My Research and Inventions ###### click here to see all content or button bellow for specific topic ### Reticle 11

Exchange Interaction

### Spin and Charge Transport

#### There are three contributions to exchange interaction between localized electrons: 1) spin-dependent electron-electron Coulomb repulsion at a short distance (antiferromagnetic); (2) spin-dependent electron-nuclear Coulomb attraction (ferromagnetic); 3) spin-dependent electron-electron Coulomb repulsion at a longer distance (ferromagnetic); There are 3 major contributions to the exchange interaction:

### 3) the weak/moderate ferromagnetic exchange interaction due to spin- dependent the Coulomb's repulsion between electrons at a longer distance due the fermion nature of electrons;

##### Due to fermion nature of an electron, the wave function of two electrons is symmetrical, when spin are antiparallel, and the wave function is antisymmetric, when spin are parallel. The Coulomb's repulsion between electrons is smaller when a distance between electrons is longer. In the case of antisymmetric wave function (parallel spins) the average distance between two electrons is a little bit longer than for the case of symmetric wave function (antiparallel spins) . Therefore, the Coulomb's repulsion energy is smaller parallel spins. It forces the spins to align ferromagnetically.

Additionally there is an asymmetric exchange interaction, which is called the Dzyaloshinskii-Moria Interaction

#### Note: The force of the exchange interaction is not some "magical" quantum-mechanical force. It is just the Coulomb interaction, but spin-dependent. The exchange interaction acts on the electron spins in order to align the spins such that the Coulomb interaction energy is minimized. There are several reasons why the Coulomb interaction becomes spin-dependent, which are:

##### (3) Features how Quantum Field of Electrons is divided into particles (electrons).

Origins of exchange interaction

1.Antiferromagnetic

very strong at short distance

due to reduction of the Coulomb repulsion force between electrons of opposite spins 2.Ferromagnetic (long distance), antiferromagnetic (short distance)

due to spin dependence of the Coulomb attraction force between electrons and atomic nuclears 3. Antiferromagnetic weak/moderate due to spin-dependence of the Coulomb repulsion force between electrons

# The origin of strong antiferromagnetic exchange interaction

### between electrons at a short distance

##### the following describes the first reason why the coulomb's repulsion between electrons depends on their spin directions

There is a strong antiferromagnetic exchange interaction due to spin- dependent the Coulomb's repulsion between electrons at a short distance. Its origin is explained as followed: Two electrons of opposite spins may occupy one quantum state. It literally means that in the case when the spins are opposite there is no Coulomb's repulsion between electrons. In contrary, when spin are parallel, the Coulomb's repulsion between electrons is very strong. It means that at very short distance the Coulomb's repulsion between electrons strongly depends on their spin directions. The Coulomb's interaction between electrons at a short distance forces the electron spins to align antiparallel.

### When two electrons of opposite spins occupy one quantum state, they become one elementary particle and there is no any the Coulomb repulsion inside of this elementary particle. Therefore, there is no any Coulomb repulsion between two electrons of opposite spins, when they occupies the same quantum state and they are at exactly same point in space.

##### Q. Why two protons of opposite spin direction can not occupy one quantum state and can not be at same point at same time, but electrons can?

A. It is because an electron is an elementary particle, which does not contain any part. In contrast, proton is not elementary particles. It consists of two up-quarks and one down-quark. Only an elementary particles have such property that two charged elementary particles can be at same point in the space.

## Why there is no Coulomb repulsion between parts of elementary particles?

A single elementary particle does not have parts and it can not interact with itself

The picture shows what would happen if an electron could interact with itself. Since there is no holding force between parts of an electron, the repelling Coulomb's force would blow up the electron. An elementary particle does not have parts, therefore there is nothing inside an elementary which could repel each other.

If we could assume that there is something inside electrons, which could repel each other, it should be another force, which should be directed opposite to the Coulomb's force and which should hold the electron together. Otherwise, the electron would be turn apart. For example, in an atomic nuclear repelling Coulomb force between protons is compensated by attractive strong force of the strong interaction. Similar inside a proton, Coulomb force is compensated by the force of the strong interaction between quarks. For electron there is no "other" force, which could hold electron against blowing up. That literally means that an electron does not have any parts, which could repel each other.

Q. Is it possible that an electron interact with itself as whole object without parts?

A. No.

If it were the case, the electron energy would depend on the electron size. For the longer electron length, the smaller interaction would be between the edges of the electron and the smaller would be the total energy of the electron. Since the length of delocalized electrons equals to the mean-free path, in a semiconductor the band gap energy would substantially depend on the crystal quality of the semiconductor. From a variety of experiments, it is very clear that it is not the case. 1) A single elementary particle can not interact with itself

2) An elementary particle does not have parts.

3) Even though an elementary particle has a finite size and it can be charged, there is no Coulomb's interaction between sizes of the electron. Otherwise, he repelling Coulomb's force would blow up the electron.

Fig.2. Coulomb repulsion energy between two electrons as function of distance between them For longer distances, the repulsion energy is reverse proportional to distance between electrons E~1/x and it is spin-independent.

For shorter distance, the energy is spin dependent. In the case of the opposite spins (blue line), the repulsion vanishes at shorter distances. In the case of the parallel spins (red line), the repulsion become infinitely large at shorter distances

Figure 2 shows the Coulomb repulsion energy between two electrons. The Coulomb interaction between two identical elementary particles depends on their mutual spin directions.

When electron spins are parallel (blue line), the Coulomb interaction between is nearly the same as the Coulomb interaction between any two charged objects. The repulsion energy sharply increases for shorter distances as 1/x, where x is the distance between electrons.

When electron spins are opposite, for longer distances the repulsion energy increases following the 1/x law, but at shorter distances the repulsion energy diverges from the 1/x law and decreases to zero. At zero distance, there is no any repulsion between electrons, because they becomes one particle.

There is an energy barrier with energy of E_b between cases when the electrons are far away and when the electrons of opposite spins are the same point.

It requires an energy larger than E_b in order to scatter an electron from a "full" state.

Q. Protons are fermions. Similar to electrons, can two protons of opposite spins occupy one quantum state.

A. No. Proton is not elementary particle. It has "parts" (quarks). Only two elementary particles can occupy one quantum state.

If two protons of opposite spins could be able to occupy one quantum state, the thermonuclear reaction inside core of the Sun would be much faster and the Sun would be burnt out already a long time ago.

Q. The electron might be a point-like particle with infinitely small size. Therefore, the Coulomb interaction inside the electron has no meaning.

A. The size of electron is not as small. The mean-free path is the effective length of delocalized conduction electrons. The radius of orbit is the effective radius (size) of the localized electrons. In a high-crystal-quality semiconductor the electron length (the mean-free path) can be as long as a few micrometers.

Besides, the size of electron can be too small, because of the Heisenberg's uncertainty principle.

Q. Proton has the charge radius, which could be measured experimentally. Does electron has the charge radius.

A. The charge radius has a little meaning for an electron. If in the case of the proton the charge is curried by quarks. Since the quarks are bounded by the strong interaction and the strong interaction sharply increase for a longer distance between quarks, there is a rather sharp boundary for charge distribution in the proton and the charge radius has clear meaning. In contrast, the electron is an elementary particle and its charge distribution is fully described by its wave function. It could be either wide (delocalized electrons) or narrow (localized electrons). The parameters like the orbital moment or the mean-free path are better describes the charge distribution of the electron.

## How Quantum Mechanics distinguish between two particles?

In Quantum mechanics a system of many electrons describes by a single wavefunction. There is Coulomb repulsion between each two electrons. Therefore, there is an interaction between parts of this function. In contrast, the an elementary particle (one electron) is described by a wavefunction as well, but this wavefunction does not describes any internal interaction. How it is possible?

A single elementary particle is described by a single wavefunction with spatial coordinates (x1,y2,z2). The wavefunction describes the interaction of the particle only with external objects, but not any intrinsic interaction.

Another elementary particle is is described by a different wavefunction with different spatial coordinates (x2,y2,z2). The Coulomb interaction between particle one and particle 2 is described by integral: ## How two elemental particles (two electrons) are transformed into one elementary particle ("full" state)?

Fig.3. Two electrons of opposite spins, when combined, form an elementary particle without spin.

Each quantum state can be occupied by two electrons of opposite spins.

When a quantum state is occupied by one electron, it is an elementary particle with charge -e and spin=1/2

When a quantum state is occupied by two electrons, it is an elementary particle with charge -2e and spin=0 Since all known elementary particles are waves, it is very common that during an interaction between them, the number of particles changes. For example, the result of interaction of an electron and a positron is only a photon.

All matter in the universe was created literally from nothing 13.8 billion years ago during the inflation period.

Creation or annihilation of particles waves is very common process. How two particles (waves) can be created from nothing could be understood as follows: When there are two absolutely identical waves, but phase shifted 180 degrees relatively each other, their sum gives zero or nothing. Therefore, two particles (for example, an electron and a positron) can combine that result will be nothing. Similarly, if some force changes the phase shift between particles from 180 degrees to any other, from nothing two particles can be created.

Mathematically the process when two electrons of opposite spins combine and create one elementary particle without spin ("full" state) can be understood as follows.

Two electrons with opposite spins, which occupy different states, are described by two spinors  They are described by different sets of coordinates (x1,y1,z1) and (x2,y2,z2). It means they are two elementary particles, which interact with other.

When these two electrons of opposite spins occupy one state, they are described by a scalar wave function, which is product of spinors (1.1) and (1.2) It is important that the scalar wavefunction (1.3) is described by one set of coordinate. It means (1.3) describes one elementary particle.

More about spin basic properties see here

# 2. Origin of moderate ferromagnetic exchange interaction

## due to spin- dependent the Coulomb's attraction between electrons and nuclears

##### the following describes the second reason why the coulomb's repulsion between electrons depends on their spin directions

There is a moderate exchange interaction between electrons due spin -dependent the Coulomb's attraction between electrons and nuclears. Usually it is ferromagnetic, but it could be antiferromagnetic at a short distance between nuclears in a crystal lattice. Its origin is explained as follows: Due to fermion nature of an electron, the wave function of two electrons is symmetrical, when spin are antiparallel, and the wave function is antisymmetric, when spin are parallel. When two localized electrons are located at two neighbor atoms, antisymmetric wave functions means that probability to find electrons is smaller at point, which is between atoms, than at point in the vicinity of nuclear. In contrast, symmetrical wave functions means that probability to find electrons is nearly the same between atom and in the vicinity of nuclear. Therefore, when spins are parallel, the probability for an electron be at vicinity of nuclear is larger and the absolute value of the negative nuclear-electron interaction energy becomes larger comparing to the case of the antiparallel spins. It forces the spins to align ferromagnetically.

## Dual nature of an electron: Wave or Particle?

How to construct a wavefunction of several electrons?

### Imaginary case 1: If electron were only a particle, not a wave

Fig.4. Wavefunction of a system of two electrons in the vicinity of two nuclears. Imaginary case of electron as only a particle, not a wave

Each electron is distributed around one nuclear. There is an overlap of electron wavefunction in the middle Possibility 1:

Each electron is an individual elementary particle, which is clear distinguished from any other electrons. In this case the electron should have an individual clear- distinguished wave function. The wavefunction of a system of electrons is a vector, each element of which is the individual wavefunction of each electrons. Possibility 2:

Probability of each electron to be at a spatial point can be independently defined Then, the probability that 1st electron will be at point r1, 2d electron will be at point r2 and 3d electron will be at point r3 will be Merits of this representation:

1) The electrons are well-separated from each other. Therefore, there is no infinite Coulomb repulsion between electrons.

2) When the nuclears are sufficiently separated from each other, this representation describes correctly two atoms, where around each nuclear only one electron is circling.

Demerits of this representation:

1) Two electrons are well- distinguished from each other.

It can be justified when the nuclears are far away from each other and one electron is near one nuclear and another is near the second nuclear. However, when two nuclears are moved into one point, there is no feature which can distinguish one electron from the second electron.

3) It contradicts the basic assumption of Quantum Mechanics that the electron is a wave. Two electrons should form a common field.

### Imaginary case 2: If electron were only a wave, not a particle

Fig.5 Wavefunction of a system of two electrons in the vicinity of two nuclears. Imaginary case of electron as only a wave, not a particle

There is no any difference between electron 1 and electron 2. Their wave functions are exactly the same. In this case the electrons form one join field of "all electrons". The field is only one and each electron is a part of this field. Therefore, each electron mimics the distribution of the common field and the spacial distributions of each electrons are the same.

It is similar to the electrical or magnetic field. The "all electrons" fields induced by different sources sum up each other. For example, the electrical field between somewhere between two electrons is the sum of the electrical field induced by 1st electron plus the electrical field induced by the second electron. At each point it can be distinguished from each source the electrical field is induced. Only the sum of the electrical field by different source are measurable.

Therefore, if we assume that there is "all electrons" field and each electron as a wave particle is a part of this field, the wavefunction of three electrons will be Merits of this representation:

1) It is following the basic assumption of Quantum Mechanics that the electron is a wave. There are a huge number of experiments, which prove that the electron is a wave (RHEED, LHEED, electron transport in a solid).

2) In the case when all nuclears are moved into a one point, one electron field correctly describes all electron features. The different electrons occupy different quantum state of different energy levels or/and different orbital moment.

Demerits of this representation:

1) The identical wave functions for two or more electrons should cause infinite Coulomb repulsion between electrons.

2) When nuclears are separated far away from each other, still each electron is circulating around both nuclears. It is unrealistic.

### Case 3: the real electron: between a wave and a particle

#### Spin. Spin symmetry. Symmetric and antisymmetric wave functions.

Fig.6 Probability to find one electron at coordinate x1 and second electron at coordinate x2 of a system of two electrons in the vicinity of two nuclears. Realistic case when electrons have properties of both a wave and a particle

Animation parameter: Distance between two nuclears. Black balls show the positions of nuclears.

# Spin=1. Asymmetric wavefunction  When two electrons approach each other, they form a common quantum field, which has a defined spin. The spin of a system of two electrons can be either 0 or 1. An object with a spin has specific properties symmetry for of its wavefunction. It can be shown (see Landau & Lifshitz "Quantum Mechanic" Vol.3, Chap. 62), that a wavefunction of a system of two identical particles is symmetric when the spin of the system is zero and it is asymmetrical when spin is 1.

Therefore, the system of two electrons is described by a spinor (which is similar to Eq.3.1, but it also includes the spin-symmetry properties). In the case of two electrons, the spinor of rank two has two wave functions: symmetrical and asymmetrical.

The symmetric wave function, which corresponds to the case of antiparallel spins of the electrons and total spin equals zero, is The antisymmetric wave function, which corresponds to the case of parallel spins of the electrons and total spin equals to one, is In the case of Figs. 4 and 5, the probability to find electron at coordinate x1 does not depend on position of the second electron. In contrast, in Fig.6 the probability of 1st electron to be at coordinate x1 depends on position of the 2d electron (coordinate x2).

Similar to Fig.5, Fig. 6 is symmetrical for exchange of x1 and x2. Therefore, as wave all electrons are fully identical and they can not be marked or distinguished from each other.

The probability for the 1st electron to be in the vicinity of the 1st nuclear is highest when the probability of the 2d electron is highest to be in the vicinity of the 2d nuclear. Two peaks can be well- distinguished for both figures of Fig.16. This means that as charge particles the electrons are not the same place. One electrons is near one nuclear and the second electron is

Merits of this representation:

1) It is following the basic assumption of Quantum Mechanics that the electron is a wave.

2) The electrons are well-separated from each other. Therefore, there is no infinite Coulomb repulsion between electrons.

3) When the nuclears in a crystal lattice are sufficiently separated from each other, this representation describes correctly two atoms, where around each nuclear only one electron is circling.

## Ferromagnetic exchange interaction between localized electrons due to Coulomb attraction to nuclear

Fig.7 (left) Probability to find one electron at coordinate x1 and second electron at coordinate x2=-x1-d, where d is distance between nuclears. (right) Coulomb attraction energy between electrons and nuclears as function of distance between nuclears.

Animation parameter: Distance between two nuclears. Black balls show the positions of nuclears. The energy of the attractive Coulomb interaction between electrons and nuclears is negative.

When the probability of electrons to be in the vicinity of nuclears is higher, the interaction between electrons and nuclears is stronger and the interaction energy is lower.

The exchange interaction forces the electron spins into the state of a lower energy.

1. Case of ferromagnetic exchange interaction. A longer distance between nuclears.

In this case maximus of the asymmetric wave function of the ferromagnetic state is at the positions of nuclears. At the middle between nuclears the asymmetric wave function becomes zero.

The the symmetric wave function of the antiferromagnetic state has two peaks and the valley between them. However, the probability for an electron between nuclears is higher for the antiferromagnetic state and the probability to be in the vicinity of the nuclears is higher for ferromagnetic state.

Therefore, the interaction is stronger and the energy is lower for the ferromagnetic state.

2. Case of antiferromagnetic exchange interaction. A shorter distance between nuclears.

For the symmetric wave function of the antiferromagnetic state, the valley disappeared and there is only one peak. Both nuclears are near maximum of this peak. The probability of electrons be in the vicinity of nuclears is high.

The asymmetric wave function of the ferromagnetic state still has two peaks with a valley between them. The nuclears are near the minimum of the valley. The probability of electrons be in the vicinity of nuclears is low.

Therefore, the interaction is stronger and the energy is lower for the antiferromagnetic state.

Q. The "full" state (a state, which is occupied by two electrons of opposite spins) has spin equal zero. Is the "full" state is a boson?

A. No. It is not. The charge of a "full" state is -2e. In the case if two "full" state are placed at same point (if they the same wavefunction), it would infinite repulsion between them.

Comparison between different representation of an electron

Probability of one electron to be at point x1 and second electron at point x2

##### black balls show position of nuclears

as a wave

as a particle

real electron

between a wave and a particle   Q. The system of two electrons have even spin: one or zero. Is it a boson?

A. No. It is not boson. The system of two electrons is just a system of two fermions. Nothing more.

The system of two electrons has a charge -2e and a charged object can not be a boson.

Two bosons can occupy one quantum state. Four electrons can not occupy one quantum state.

# 3.Origin of weak moderate ferromagnetic exchange interaction

### due to spin-dependent Coulomb's repulsion between electrons

##### the following describes the third reason why the coulomb's repulsion between electrons depends on their spin directions

There is a moderate/small ferromagnetic exchange interaction between electrons due spin- dependent the Coulomb's repulsion between electrons at a longer distance Its origin is explained as follows: Due to fermion nature of an electron, the wave function of two electrons is symmetrical, when spin are antiparallel, and the wave function is antisymmetric, when spin are parallel. The Coulomb's repulsion between electrons is smaller when a distance between electrons is longer. In the case of antisymmetric wave function (parallel spins) the average distance between two electrons is a little bit longer (red line of Fig. 7 ) than for the case of symmetric wave function (antiparallel spins) (blue line of Fig. 7 ) . Therefore, the Coulomb's repulsion energy is smaller parallel spins. It forces the spins to align ferromagnetically.

Each electron should not be considered as an individual object. Only the Quantum Field of All Electrons is an object, which can divided into a number of electrons. There are many possibilities how the Quantum Field of Electrons can be divided into particles (electrons).

Similar, the electromagnetic field could also be divided into particles (phonons) and there are many possibilities how to divide the field into the photons.

In contrast to photons, which do not interact with each other, the electrons repels each other. Therefore, the energy of the Quantum Field of Electrons depends how the field is divided into the particles (electrons). The lowest energy correspond to the ground state.

Under different conditions (different spin direction or different external field), the division the Quantum Field of Electrons into particles (electrons) corresponding to a lowest energy might be different.

Therefore, a different total spin of a system of several electrons corresponds to a different division the Quantum Field of Electrons into particles (electrons) and a different distribution of wave functions of these particles (electrons). Since for the different spin the electron distributions become different, the Coulomb's interaction between the electrons become different as well. It makes the Coulomb's interaction to be spin-dependent.

### Bethe–Slater curve

 Bethe–Slater curve It is empirical curve, which represents the measured exchange interaction as distance between localized electrons. from Wikipedia wiki page is here

This empirical curve shows that for a shorter distance between the localized electrons of metals are ferromagnetic and at a shorter distance it is antiferromagnetic.

## localized and conduction (delocalized) electrons

. The electrons can be either localized or delocalized. The localized electrons usually (but not necessarily) have d- or f- spatial symmetry. Their length is short and it is about the interatomic length. In contrast, the length of the delocalized or conductive electrons is longer. In average, it equals to the electron mean-free path. The spatial symmetry of delocalized electrons is s- or p-. The transport mechanisms for localized and delocalized electrons are different. If the localized electrons move only because of scattering from a state to  a state, the transport of delocalized (conduction) electrons is due their movement between scatterings.

Additionally, the delocalized (conduction) electrons can be divided into two groups. The electrons of one group can freely move in bulk of a metal. The electrons of the second group are bound to a defect or an interface. Since the bounded electrons cannot move, they contribute only to the transport due to the scatterings.

### Different types of the exchange interaction

(1) Localized -Localized -----> neighbor to neighbor exchange interaction

d-electron to d-electron

The spin direction of each localized electron is fixed in time.

The exchange interaction can be either ferromagnetic or antiferromagnetic.

Each neighbor electron may have individual magnetization, which is fixed and which does not change in time.

For example, in a ferromagnetic material each delocalized neighbor electron has parallel spin direction.

in a antiferromagnetic material each delocalized neighbor electron has parallel spin direction.

(2) Delocalized -delocalized ----->scatterings

conduction electron to conduction electron

The spin direction of each delocalized (conduction) electron is not fixed in time. The spin direction of delocalized (conduction) electrons very frequently changes in time with about picosecond time scale..

#  Spin waves are waves of magnetization direction of localized electrons. They may propagate a long with only a weak absorption. The red arrows show spins of localized d-electrons. Click on image to enlarge it

## Spin Waves

Due to the exchange interaction between localized electrons, the waves of magnetization direction can propagate.

The spin waves is a feature of localized electrons. The localized d-electrons can support the spin waves very well.

The conduction electrons may support a spin waves, but only with a longer wavelength. Also, a spin wave of spins of conduction electrons has a substantial loss.

The reasons why the spins of conduction electrons are not so good for the propagation of a spin wave are following:

(1) substantially larger size of a conduction electron;
(2) the conduction electrons experience very-frequent scatterings
(3) a conduction electron moves in space with a speed, which may be faster than the speed of the spin wave.
(4) Conduction electrons significantly overlap each other. Each conduction electron are fully overlapped with millions and more other conduction electrons. It limits a free movement of its spin.

Q. Is it possible that an elementary particle without charge be a fermion?

Majorana fermion is a fermion particle, which does not have antiparticle. The Majorana fermions should be uncharged.

A. No. Only charged particles can be fermions. The Majorana fermion can not exist.

Only one fermion (or two fermion with opposite spins) can occupy one quantum state. In order for an elementary particle to be a fermion, there should be a force, which prevents two or more fermions to occupy one quantum state. In case of a charge particle, the Coulomb repulsion prevents two or more identical particles be at the same place at the same time (occupy the same quantum state).

There should be some force, which repels fermions from each other to prevent occupation of a quantum state by two fermions. The force should be one from known forces of the nature. There is no any "special" "quantum-mechanical" force to do this.

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