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Ordinary Hall Effect

### General Notes about the Ordinary Hall effect in a metal

#### (Q4) about influence of standing-wave electrons on the Hall effect due to defects and interfaces

##### () standing-wave electron vs localized electron ......... ### Ordinary Hall effect (OHE)

origin of OHE: Lorentz force  (definition) The OHE describes the fact that charge is accumulated at sides of metallic wire, when an external magnetic field H is applied perpendicularly to the wire. The origin of the OHE is the Lorentz force The Lorentz force turns the electron (green bowl) from a straight movement. The electron, which moves in a magnetic field H, experience an electrical field perpendicularly to its movement (relativistic effect). This electric field interacts with the electron charge and turns the electron movement direction. It creates an electron current (Hall current) perpendicularly to the bias current.
The movement of the conduction electrons (green balls) turns from a straight path due the Lorentz force induced by the magnetic field H. As a result, the electrons are accumulated at the side of the wire.
click on image to enlarge it click on image to enlarge it

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

(interact with) Electron Charge of conduction electrons OHE is linearly proportional to an external magnetic field H

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

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

1

### metallic wire with a Hall probe

OHE in Ru   Fig. 10a An electron current flows from down to up. The OHE creates the Hall current flowing perpendicularly to the wire. The Hall current creates a charge accumulation at sides of the wire and therefore the Hall voltage, which is measured by a nanovoltmeter.

Fig. 10b A conventional measurement setup. A metallic nanowire with a pair of Hall probe. Two metal contacts contact the opposite sides of nanowire to measure the Hall voltage.

Fig. 10 c. Hall angle vs an applied magnetic field H measured in ruthenium Ru . The Ru thickness is 25 nm. The dependence is nearly perfectly linear. There are a very weak deviation from linear dependence (See below)
click on image to enlarge it

. DC measurement using a Hall Bar

The ordinary Hall effect is evaluated by measuring a voltage on side of the wire (the Hall voltage VHall). From the measured Hall voltage, the Hall angle αHall is evaluated as ( See HallAMRbasic.pdf): where V is the voltage applied to the wire, w is the width of the wire and L is the length of the wire.

(features of measurements) (a non-magnetic metal) : The OHE is only one Hall effect from the family of the Hall effects, which occurs in a non-magnetic metal and a semiconductors. The dependence of the Hall angle vs external magnetic field H is is nearly perfectly linear. There are a very weak deviation from linear dependence (See below)

(a ferromagnetic metal): Additionally to OHE, the Anomalous Hall effect (AHE) and the Inverse Spin Hall effect (ISHE) also contribute to the Hall voltage. The distinguish feature of OHE is its linear dependence on an external magnetic field H (see here) . At a low external magnetic field, the OHE contribution is substantially smaller than AHE and ISHE contribution. At a high magnetic field, the OHE contribution becomes dominated.

## Relativistic origin of the Hall effect (the Lorentz force) Relativistic origin of the Lorentz force (Hall effect). An object (red) is moving in a static magnetic field. In the coordinate system moving together with the object, the static magnetic field is relativistically transformed into the effective magnetic field Heff and the effective electrical field Eeff. In case if the particle is charged, it experiences the force from the effective electrical field Eeff (the Lorentz force).

#### When an electron moves in a magnetic field, it experiences the Lorentz force. The Lorentz force has a relativistic origin. The Theory of Relativity states that a particle moving in a magnetic field experiences an effective electrical field, which is directed perpendicularly to the magnetic field and perpendicularly to the particle movement direction. It is important to emphasize that the direction and magnitude of the effective electrical field does not depend either on the particle charge or on the particle spin.

According to the Theory of The Relativity the electric and magnetic field mutually transformed into each other depending on the speed of an observer. For example, if in a coordinate system of static observer there is only a magnetic field, a movable observer will experience this field as both an electrical field and a magnetic field.

A particle moving in a static magnetic field experiences an effective electric field. The effective electrical field acts on the particle charge (the Lorentz force, Hall effect) and forces the particle to move along this field.

A particle moving in a static electrical field experiences an effective magnetic field. The effective magnetic field acts on the particle magnetic moment (spin-orbit interaction) and causes the precession of the magnetic moment around the direction of the effective magnetic field.

### The Hall Effect and the Spin-Orbit interaction are close cousins

the Hall effect ==== results in ====> an effective electrical field

The Lorentz force- tween effect with the effect of the spin-orbit interaction The Lorentz force (Hall effect) and the spin-orbit interaction, both are relativistic effects. The both effects are originated from fact that an electrical and magnetic field are relative to the velocities of observers. A particle, which moves in a static magnetic field, experiences an effective electrical field (The Lorentz force). A particle, which moves in a static electric field, experiences an effective magnetic field. The spin-orbit interaction has a similar relativistic origin. In case if the particle has a magnetic moment (spin), there will be a spin precession around the effective magnetic field. the Spin-Orbit interaction ===results in=====> an effective magnetic field

The effective electrical field due to the Hall effect does not depend either on the particle charge or the particle spin. it is only depend on the particle movement direction and the direction of the magnetic field !!!!

#### Ordinary Hall effect describes a generation of an electrical current, which flowing perpendicularly to the main current due an perpendicular external magnetic field ##### See HallAMRbasic.pdf for basic relation of Ordinary Hall effect

Results in short: Ordinary Hall in semiconductors and metals   Hall effect in n-type semiconductor. There is a spin accumulation at the right side, where there is an accumulation of the negative charge. There is no spin accumulation at the right side of the sample. Hall effect in p-type semiconductor. There is a spin accumulation at the right side, where there is an accumulation of the positive charge. There is no spin accumulation at the right side of the sample. Hall effect in metal. There is a spin accumulation at the right side. However. there is almost no charge accumulation!!

 The Origin of ordinary Hall effect The length of a vector from axis origin to sphere is proportional to the number of electrons moving in the vector direction. state 1: (no current & no magnetic field): Sphere is symmetrical with respect to the axis origin. state 2: (there is an electrical field in y-direction, but no magnetic field): Sphere is shifted along y-axis. There are more electrons moving in y-direction than in opposite direction state 3: (there are an electrical field in y-direction and magnetic field in z- direction): Sphere is shifted along y- and x- axes. Magnetic field rotates electrons movement direction slightly in CCW direction. Since there are more electrons moving in x-direction (due to the electrical field), the slight change of their movement direction makes number of electrons moving in x-direction is larger than in opposite direction. As a result, the magnetic field creates an electrical current along x-direction click on image to enlarge it

## Calculation of Ordinary Hall effect using the Boltzmann Equations

#### oversimplified classical calculations

##### Note: Calculations are for the band current of the conduction electrons (See here) How the calculations are done:

(step 1) Without an electrical field, the distribution of electron movement direction is a sphere Fi,0.

(step 2) In an electrical field, the electron are accelerating and gaining energy. As a result, there are more electrons moving along the electrical field than opposite to the field. The distribution sphere is shifted along the electrical field.

(step 3) In a magnetic field, the electron direction is turn slightly by the Lorentz force with respect to direction of magnetic field (Eq.(1.2)). This causes a rotation of the distribution of electron movement directions with the pivot point at the axes origin.

(step 4) When there is no electrical field and the distribution field is a sphere at center of coordinate origin. Under a magnetic field, the sphere is rotates. However, since the rotation center is at the center of the sphere, the rotation does not change the distribution. (Eq. (1.4))

(step 5) When there is an electrical field, the distribution is shifted from the center along along the electrical field. Under a magnetic field the distribution is rotated. Since the center of the sphere is shifted from the rotation center, . However, since the rotation center is at the center of the sphere, the rotation does not change the distribution. (Eq. (1.4))

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The Hall effect can be described by the force term (See Fig.7b) in the Boltzmann transport equations where In an equilibrium in the electron gas the electrons move in all direction. Since the Lorentz force only changes the electron movement direction, the Lorentz force does not change the equilibrium electron distribution Fi,0 Eq. (1.3) can be verified as follows (See Fig.7d) Only if there is an electron current , the Lorentz force may change the electron distribution. The change of the electron when there is an electron current, was calculated here. Substituting this distribution ( Eq. (14.7) ) into Eq.(1.1) gives Using the relaxation-time approximation Eqs. (9),(12), ignoring the term of order H2 and substituting Eq. (1.5) into the Boltzmann equations Eqs. (18) gives Only Magnetic field has no influence on distribution There is no change of electron distribution, when only a magnetic field is applied. Even though each electron experiences the Lorentz force, it does not affects the whole distribution. Even though the movement of each electron is turn from straight by magnetic field, it does not cause any current. Because the change is equal in all direction. E.g. number of electrons turned from +y - direction to +x - direction is equal to number of electrons turned from -y - direction to -x - direction. Everything is balanced and there is no change. click on image to enlarge it

From Eq. (1.6) the Hall current can be calculated as (See Eq.(11.3) here) or Hall current can be calculated as ### Change of distribution of electron movement directions due to the the Lorentz force

Influence of magnetic field and electrical current on distribution of electron movement directions
no current, no magnetic field current, no magnetic field both current & magnetic field magnetic field, but no current    Fig.7a. The distribution is a perfect sphere with center at axis origin. In any direction the numbers of electrons moving in opposite directions are exactly the same Fig. 7b. The center of the distribution is shift along y- direction. As a result, there are more electron moving along y-axis than electrons moving in opposite to y-axis. Therefore, there is an electrical current along the y- axis. (distribution is obtained here) Fig. 7c. The center of the distribution is shift in both y- and x- directions. Additionally to the electron current J along y-axis, there is an electrical current along x- axis. Fig. 7d. The distribution is not affected by the magnetic field. Even though each electron experiences the Lorentz force, the distribution does not change. Eq (2.4) proves that fact.
The length of a vector from axis origin to sphere is proportional to the number of electrons moving in the vector direction.
The red sphere shows the distribution of electron movement directions. The red sphere shows the distribution in the of no current and no magnetic field
click on image to enlarge it What an electrical field do for the movement-direction distribution: The distribution is shifted along the electrical field direction In an electrical field, an electron is accelerating along electrical field. Therefore, the speed and energy increase for electron moving along the electrical field and decrease for electron moving opposite to the field. E.g. an electrical field Ex along x-axis makes the scattering probability towards +x-direction higher than in -x- direction p(+x)>p(-x). As a result, the number of electrons moving in the +x- directions becomes larger. What an magnetic field do for the movement-direction distribution: The distribution is rotated around magnetic field, The rotation canter is the axis origin #### What is the electron distribution graph?

The graph describes the directional distribution of electrons. The length of a vector from axis origin to sphere is proportional to the number of electrons moving in the vector direction. Usually the distribution is a sphere, which center is at the axis origin (Fig.7.a). In this case, the electrons are equally distributed in all directions and the number of electrons moving in each direction is equal.

#### Why the distribution is spherical in the absence of an electrical current?

The conduction electrons experience very frequent scatterings. At room temperature an electron experiences more than one scattering in one picosecond (1E-12 s). The scatterings are directional independent and they mix up electrons in all directions. Due the scatterings, the number of electrons moving in each direction is equal for any direction.

#### Why does the electrical field move the electron distribution towards its direction?

A conduction electron is accelerated in an electrical field along the direction of the field.

#### Can we assign a particular speed for an accelerating electron?

Yes. The electron scatterings in a solid are frequent (~one scattering per 1ps at room T).

#### Is the electron directional distribution always a sphere?

No. E.g. in case of quantum well (QW), the electron cannot move perpendicular to QW, therefore the distribution is 2D circle. In case of an anisotropic material the distribution can be of ellipse shape. However, in most of materials the distribution is a sphere ( or close to a sphere).

It is important. When there is no current, the numbers of electrons moving in any two opposite directions are equal.

### Ordinary Hall effect in semiconductors and metals.

#### case of spin-polarized electron gas

In the case when a magnetic field is applied perpendicularly to the flow direction of a drift current, the charge carriers experience the Lorentz force in the direction perpendicular to both the magnetic field and the current. Because of this force, the carriers are accumulated at the edges of the sample and a voltage transverse to the drift current is built up. This voltage is called the Hall voltage. In n- and p-type semiconductors the Hall voltage is of opposite sign, because the holes and electrons are accumulated at the same side of sample, but they have opposite charge (Figs.14-15). Because of a nearly- equal number of holes and electrons in a metal, the Hall voltage in the metal is small and it is proportional to the gradient of the density of states at the Fermi level. When the electron gas in a metal is spin-polarized, both the electrons and holes are spin-polarized. Because of the Hall effect, the spin-polarized electrons and holes are accumulated at the same side of the sample. Therefore, at this side of sample a significant spin accumulation occurs. (Fig.16)

### Ordinary Hall effect in n-type semiconductors. Case of spin-polarized electrons Fig. 14. The Hall effect in a n-type semiconductor. The magnetic field is applied perpendicular to the film (blue arrows). There is only an electron current. Dark-green arrows show path of the electrons, which transfer a negative charge and spin. The negative charge and spin are accumulated at the right side of the wire. At the left side of the wire electrons are depleted and amount of spin-polarized electrons is depleted as well

In a n-type semiconductor there is only one type of carries. It is the electrons, which energy is above the Fermi energy.

#### Features of the Ordinary Hall effect in a n-type semiconductor, in which the conduction electrons are spin-polarized:

The electrons diffuse from a "-" source toward a "+" drain. As they diffuse, the magnetic field turns them to the left with respect to their movement direction.

As a result, both the spin and the negative charge are accumulated at the right side of the wire.

At the left side of the wire, both the spin and the negative charge are depleted It is relatively safe to use oversimplified classical description of the Ordinary Hall effect for a n-type semiconductor.

In the case of a n-type semiconductor:

(no limitation 1) the most of conduction electrons solely occupy one quantum state. There are a very few states, which are occupied by two electrons of opposite spins (See here)

(no limitation 2) there are a lot of unoccupied quantum states above the Fermi level (See here). Therefore, there are no limitation for an electron to change its speed, movement direction or energy.

### Ordinary Hall effect in p-type semiconductors. Case of spin-polarized holes

In a p-type semiconductor, there are two types of carries:

(1st type of carries): Holes are the carriers for the Spin and the Charge.

The holes are the conduction electrons, which energy below EFermi. They move from a positive- voltage source to negative-voltage drain similar as a positively- charged particle does (See here details)

#### Features of the Ordinary Hall effect in a p-type semiconductor, in which the holes are spin-polarized:

The holes diffuse from a "+" source toward a "-" drain. As they diffuse in the magnetic field they turn to the right.

note about the current of "full" states (click to expand)

The current of "full" states is the normal current.

This means that the negatively charged "full" states diffuses from a "-" source toward a "+" drain.

Due to the Hall effect they should turn to the left (similar to the electrons shown in Fig. 14).

However, the most of the electrons, which occupies the "full" states, are the standing-wave electrons. (See Fig.9 here)

The standing-wave electrons do not move, therefore they do not contribute to the Hall current.

In contrast, the electrons, which occupies the "spin" states, are the running-wave electrons.

For this reason, mostly the "spin" states contribute to the Hall effect in the case of the hole current.  Fig. 15. Hall effect in a p-type semiconductor. Magnetic field is applied perpendicular to the film (blue arrows). There is only an electron current green arrows show path of holes, which transfer a positive charge and spin. The positive charge and spin are accumulated at the right side of the wire. At left side of the wire, a negative charge accumulated and the spin is depleted

### Ordinary Hall effect in metals. Case of spin-polarized conduction electrons Fig. 16. The Ordinary Hall effect in metal. The electron current (green balls) flows from “-” to “+” . In a magnetic field the electrons turn left in respect to direction of their movement. The holes current (blue balls) flows from “+” to “-” . In a magnetic field the holes turn right in respect to direction of their movement. Therefore, the charge does not accumulate. In contrast, the spin is accumulated at one edge of the sample.

The drift current in a metal consists of the electron and hole currents, which flow in the opposite direction. Since the charge of an electron and a hole is opposite, the direction of the charge transport is the same for the electron and hole current. This is the transport of the "-" charge from "-" to "+". Or what is the same, the transport of the"+" charge from "+" to "-".

Since the electrons and holes move in the opposite directions, in the magnetic field the electrons turn to the left and holes turn to the right.

As result, both the electrons and the holes are accumulated at the same side of the wire. At the opposite side of the wire, both the electrons and holes are depleted.

Since the charge of an electron and a hole is opposite, the charge is not accumulated. This is the reason why the Hall voltage in a metal is small.

In contrast, the spin direction of electrons and holes is the same, therefore there is a significant spin accumulation at opposite side of the wire due to the Ordinary Hall effect in a metal. Fig. 17. Ordinary Hall effect in a metal. Magnetic field is applied perpendicular to the film (blue arrows). Green arrows, which are directed from "-" to "+", show path of spin-polarized electrons. They accumulate the negative charge and up- spin at the right side of the metallic wire. Green arrows, which are directed from "+" to "-", show path of spin-polarized holes. They accumulate the positive charge and up- spin at the right side of the metallic wire. As a result, there is a spin accumulation, but there is no charge accumulation at

###   Spin accumulation due to Ordinary Hall effect in a ferromagnetic metal

###### details are here Zayets JMMM (2014)

(What it is about) The conduction electrons in a ferromagnetic metal are spin-polarized. Due to the Ordinary Hall effect, the spin polarization becomes larger than the equilibrium spin polarization at one side of ferromagnetic wire and smaller at another side.

(Origin of the effect) In ferromagnetic metal, both the electrons and holes are spin- polarized. Due to the Ordinary Hall effect the electrons and holes are accumulated at the same side of the ferromagnetic wire. The accumulation of the spin-polarized electrons and holes creates the spin accumulation at a side of a ferromagnetic metal due to the Ordinary Hall effect.

(Direction of the created spin accumulation)    The direction of spin polarization, which is created by the Ordinary Hall effect, is exactly the same as the spin direction of the equilibrium spin polarization in the nanomagnet

(Why is it relevant?) The conduction electrons in a ferromagnetic wire are already spin- polarized. Even though the Ordinary Hall effect makes the spin polarization slightly smaller and slightly larger than the equilibrium spin polarization in some places, what difference does it make? In the average the spin polarization is still nearly the same as in the equilibrium?

(Answer: it creates the pump torque for a spin precession and magnetization reversal) An increase or a decrease of spin polarization at an interface of a nanomagnet influences the magnetic properties of the nanomagnet such as the Perpendicular Magnetic Anisotropy (PMA) . Also, additional spin polarization increase the magnetic field, which is induced by the spin- polarized electrons (See here). It is important that both changes are created and can modulated by the electrical current. This featured makes possible the parametric enhancement of magnetization precession and the parametric magnetization reversal in a nanomagnet. For parametric magnetization reversal due to modulation of the anisotropy field (see here) and due to modulation of the magnetic field, which is induced by the spin polarization, see here.

(comparison with another mechanism of spin accumulation)           The distinguish feature of the spin accumulation due the Ordinary Hall effect is that it is substantially larger than another known types of the spin accumulation such as the Spin Hall effect and the spin- dependent scatterings across an interface.

Since the charge of electrons and holes is opposite and there are nearly- equal numbers of the holes and electrons in metal, the charge accumulation and therefore the Hall voltage are very small in a metal. In contrast, the spin direction of the electrons and the holes is the same in a ferromagnetic metal. Spin-polarized electrons accumulated at sides of wire due to spin accumulation of the spin-polarized electrons and holes is the the same. .

#### (facts supporting the effect of the spin accumulations induced by the Ordinary Hall effect):

The spin-polarized electrons and spin-polarized holes are the similar objects. Both are the electrons which occupies solely one quantum state (max occupance is two electrons of opposite spins)

Only difference between electrons and holes is their energy with respect to the Fermi energy EFermi.

The energy of electrons is higher than EFermi and the energy of holes is lower than EFermi.

The number of electrons, which solely occupies one quantum state, increases below EFermi. and decreases above EFermi.. This causes that the electrons moves from "-" to "+" voltage similar to negatively-charged particles. In contrast, the holes moves from "+" to "-" voltage similar to positively-charged particles. See more details here.

Since the electrons and holes move in the opposite directions, they turn in the opposite directions due the Lorentz force and contribute oppositely to the Ordinary Hall effect.

The electrons turn to the left and the holes turn to the right, but they move in opposite directions. As a result, both the positive and negative charge is accumulated at the same side of wire and in total there is no charge accumulation. It is different for the spin comparing to the charge. If the electrons and holes have the opposite charge, they have the same direction of the spin for spin-polarized electrons. Therefore, the spin is accumulating at side of a metallic wire due to the Ordinary Hall effect and this type of the spin accumulation is very effective.