anomalous Hall effect is proportional to the magnetization of the material (not intrinsic magnetic field).


Note: All data of this page represent only my personal point of view, which are based on my experimental and theoretical research.

Origins of Hall effects

Ordinary Hall effect (OHE)

Anomalous Hall effect (AHE)

Lorentz force

Spin-dependent scatterings

When an electron moves between scatterings in a magnetic field, it experiences the Lorentz force. The Lorentz force turns out the electron from a straight path. Therefore, there is a flow of electrons towards left

When a conduction electron is scattered on a magnetized defect or an interface boundary, the probability of scattering may be larger towards the left side than towards the right side. In this case, there is a flow of electrons towards left

Hysteresis loop of Hall voltage in the case when major contribution to Hall effect is OHE. It is the case of a paramagnetic metal. The loop is due to AHE. The slope is due to OHE. Hysteresis loop of Hall voltage in the case when major contribution to Hall effect is AHE. It is the case of a ferrimagnetic metal like Fe or Co

Ordinary Hall effect (OHE) vs Anomalous Hall effect (AHE)

Origin

-OHE is induced by the Lorentz force

-AHE is due to the spin-dependent scatterings of the conduction electrons

M or H?

- OHE is linearly proportional to the intrinsic magnetic field H (or B)

- AHE is linearly proportional to the magnetization M

Dependence on spin polarization

- OHE does not depend on the spin polarization

-AHE is linearly proportional to the spin polarization

Bulk and interface contributions

- OHE has only the bulk contribution

-AHE has two very different contributions: from bulk and from interface

 

 

 

 

 

 

 


 

Origin of Anomalous Hall effect (AHE)

Origins of Anomalous Hall effect: Spin-dependent & direction dependent scatterings

 

in both cases, the spin-orbit interaction (SO) makes scattering probability different between to the left and to right directions. Also, the SO makes scatterings to be spin-dependent.

Type 1: Bulk type

Type 2: Interface type

Origin: defect scatterings: Origin: scattering across interface

Fig. 5 (a). Scattered of electrons between delocalized states. The view point moves together with electrons. Distribution of the wavefunction of delocalized electrons is shown as the dark-yellow ellipses. The delocalized electron is shown as the green ball. The blue balls are defects. The green halo shows the distribution of the electrical field around defect. The green arrows shows the direction of the electrical field. The red arrows shows the direction and the magnitude of the effective magnetic field of the spin-orbit interaction. Click on image to enlarge it.

Fig. 5 (b) The electron current flows along a contact between two metals (the blue wall) . There is a charge accumulation at the contact (the violet balls).Distribution of the wavefunction of delocalized electrons is shown as the dark-yellow ellipses. The delocalized electron is shown as the green ball. The violet balls are charge accumulated at the interface. The yellow halo shows the distribution of the electrical field around the charge. The yellow arrows shows the direction of the electrical field. The red arrows shows the direction and the magnitude of the effective magnetic field of the spin-orbit interaction. Click on image to enlarge it.

The origin of Anomalous Hall effect is spin-dependent scatterings

note: AHE has the same origin as the Spin Hall effect.

What is the spin dependent scattering?

A. it is a scattering, which probability is different between left and right directions with respect to the electron movement direction and which probability depends depends on the electron spin.

 

Which effect makes the scatterings of conduction electrons to be direction and spin-dependent?

A. The spin-orbit (SO) interaction (details see here)

How the SO interaction makes a scattering to be direction-dependent and spin-dependent?

There is an electrical field around the defect. It is directed to the left from left side of defect and to right from right side. As a consequence, the effective magnetic field of SO interaction is opposite directions at the left and right sides of the defect. The effective magnetic field of SO interaction affects the energy and symmetry of electron. Since the field is different, the energy and symmetry depends on the direction and the spin. As a result, the scattering probability depends on the direction and spin.

 

Physical mechanisms of spin depend scatterings

How does the spin-orbit (SO)interaction affect the scattering

 

 

An electron at different sides from a defect experiences the opposite magnetic field of SO. As a result, it has a different energy and symmetry. It causes a different scattering probability towards left and right. Since only the electron spin interacts with the magnetic field of SO, the difference of scattering probabilities is spin-dependent. The first origin of direction- dependency and spin- dependency of the electron scattering is the direction dependency of electron energy, which is induced by SO. More details see here. The second origin is the direction dependency of the electron symmetry.
click on image to enlarge it

Bulk-type spin-dependent scatterings on defects:

The electrical field around a defect induces the effective magnetic field due to the spin-orbit interaction (Fig.5(a)). The effective magnetic field is in opposite directions when a delocalized electron moves from the left or from the right side from the defect. Because of the effective magnetic field of the spin-orbit interaction, the scattering probability is different whether the electron is scattered from left to the right side or from the right to the left side of the defect.

Interface-type spin-dependent scatterings across interface:

There is a charge accumulation at the contact (the violet balls of Fig. 5(b)). For example, the charge may be accumulated because of the different work functions of the metals. The electrical field of the accumulated charge induces the effective magnetic field of the spin-orbit interaction for electrons flowing along the interface. The effective magnetic field is of opposite sign for electrons flowing at each side of the contact. The electron scattering probability across contact interface is different whether an electron is scattered from the left to the right metal or in the opposite direction. Because of this difference, a spin current flows across the contact interface when a charge current flows along the contact.

 

 

 


Measurement of the Ordinary & Anomalous Hall effects. The Hall angle αHall

Measurement of Anoumoulous Hall effect

Measurement setup

 
 
A bias current through the nanowire. A nanovoltmeter measures the Hall voltage  

click on image to enlarge it

The OHE and AHE is measured by the Hall angle αHall. The contributions from AHE and OHE are independent: αHall=αAHE+αOHE

What is better to use the Hall resistance or Hall angle αHall

A. The Hall angle αHall. It is only correct parameter characterizing the AHE and OHE.

What are merits to use the Hall resistance for measurement of AHE and OHE?

None. It is just a number in the Ohm unit. It has no physical meaning. It depends on the device geometry. It is not a material intrinsic parameter.

What are merits to use Hall angle αHall of measurement of AHE and OHE?

merit 1: The αHall has a direct physical meaning. It the angle of deviation of electron movement from a straight line along an applied electrical field.

merit 2: The αHall is an intrinsic parameter of a material. It does not depend on the device geometry or film structure. The magnitude of the OHE and AHE in different devices and different films should be only compared by comparing their αHall.

Measurement of αHall

The Hall angle αHall is defined as

where σxx and σxy are diagonal and off-diagonal components of the conductivity tensor

nanowire with two pairs of Hall probes

Measurement setup

 
 
Backside Hall probe is connected to a nanomagnet. FeB is thicker in this region and top of FeB is covered by MgO. The Front side Hall probe is connected at side of a nanomagnet. FeB is thinner in this region and at its top covered by SiO2. The distance between two Hall pairs is 11 μm.  

click on image to enlarge it

Single layer film:

In this case the Hall angle αHall is calculated as

where the Hall voltage VHall, is the Hall voltage L is wire length and w is wire width. The Hall resistance RHall can be calculated

where R is wire resistance

Double-layer metallic wire:

This case when metallic wire consists of two layers: The first layer is made of a ferromagnetic metal. The second layer is made of a non-magnetic metal. In this case the Hall angle αHall of the ferromagnetic metal is calculated as

where tferro,tisot, σferro,σisot are thicknesses and conductivities of ferromagnetic and non-magnetic metals.

 

to see how to obtain Eqs.(4.15),(4.24), click here to expand it

Single-layer metallic wire

Due to the Hall effect there is an electrical current current across the metallic wire, which can be calculated as

where V is the bias voltage, L is wire length, j|| and j is current density along and across the wire .

The current j makes a charge accumulation at walls of the metallic wire. This charge induces the voltage, which is called the Hall voltage VHall. The Hall voltage induces the current, which is opposite to j. The current density of this current j⊥,comp can be calculated as

where w is the width of the wire. In total, there is no current across the wire, σ=σxx is conductivity of the wire.

Since in total there is no current flow across the wire, we have

Substituting Eqs.(4.2),(4.3) into Eq.(4.4) gives

or

The Hall resistance RHall is defined as

where J|| is the current flowing throw the nanowire

where thick is wire thickness


Double-layer metallic wire

This case when metallic wire consists of two layers: The first layer is made of a ferromagnetic metal. The second layer is made of a non-magnetic metal. The Hall current is generated only inside ferromagnetic

The Hall current J, which flows across wire, can be calculated as

where j is the Hall current density , tferro is the thickness of ferromagnetic layer, w is the wire width, L is the wire length and V is the bias voltage. αHall is the Hall angle of the ferromagnetic metal. It is assumed that the Hall angle αHall=0 in the non-magnetic metal.

Due to the charge accumulation, the current flows in the opposite direction in both layers

tisot is the thickness of layer of the non-magnetic metal, σferro,σisot are conductivities of ferromagnetic and non-magnetic metals..

Since in total there is no current flow across the wire, we have

Substituting Eqs. (4.21),(4.22) into Eq.(4.22a) we have

Simplifying Eq.(4.23) gives the Hall angle in case of wire consisted of two layers as

 

 

 

 

 

 

 

 

Shape of hysteresis loop for Anomalous Hall effect. Experiment

 

Case of only Lorentz force contribution

Au film.

with injected spin-polarized electrons

Rhombic shape

click to enlarge

Case of only contribution due to spin-dependent scatterings

FeBTb film

Square shape

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Au(15nm)/FeBTb(9 nm). Spin-polarized electrons are injected into from FeBTb into Au due to the spin proximity effect.

Measured conductivity is 1.1E7 S/m2. Because of a low conductivity of FeBTb layer, contribution of this layer to Hall effect is negligible. The high conductivity in Au layer indicates that the current of running-wave electrons is the transport mechanism in this layer.

FeBTb (20 nm)

Measured conductivity is 0.06E7 S/m2. The low conductivity indicates that the scattering current is the transport mechanism.

 

 

 

 

 

 

 

 

 

Notice: Polarities of loops in Au and FeBTb are different.

 

 

 

 

 

 

 

 

 

 

 

 

Dependence of loop of Anomalous Hall effect on Hall angle

 

Case of only Lorentz force contribution

area of loop significantly changes

Case of only contribution due to spin-dependent scatterings

area of loop practically does not change

 

 

 

How to distinguish different contributions to the Anomalous Hall effect?

A. From hysteresis loop of measured Hall effect, it can be distinguished which is the major contribution to the Anomalous Hall effect.

 

 

 

 

 

 

 

 

 

 

 

 

Ordinary Hall effect due to the Lorentz force in a ferromagnetic metal

Hall effect due to the Lorentz force

 

Hole-mediated conductance. Positive Hall effect.

Ni, Co (fcc), Cu, Ag, Au, Al, Pt, Mg, Pd, Gd, Tb

Positive exchange or Positive V_Hall0

Negative exchange or Negative V_Hall0

Electrons-mediated conductance. Negative Hall effect

Fe, FeCo (bcc), Ta, Ru, Cr, Rh, W, V

Positive exchange or Positive V_Hall0

Negative exchange or Negative V_Hall0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Anomalous Hall effect due to the spin-dependent scatterings

 

Anomalous Hall effect & Ordinary Hall effect:

The Anomalous Hall effect describes describes the fact that a change can be accumulated at the edges of a metallic wire when a magnetic field is applied perpendicularly to the drift spin-polarized current.

Spin Hall effect:

The Spin Hall effect describes the fact that spin may be accumulated at the edges of a metallic wire when spin-unpolarized drift current flows in the wire.

Inverse Spin Hall effect:

The Inverse Spin Hall effect describes the fact that the spin may be accumulated at the edges of a metallic wire when diffusive spin current flows in the wire.

 

The effects which are originated by spin-dependent scatterings

The effective electrical field, which is induced by the defect, is shown by the green arrows. The electrical field induces the spin-orbit interaction. The direction of the effective magnetic field of the spin-orbit interaction is different for electrons scattered into the left and into the right. This makes different the probabilities of scattering into the left and into the right.

Anomalous Hall effect

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

click here or on picture to enlarge it

Spin Hall effect

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

click here or on picture to enlarge it or another version

Inverse Spin Hall effect

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

click here or on picture to enlarge it

Under an applied voltage the 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 (spin directed up (TIA assembly)) flow from back to front of the wire. In the opposite direction the spin-unpolarized electrons (spin directed in all directions (TIS assembly)) 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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Questions Answers

Q. Do the spin Hall effect, anomalous Hall effect and inverse spin Hall effect occur because of the electron scattering on magnetized defects (defects with non-zero magnetic moment). Are the defect should be magnetized ?

A. It is not necessary. The defect can be either magnetized or non-magnetized. It is important the electrical field around the defect, which induces spin-orbit interaction and which makes a scattering to be spin-dependent.

Q. The spin Hall effect, anomalous Hall effect and inverse spin Hall effect are very similar. How to distinguish between these effects?

A. These three effects are very similar. It is difficult to distinguish between them. For example, the inverse Spin Hall effect can be explained as the sum of the Spin Hall effect and the anomalous hall effect. The spin-polarized part of the spin diffusion current experiences the anomalous hall effect, which makes a charge accumulation at edges of a wire. The spin-unpolarized part of the spin diffusion current experiences the Spin Hall effect, which makes (changes) a spin accumulation at edges of the wire.

Q. The spin Hall effect, anomalous Hall effect and inverse spin Hall effect can occur only in "bad" metals with a large number of defects?

A. The magnitude of these effects may be large only in case of metal is a sufficient number of defects. Therefore they can be observed easily in “ bad” metals with a small conductivity. For example, FeBTb, which conductivity is 50 times smaller than the conductivity of gold and 12 times smaller than the conductivity of monocrystal iron, has a large Hall angle (~0.5 deg) of the anomalous Hall effect and a large Spin-Hall torque.

In the case when the density of defect is small, the current the running-wave electrons is large and the contribution of the scattering current into the electron transport is negligible. It makes all scattering-originated effects to be screened and magnitude of the will be not large.

In the case when the density of defect is larger, the electron wave function overlaps several defect simultaneously, it would make all scatterings spin-independent and there will be no spin Hall effect, anomalous Hall effect and inverse spin Hall.

Other important condition for existence of spin-dependent scatterings is that the electron mean-free path should be shorter than the effective radius of electrical field around a defect.

For a larger magnitude of the spin Hall effect, anomalous Hall effect and inverse spin Hall effect, the number of defects should be not very large and not very small, but it should be an optimum.

 

 

Anomalous Hall effect

Q. For the anomalous Hall effect, is it essential for electron gas to be spin-polarized?

A. The anomalous Hall effect, which is originated due to the spin-dependent scatterings, occurs only when the electron gas is spin-polarized. The Hall angle is linearly proportional to the spin polarization of the electron gas.

Q. Why there is no anomalous Hall effect in case when a magnetic field is applied along the wire?

A. The spin polarization of electron gas is directed along an applied magnetic field. The defect induces an electric field around itself. An electron, which moves in this field, experiences the effective magnetic field of spin-orbit interaction , which is directed either up or down, when for electrons scattered into the left or right. Because of different directions of the effective magnetic field of the spin-orbit interaction, the probability of electron scattering toward left and right may be different and it this difference is the spin-dependent. In the case when the effective magnetic field is along the spin of electron, electron energy is larger. The energy difference for electrons with spin up and spin down is the reason for the spin-dependent scatterings.

When the magnetic field is applied along the wire, the spin direction of the spin-polarized electrons (electrons of TIA assembly) is along the wire as well. The electrons of this spin direction have the same energy in magnetic field directed up and down. Therefore, the effective magnetic field of the spin-orbit interaction, which is directed up and down, does not make scattering probabilities into the left and into the right to be different and scattering to be spin-dependent. This is the reason why there is no anomalous Hall effect when a magnetic field is applied along a wire.

 

Q. The electrical field around defect is distributed in all directions. Why there is difference of scattering probabilities only between left and right directions as it is shown in above Fig.?

The scattering current may flow only if there is a difference in scattering probabilities into two opposite direction. The spin-orbit interaction makes this difference. The effective field of the spin-orbit interaction is directed up and down for electrons scattered into the left and into the right. Since the electrons with spin directed up and down have different energies in this field, their scattering probability is different. Similar, the electrons, which spin is directed left and right, have different scattering probabilities into up and down directions.

Q. In above Figure, spin-up electrons are scattered into the right and spin-down electrons are scattered into the left. Why not in opposite direction?

A. This direction is chosen as an example . In which direction electrons are scattered depends on a material. And specifically it depends whether the density of states in a metal increases or decreases with energy at the Fermi surface. When electron has been scattered, it experiences the electrical field at back side of the defect, which directed from front to back of the wire. This electrical field induces the effective magnetic field of the spin-orbit interaction, which is directed upward when the electron moves to the left (See here) and it is directed downward when the electron moves to the right. In the case when the effective magnetic field is along the spin of electron, electron energy is larger. When the density of states in a metal increases with increasing electron energy, the scattering probability is larger when electron spin is along the effective magnetic field. Therefore, the case shown in above Figure corresponds to a metal, in which the density of states decreases with energy.

Q. In Figure 1 hit it is only shown that only the electrons contribute into the effects. Is there any contribution of holes into the effects??

The holes (electrons of the energy lower than the Fermi energy) contributes nearly equally as the electrons (electrons of the energy higher than the Fermi energy). Importantly, the polarity of the hole and electron contributions are the same. This is the reason why the spin Hall effect, anomalous Hall effect and inverse spin Hall effect are effects with a large magnitude. It is similar to the case of the ordinary Hall effect

 


Why do scatterings become spin-dependent?

Reason why a scattering become spin-dependent

click here to enlarge it

When a spin-up electron moves forward (central), its scattering probability is different towards left and towards right.

The Fermi-Dirac electron distributions before scattering (center), after scattering to the left (left) and to the right (right) are shown in yellow. The vertical axis is the electron energy. The horizontal axis is the occupation probability.

The electron energy depends on mutual directions of the electron spin and the effective field of the spin-orbit interaction Hso

The scattering probability is highest, when at the energy of high occupation probability before scattering (center) there is sufficient number of unoccupied states after scattering (left or right). It is the case of scattering towards left.

 

Notes:

1. Scattering can be spin-dependent on both the magnetized and non-magnetized defect.

2. In electrical field around a defect an electron experiences the spin-orbit interaction, which makes the electron scattering on the defect spin-dependent.

 

 

When an electron is scattered on a defect, behind the defect the electrical field of the defect is directed along the electron movement. A moving electron experiences the effective magnetic field of the spin-orbit interaction only when electron moves across an electrical field.

When the electron is scattered towards left, the effective magnetic field of the spin-orbit interaction Hso is directed up and along the electron spin. Therefore, the energy the scattered electron becomes smaller.

When the electron is scattered towards right, the Hso is directed down and opposite to the electron spin. Therefore, the energy the scattered electron becomes larger.

As can be seen from Figure, for scattering towards left there are many unoccupied quantum states, therefore the scattering probability towards left is higher. In the contrast, there are almost no unoccupied quantum states for scatterings toward right, therefore the scattering probability towards right is lower.

 

 

 

 

 

 

 

 

 

 

 

 



 

 

Anisotropic magnetoresistance (AMR)

AMR of FeB film

click here to enlarge it

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.

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.

 

The magnitude of the AMR effect is linearly proportional to the spin-polarization of the electron gas in a ferromagnetic metal.

 

 

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.

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

click here to enlarge it

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

click here to enlarge it

 

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?

A. There is some correlation, but it is not very straight and obvious. All effects are originated from the spin-dependent scatterings. However, if the Anomalous Hall effect, the Spin Hall effect and Inverse Spin Hall effect occur because of a difference of scattering probabilities towards left and right, the AMR effect occurs because average probability for both scattering into left and right changes as well.

 

 

Unresolved issue

It is still unclear relation between magnitude of the AMR and magnitude of AHE & OHE. For discussion on experimental data on this topic See T.R. Mcguire and R.I. Potter, IEEE Trans. Magn. (1975).

 

 

 

 

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