Section 3 Applications

coherent optical processing techniques to synthetic-aperture radar. Proceedings

communication theory. Journal of the Optical Society of America. 1962;**52**:1123

[24] Yu FTS. Introduction to Diffraction,

[25] Wiener N. Cybernetics. Cambridge,

Stationary Time Series. Cambridge, MA:

[27] Zimmerman R. The Universe in a Mirror: The Saga of the Hubble Space Telescope. Princeton, NJ: Princeton

Holography, Chapter 10. Cambridge, Mass: MIT Press; 1973. pp. 91-98

of the IEEE. 1966;**54**:1026

*Quantum Mechanics*

[23] Leith EN, Upatniecks J. Reconstructed wavefront and

Information Processing and

[26] Wiener N. Extrapolation, Interpolation, and Smoothing of

MA: MIT Press; 1948

MIT Press; 1949

Press; 2016

**188**

**Chapter 10**

**Abstract**

ödinger solver.

**1. Introduction**

**191**

Analysis of Quantum

Quantum Mechanics

*Aynul Islam and Anika Tasnim Aynul*

Confinement and Carrier

Transport of Nano-Transistor in

Quantum mechanics is the branch of physics that consists of laws explaining the physical properties of the nature of nano-particles and their characteristics on an atomic scale. The study of nano-particles significantly challenges our current perception of the universe and the fabric of reality itself. Quantum particles have both wave-like and particle-like characteristics. The fundamental equation that predicts the physical behaviour of a quantum system is the Schrödinger equation and the Poisson equation using Monte Carlo simulations. This gives rise to the *wavefunction, electron and hole densities, energy levels and band structure* of the system which contains all the measurable information about the particle such as time and position, where position is represented using probabilities. This is because particles do not have one definite position during the time before measurement. In fact, they exist as a fuzzy distribution of all possible states where the likelihood of finding the particle in some states is more probable than others. This is known as being in a *superposition* of all states. When the quantum system is observed, however, its wavefunction *collapses* so it consequently falls into one specific position. Moreover, in this chapter we present the simulation results of conduction band profile, electron density (classical and quantum mechanical), eigenstate and eigenfunctions for Si, SOI and III-V MOSFET structures at bias voltage 1.0 V using 1D Poisson-Schr-

**Keywords:** nano-devices, nano-particles, MOS, SOI and III-V structures, 1D Poisson-Schrödinger solver, conduction and valence band profile, carrier density

In this chapter, a connection between the band structure and quantum confinement effects with device characteristics in nano-scale devices is established. Three different devices are presented: a 25 nm gate length Si MOSFET, a 32 nm SOI MOSFET and a 15 nm In0.3 Ga0.7As channel MOSFET. We use a 1D Poisson-Schrödinger solver across the middle of the gate along the channel of the devices. The goal is to obtain the calculations of an energy of bound states and associated carrier wavefunctions which are carried out self consistently with electrostatic potential.

and wavefunctions in the potential well, wave-particle duality

#### **Chapter 10**

## Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum Mechanics

*Aynul Islam and Anika Tasnim Aynul*

### **Abstract**

Quantum mechanics is the branch of physics that consists of laws explaining the physical properties of the nature of nano-particles and their characteristics on an atomic scale. The study of nano-particles significantly challenges our current perception of the universe and the fabric of reality itself. Quantum particles have both wave-like and particle-like characteristics. The fundamental equation that predicts the physical behaviour of a quantum system is the Schrödinger equation and the Poisson equation using Monte Carlo simulations. This gives rise to the *wavefunction, electron and hole densities, energy levels and band structure* of the system which contains all the measurable information about the particle such as time and position, where position is represented using probabilities. This is because particles do not have one definite position during the time before measurement. In fact, they exist as a fuzzy distribution of all possible states where the likelihood of finding the particle in some states is more probable than others. This is known as being in a *superposition* of all states. When the quantum system is observed, however, its wavefunction *collapses* so it consequently falls into one specific position. Moreover, in this chapter we present the simulation results of conduction band profile, electron density (classical and quantum mechanical), eigenstate and eigenfunctions for Si, SOI and III-V MOSFET structures at bias voltage 1.0 V using 1D Poisson-Schrödinger solver.

**Keywords:** nano-devices, nano-particles, MOS, SOI and III-V structures, 1D Poisson-Schrödinger solver, conduction and valence band profile, carrier density and wavefunctions in the potential well, wave-particle duality

#### **1. Introduction**

In this chapter, a connection between the band structure and quantum confinement effects with device characteristics in nano-scale devices is established. Three different devices are presented: a 25 nm gate length Si MOSFET, a 32 nm SOI MOSFET and a 15 nm In0.3 Ga0.7As channel MOSFET. We use a 1D Poisson-Schrödinger solver across the middle of the gate along the channel of the devices. The goal is to obtain the calculations of an energy of bound states and associated carrier wavefunctions which are carried out self consistently with electrostatic potential.

The obtained wavefunctions are then used to calculate a carrier density which allows to obtain a sheet density across the structure at given bias.

One of the architectures seriously considered the Silicon-On-Insulator (SOI) transistor as a replacement for bulk MOSFETs. SOI transistors have many advantages compared with the conventional bulk MOSFET architecture. One of the most important is a better electrostatic integrity. SOI devices tolerate thicker gate oxides and low channel doping, allowing scaling to sub-10 nm channel lengths without substantial loss of performance.

*2.1.1 Wave and photon energy*

*DOI: http://dx.doi.org/10.5772/intechopen.93258*

*2.1.2 Wave-particle duality*

momentum of electrons.

**Figure 1.**

*frequency.*

**193**

ematically with more details in the next section [1, 2].

In general, a wave is a 'perturbation' from the surrounding or 'collision' between particles that travel from one position to another position over time. As we know a wave like classically or electromagnetic wave, which carries momentum and energy during changing the position, while in quantum mechanically a wavefunction can be applied to find out probabilities. We can say the equation for such wavefunction, so called Schrödinger wave equation, which will be described and developed math-

*Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum…*

We will now explore the physics behind the photoelectric effect. If light (monochromatic) falls on a smooth and clean surface of any material, then at some specific conditions, electrons are emitted from the surface. According to classical physics, the high intensity of light where the work function of the material will be overcome and an electron will be emitted from the surface, does not depend on the incident frequency, which is not observable. The observed effect is that, at a constant intensity of the incident light, the kinetic energy of the photoelectron increases linearly with frequency, start at specific frequency *ν***0**, below this frequency we did not observe any emission of photoelectron, as shown in **Figure 1(a)** and **(b)**.

After heating the surface, from that surface, thermal radiation will be emitted continuously (Planck), which form in discrete packets of energy called 'quanta'. The energy of these quanta is generally described by *E* ¼ *hν*, where *ν* is the frequency of radiation and *h* is a Planck's constant. However, Einstein explained that the energy in a light wave also contains photon or quanta, whose energy is also given by *E* ¼ *hν*. A photon with high energy can emit an electron from the surface of the material. The required energy to emit an electron is equal to the work function of the material, and rest of the incident photon energy can be converted into the kinetic energy of the photoelectron [1–3]. The maximum kinetic energy of

As we know the light waves in the photoelectric effect behave like particles. In the Compton effect experiment, an X-ray beam was incident on a solid - individual photons collide with single electrons that are free or quite loosely bound in the atoms of matter, as a result colliding photons transfer some of their energy and

*(a) The photoelectric effect and (b) the kinetic energy of the photoelectron as a function of the incident*

**<sup>2</sup>** *mv***<sup>2</sup>** <sup>¼</sup> *<sup>h</sup><sup>ν</sup>* � *<sup>h</sup>ν***<sup>0</sup>** (1)

the photoelectron can be written below as in the equation form:

*Tmax* <sup>¼</sup> **<sup>1</sup>**

However, the transition to this new device architecture and the eventual introduction of new materials in order to further boost device performance is a challenging task for the industry. However, the simulation of UTB transistors has to consider the impact of quantum confinement effects on the device electrostatics. The confinement effects can be induced into classical device simulation approaches using various approximations which include the density gradient approach [1, 2], the effective potential approach [3] and 1D Poisson-Schrödinger solver acting across the channel [4].

This idea leads onto another fundamental quantum superpower called quantum tunnelling. Quantum tunnelling causes particles to simply pass through physical barriers. If a particle was trapped in a well where it has not got enough kinetic energy to escape the well, it would stay in the well as one would expect, however, there is a slight difference. There will also be an exponentially decaying probability that the particle is found outside the well (under specific conditions, that is)! This has to do with the fact that the particles have a 'wave' of probable locations it can be in which extends beyond the well.

#### **2. Quantum mechanics in a semiconductor**

The purpose of this chapter is to understand the behaviour and properties of the particles of semiconductor material and devices. In order to get a conception of conduction band and valence band profile, drift velocity, energy, characteristics of the electrical field, wavefunction, carriers density and the mobility of carriers, we need to have an idea on the behaviour of carriers and then proper analysis about semiconductor materials which is related to the different potential. For more understanding about the particles in the theory of semiconductor physics we need to increase our knowledge extensively on the area of quantum mechanical wave theory. Furthermore, we will get idea about the physical behaviour of the materials in semiconductor physics whose electrical properties are related to the behaviour of the carriers in the crystal lattice structure. We will do an analysis of these carriers with the formulation of quantum mechanics, so called 'wave mechanics'. One of the most important parts to describing wave mechanics is 'Schrödinger wave equation'. The gradient of Poisson equation describes about the carriers density of the materials in the semiconductor devices. More details about the characteristics and behaviour of carriers of the semiconductor materials related with the quantum mechanical behaviour are described in this chapter.

#### **2.1 Action of quantum mechanics**

Generally, in quantum mechanics, we need to know the basic idea about the principle of tiny energy behaviour (photon), the wave-particle duality, wavefunction behaviour in the potential well, Heisenberg uncertainty principle, and the Schrödinger and Poisson equation.

*Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum… DOI: http://dx.doi.org/10.5772/intechopen.93258*

#### *2.1.1 Wave and photon energy*

The obtained wavefunctions are then used to calculate a carrier density which

One of the architectures seriously considered the Silicon-On-Insulator (SOI) transistor as a replacement for bulk MOSFETs. SOI transistors have many advantages compared with the conventional bulk MOSFET architecture. One of the most important is a better electrostatic integrity. SOI devices tolerate thicker gate oxides and low channel doping, allowing scaling to sub-10 nm channel lengths without

However, the transition to this new device architecture and the eventual introduction of new materials in order to further boost device performance is a challenging task for the industry. However, the simulation of UTB transistors has to consider the impact of quantum confinement effects on the device electrostatics. The confinement effects can be induced into classical device simulation approaches using various approximations which include the density gradient approach [1, 2], the effective potential approach [3] and 1D Poisson-Schrödinger solver acting

This idea leads onto another fundamental quantum superpower called quantum tunnelling. Quantum tunnelling causes particles to simply pass through physical barriers. If a particle was trapped in a well where it has not got enough kinetic energy to escape the well, it would stay in the well as one would expect, however, there is a slight difference. There will also be an exponentially decaying probability that the particle is found outside the well (under specific conditions, that is)! This has to do with the fact that the particles have a 'wave' of probable locations it can be

The purpose of this chapter is to understand the behaviour and properties of the particles of semiconductor material and devices. In order to get a conception of conduction band and valence band profile, drift velocity, energy, characteristics of the electrical field, wavefunction, carriers density and the mobility of carriers, we need to have an idea on the behaviour of carriers and then proper analysis about semiconductor materials which is related to the different potential. For more understanding about the particles in the theory of semiconductor physics we need to increase our knowledge extensively on the area of quantum mechanical wave theory. Furthermore, we will get idea about the physical behaviour of the materials in semiconductor physics whose electrical properties are related to the behaviour of the carriers in the crystal lattice structure. We will do an analysis of these carriers with the formulation of quantum mechanics, so called 'wave mechanics'. One of the most important parts to describing wave mechanics is 'Schrödinger wave equation'. The gradient of Poisson equation describes about the carriers density of the materials in the semiconductor devices. More details about the characteristics and behaviour of carriers of the semiconductor materials related with the quantum

Generally, in quantum mechanics, we need to know the basic idea about the

wavefunction behaviour in the potential well, Heisenberg uncertainty principle,

principle of tiny energy behaviour (photon), the wave-particle duality,

allows to obtain a sheet density across the structure at given bias.

substantial loss of performance.

in which extends beyond the well.

**2. Quantum mechanics in a semiconductor**

mechanical behaviour are described in this chapter.

**2.1 Action of quantum mechanics**

**192**

and the Schrödinger and Poisson equation.

across the channel [4].

*Quantum Mechanics*

In general, a wave is a 'perturbation' from the surrounding or 'collision' between particles that travel from one position to another position over time. As we know a wave like classically or electromagnetic wave, which carries momentum and energy during changing the position, while in quantum mechanically a wavefunction can be applied to find out probabilities. We can say the equation for such wavefunction, so called Schrödinger wave equation, which will be described and developed mathematically with more details in the next section [1, 2].

We will now explore the physics behind the photoelectric effect. If light (monochromatic) falls on a smooth and clean surface of any material, then at some specific conditions, electrons are emitted from the surface. According to classical physics, the high intensity of light where the work function of the material will be overcome and an electron will be emitted from the surface, does not depend on the incident frequency, which is not observable. The observed effect is that, at a constant intensity of the incident light, the kinetic energy of the photoelectron increases linearly with frequency, start at specific frequency *ν***0**, below this frequency we did not observe any emission of photoelectron, as shown in **Figure 1(a)** and **(b)**.

After heating the surface, from that surface, thermal radiation will be emitted continuously (Planck), which form in discrete packets of energy called 'quanta'. The energy of these quanta is generally described by *E* ¼ *hν*, where *ν* is the frequency of radiation and *h* is a Planck's constant. However, Einstein explained that the energy in a light wave also contains photon or quanta, whose energy is also given by *E* ¼ *hν*. A photon with high energy can emit an electron from the surface of the material. The required energy to emit an electron is equal to the work function of the material, and rest of the incident photon energy can be converted into the kinetic energy of the photoelectron [1–3]. The maximum kinetic energy of the photoelectron can be written below as in the equation form:

$$T\_{\text{max}} = \frac{1}{2}\ m\nu^2 = h\nu - h\nu\_0 \tag{1}$$

#### *2.1.2 Wave-particle duality*

As we know the light waves in the photoelectric effect behave like particles. In the Compton effect experiment, an X-ray beam was incident on a solid - individual photons collide with single electrons that are free or quite loosely bound in the atoms of matter, as a result colliding photons transfer some of their energy and momentum of electrons.

**Figure 1.**

*(a) The photoelectric effect and (b) the kinetic energy of the photoelectron as a function of the incident frequency.*

A portion of the X-ray beam was deflected and the frequency of the deflected wave had shifted compared to the incident wave as shown in **Figure 2**. The observed change in frequency and the deflected angle corresponded exactly to the collision between an X-ray (photon) and an electron in which both energy and momentum are conserved [1, 2].

In 1924, de Broglie observed that just as the waves exhibit particle-like behaviour, the particles also show wave-like characteristics. So, the assumption of the de Broglie was the existence of a wave-particle duality principle. The momentum of a photon is given by:

$$p = \frac{h}{\lambda}; = \frac{h}{p},\tag{2}$$

One consequence of the uncertainty principle is that we cannot, for example,

*Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum…*

Generally, the Schrödinger equation description depends on the physical situation. The most common form is the time-dependent Schrödinger equation which gives an explanation of a system related with time and also predicts that wave functions can form standing waves or stationary states. The stationary states can also be explained by a simpler form of the Schrödinger equation, the timeindependent Schrödinger Equation [7–9]. We will explain here the motion of electrons in a crystal by theory, which is described by Schrödinger wave equation.

*2.2.1 Time dependent and time independent Schrödinger wave equation and the density*

The Schrödinger equation is one of the fundamental tools for the understanding and prediction of nano-scaled semiconductor devices. For the case of one dimension the wave vector and momentum of a particle can be considered as scalars, so

We use these equation and properties of classical waves to set up a wave equation, known as the Schrödinger wave equation. We solve this equation for the particles which are confined to a potential well, and also to find the solution for particular discrete values of the total energy. However, we develop a theory by considering a particle, moving under the influence of a potential, V (x, t). For this case, the total energy E is equal to the sum of the kinetic and potential energies

> **2***m* þ *V*

*<sup>∂</sup><sup>x</sup>* , *<sup>E</sup>* <sup>¼</sup> *<sup>i</sup>*<sup>ℏ</sup> *<sup>∂</sup>***<sup>Ψ</sup>** ð Þ **<sup>x</sup>**,**<sup>t</sup>**

*<sup>∂</sup>***<sup>2</sup>***ψ*ð Þ *<sup>x</sup>*,*<sup>t</sup>*

where *ψ*ð Þ *x*,*t* is the wave function, which describes the behavior of an electron in the device and V(x) is the potential function assumed to be independent of time, and *m* is the mass of the particle. Assume that the wave function can be written in the form

where *ψ*ð Þ *x* is a function of the position *x* only and *ϕ*ð Þ*t* is a function of time *t*

*<sup>E</sup><sup>ψ</sup>* <sup>¼</sup> *<sup>H</sup><sup>ψ</sup>* <sup>¼</sup> *<sup>p</sup>***<sup>2</sup>**

After solving Eqs. (6) and (7), we can develop the one-dimensional time-

**2***m*

As we know the momentum operator, and energy are given by

*<sup>p</sup>* ¼ �*i*<sup>ℏ</sup> *<sup>∂</sup>*

*<sup>∂</sup><sup>t</sup>* ¼ � <sup>ℏ</sup>**<sup>2</sup>**

only. Substituting this form in the Schrödinger wave Eq. (8), we get

dependent Schrödinger equation, which can be written as

*<sup>i</sup>*<sup>ℏ</sup> *<sup>∂</sup><sup>ψ</sup>* ð Þ *<sup>x</sup>*,*<sup>t</sup>*

*E* ¼ ℏ*ω p* ¼ ℏ*k* (5)

*<sup>ψ</sup>* (6)

*<sup>∂</sup>x***<sup>2</sup>** <sup>þ</sup> *V x*ð Þ*ψ*ð Þ *<sup>x</sup>*,*<sup>t</sup>* , (8)

*ψ*ð Þ¼ *x*,*t ψ*ð Þ *x ϕ*ð Þ*t* , (9)

*<sup>∂</sup><sup>t</sup>* (7)

determine the exact position of an electron. We, instead, will determine the

probability of finding an electron at a particular position [5, 6].

**2.2 Basic principle of Schrödinger and Poisson equation**

*DOI: http://dx.doi.org/10.5772/intechopen.93258*

*probability function*

which can be written as,

**195**

relating the de Broglie equation, we can write as

where *p* is the momentum of the particle and *λ* is known as the de Broglie wavelength of the matter wave. In general, electromagnetic waves behave like particles (photons), and sometimes particles behave like waves. This wave-particle duality principle quantum mechanics applies to small particles such as electrons, protons and neutrons. The wave-particle duality is the basis on which we will apply wave theory to explain the motion and behaviour of electrons in a crystal.

#### *2.1.3 Uncertainty principle*

The uncertainty principle describes with absolute accuracy the behaviour of subatomic particles, which makes two different relationships between conjugate variables, including position and momentum and also energy and time [1, 2].

For the case 1, it is impossible to simultaneously explain with accuracy the position and momentum of a particle. If the uncertainty in the momentum is Δ*p*, and the uncertainty in the position is Δ*x*, then the uncertainty principle is stated as

$$
\Delta p \Delta \mathfrak{x} \geq \frac{\hbar}{2} \text{ or } \hbar,\tag{3}
$$

where <sup>ℏ</sup> is defined as <sup>ℏ</sup> <sup>¼</sup> *<sup>h</sup>* **<sup>2</sup>***<sup>π</sup>* = 1.054 x10�<sup>34</sup> J-s and is called international Planck'<sup>s</sup> constant.

For the case 2, it is impossible to simultaneously describe with accuracy the energy of a particle and the instant of time the particle has this energy. So, if the uncertainty in the energy is given by Δ*E* and the uncertainty in the time is given by Δ*t*, then the uncertainty principle is stated as

$$
\Delta E \Delta t \geq \hbar \tag{4}
$$

**Figure 2.**

*Compton scattering diagram showing the relationship of the incident photon and electron initially at rest to the scattered photon and electron given kinetic energy.*

*Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum… DOI: http://dx.doi.org/10.5772/intechopen.93258*

One consequence of the uncertainty principle is that we cannot, for example, determine the exact position of an electron. We, instead, will determine the probability of finding an electron at a particular position [5, 6].

#### **2.2 Basic principle of Schrödinger and Poisson equation**

A portion of the X-ray beam was deflected and the frequency of the deflected

In 1924, de Broglie observed that just as the waves exhibit particle-like behaviour, the particles also show wave-like characteristics. So, the assumption of the de Broglie was the existence of a wave-particle duality principle. The momentum of a

*<sup>λ</sup>* ; <sup>¼</sup> *<sup>h</sup>*

*<sup>p</sup>* , (2)

**<sup>2</sup>** *or* <sup>ℏ</sup>, (3)

**<sup>2</sup>***<sup>π</sup>* = 1.054 x10�<sup>34</sup> J-s and is called international Planck'<sup>s</sup>

Δ*E*Δ*t* ≥ ℏ (4)

wave had shifted compared to the incident wave as shown in **Figure 2**. The observed change in frequency and the deflected angle corresponded exactly to the collision between an X-ray (photon) and an electron in which both energy and

*<sup>p</sup>* <sup>¼</sup> *<sup>h</sup>*

wave theory to explain the motion and behaviour of electrons in a crystal.

Δ*p*Δ*x*≥

where *p* is the momentum of the particle and *λ* is known as the de Broglie wavelength of the matter wave. In general, electromagnetic waves behave like particles (photons), and sometimes particles behave like waves. This wave-particle duality principle quantum mechanics applies to small particles such as electrons, protons and neutrons. The wave-particle duality is the basis on which we will apply

The uncertainty principle describes with absolute accuracy the behaviour of subatomic particles, which makes two different relationships between conjugate variables, including position and momentum and also energy and time [1, 2]. For the case 1, it is impossible to simultaneously explain with accuracy the position and momentum of a particle. If the uncertainty in the momentum is Δ*p*, and the uncertainty in the position is Δ*x*, then the uncertainty principle is stated as

ℏ

For the case 2, it is impossible to simultaneously describe with accuracy the energy of a particle and the instant of time the particle has this energy. So, if the uncertainty in the energy is given by Δ*E* and the uncertainty in the time is given by

*Compton scattering diagram showing the relationship of the incident photon and electron initially at rest to the*

momentum are conserved [1, 2].

photon is given by:

*Quantum Mechanics*

*2.1.3 Uncertainty principle*

where <sup>ℏ</sup> is defined as <sup>ℏ</sup> <sup>¼</sup> *<sup>h</sup>*

Δ*t*, then the uncertainty principle is stated as

*scattered photon and electron given kinetic energy.*

constant.

**Figure 2.**

**194**

Generally, the Schrödinger equation description depends on the physical situation. The most common form is the time-dependent Schrödinger equation which gives an explanation of a system related with time and also predicts that wave functions can form standing waves or stationary states. The stationary states can also be explained by a simpler form of the Schrödinger equation, the timeindependent Schrödinger Equation [7–9]. We will explain here the motion of electrons in a crystal by theory, which is described by Schrödinger wave equation.

#### *2.2.1 Time dependent and time independent Schrödinger wave equation and the density probability function*

The Schrödinger equation is one of the fundamental tools for the understanding and prediction of nano-scaled semiconductor devices. For the case of one dimension the wave vector and momentum of a particle can be considered as scalars, so relating the de Broglie equation, we can write as

$$E = \hbar a \ p = \hbar k \tag{5}$$

We use these equation and properties of classical waves to set up a wave equation, known as the Schrödinger wave equation. We solve this equation for the particles which are confined to a potential well, and also to find the solution for particular discrete values of the total energy. However, we develop a theory by considering a particle, moving under the influence of a potential, V (x, t). For this case, the total energy E is equal to the sum of the kinetic and potential energies which can be written as,

$$E\Psi = H\Psi = \left(\frac{p^2}{2m} + V\right)\Psi\tag{6}$$

As we know the momentum operator, and energy are given by

$$p = -i\hbar \frac{\partial}{\partial \mathbf{x}},\\E = i\hbar \frac{\partial \Psi'(\mathbf{x}, \mathbf{t})}{\partial t} \tag{7}$$

After solving Eqs. (6) and (7), we can develop the one-dimensional timedependent Schrödinger equation, which can be written as

$$i\hbar\frac{\partial\boldsymbol{\mu}\,(\boldsymbol{\varkappa},\boldsymbol{t})}{\partial\boldsymbol{t}} = -\frac{\hbar^2}{2m}\frac{\partial^2\boldsymbol{\mu}(\boldsymbol{\varkappa},\boldsymbol{t})}{\partial\boldsymbol{\varkappa}^2} + \mathbf{V}(\boldsymbol{\varkappa})\boldsymbol{\nu}(\boldsymbol{\varkappa},\boldsymbol{t}),\tag{8}$$

where *ψ*ð Þ *x*,*t* is the wave function, which describes the behavior of an electron in the device and V(x) is the potential function assumed to be independent of time, and *m* is the mass of the particle. Assume that the wave function can be written in the form

$$
\Psi(\mathfrak{x}, \mathfrak{t}) = \mathfrak{y}(\mathfrak{x}) \mathfrak{g}(\mathfrak{t}), \tag{9}
$$

where *ψ*ð Þ *x* is a function of the position *x* only and *ϕ*ð Þ*t* is a function of time *t* only. Substituting this form in the Schrödinger wave Eq. (8), we get

$$i\hbar \left. \psi(\mathbf{x}) \frac{\partial \phi(\mathbf{t})}{\partial \mathbf{t}} = -\frac{\hbar^2}{2m} \phi(\mathbf{t}) \frac{\partial^2 \psi(\mathbf{x})}{\partial \mathbf{x}^2} + \mathbf{V}(\mathbf{x}) \left. \psi(\mathbf{x}) \right| \phi(\mathbf{t}) \tag{10}$$

Now, if we divide both sides of the Eq. (10) by the wave function, *ψ*ð*x*Þ *ϕ*ð Þ*t* , we get

$$i\hbar\frac{\mathbf{1}}{\phi(t)}\frac{\partial\phi(t)}{\partial t} = -\frac{\hbar^2}{2m}\frac{\mathbf{1}}{\Psi(\mathbf{x})}\frac{\partial^2\Psi(\mathbf{x})}{\partial\mathbf{x}^2} + \mathbf{V}(\mathbf{x})\tag{11}$$

After solving the Eq. (11) (using differential equation), the solution of the above equation can be written as

$$\boldsymbol{\phi}(\mathbf{t}) = \mathbf{e}^{-\boldsymbol{i}\left(\frac{\boldsymbol{k}}{\hbar}\right)\mathbf{t}} \tag{12}$$

*2.2.2 Wave function behaviour: finite square well, infinite square well, and tunnelling*

*Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum…*

In quantum mechanics, finite square well is an important invention to explain the particle wave function behaviour in the crystal. It is a further development of the infinite potential well, in which particle is confined in the square well. The finite potential well, there is a probability to find the particle outside the box. The idea in quantum mechanics is not like the classical idea, where if the total energy of the particle is less than the potential energy barrier of the walls it is not possible to find the particle outside the box. Alternatively, in quantum mechanics, there is a probability of the particle existing outside the box even if the particle energy is not enough by comparing the potential energy barrier of the walls [1, 8, 9].

We apply here the time independent Schrödinger equation for the case of an electron in free space. Consider the potential function *V x*ð Þ will be constant and energy must have the condition *E > V x*ð Þ. For analysis, we assume that the potential function *V x*ð Þ¼ **0** for the region II inside the box, as shown in **Figure 3**, and then

**2***mE*

*<sup>∂</sup>x***<sup>2</sup>** ¼ �*k***<sup>2</sup>**

After solving the Eq. (19) using differential equation, the general solution becomes

<sup>ℏ</sup>**<sup>2</sup>** ð Þ *<sup>E</sup>* � *V x*ð Þ *<sup>ψ</sup>*ð Þ¼ *<sup>x</sup>* **<sup>0</sup>**

*<sup>δ</sup>x***<sup>2</sup>** <sup>¼</sup> ð Þ *<sup>E</sup>* � *<sup>V</sup>***<sup>0</sup>** *<sup>ψ</sup>*ð Þ *<sup>x</sup>* (20)

*ψ*ð Þ¼ *x A* **sin** ð Þþ *kx B* **cos** ð Þ *kx* ,

Now, for the region I and region III, outside the box, where the potential

*Potential function of the finite potential well for different regions along the x-direction, with three discrete*

*=***2***m*, then Eq. (18) leads to

<sup>ℏ</sup>**<sup>2</sup>** *<sup>ψ</sup>*ð Þ¼ *<sup>x</sup>* **<sup>0</sup>** (18)

*ψ*ð Þ *x* (19)

the time-independent wave equation can be written as from Eq. (13) as

*k***2**

*<sup>∂</sup>***<sup>2</sup>***ψ*ð Þ *<sup>x</sup>*

where *A* and *B* are complex numbers, and *k* is any real number.

**2***m*

assumed to be constant, *V x*ð Þ¼ *V***0**, and Eq. (13) becomes

� ℏ**2 2***m δ***2** *ψ*ð Þ *x*

*<sup>∂</sup>***<sup>2</sup>***ψ*ð Þ *<sup>x</sup> ∂x***<sup>2</sup>** þ

*<sup>∂</sup>***<sup>2</sup>***ψ*ð Þ *<sup>x</sup> ∂x***<sup>2</sup>** þ

**<sup>2</sup>***mE* <sup>p</sup> *<sup>=</sup>*<sup>ℏ</sup> or *<sup>E</sup>* <sup>¼</sup> <sup>ℏ</sup>**<sup>2</sup>**

*behaviour*

*DOI: http://dx.doi.org/10.5772/intechopen.93258*

Letting *<sup>k</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi

**Figure 3.**

**197**

*energy levels and corresponding wave function.*

We notice the solution of Eq. (12) is the classical exponential form of a sinusoidal wave. As we see from Eq. (11) on the left hand side with function of time, which is equal to the constant of total energy of the particle, and the right hand side is a function of the position *x* only. After simplification of the above equation the time-independent portion of the Schrödinger wave equation can now be written as

$$-\frac{\hbar^2}{2m}\frac{1}{\Psi(\mathbf{x})}\frac{\partial^2 \Psi(\mathbf{x})}{\partial \mathbf{x}^2} + \mathbf{V}(\mathbf{x}) = E$$

$$\frac{\partial^2 \Psi(\mathbf{x})}{\partial \mathbf{x}^2} + \frac{2m}{\hbar^2} \left(E - \mathbf{V}(\mathbf{x})\right)\Psi(\mathbf{x}) = \mathbf{0} \tag{13}$$

Now, the total wave function can be written as in the form of product of the position or time independent function and the time dependent function,

$$
\Psi(\mathfrak{x}, \mathfrak{t}) = \mathfrak{y}(\mathfrak{x}) \,\, \phi(\mathfrak{t}) = \mathfrak{y}(\mathfrak{x}) \mathbf{e}^{-i \left(\frac{\mathfrak{x}}{\mathfrak{t}}\right) \mathfrak{t}} \tag{14}
$$

According to the Max Born, the function j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>*,*<sup>t</sup>* <sup>2</sup> *dx* is the probability of finding the particle between *<sup>x</sup>* and *<sup>x</sup>* <sup>þ</sup> *dx* at a given time, we can express also j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>*,*<sup>t</sup>* <sup>2</sup> *dx* as a probability density function.

$$\left|\boldsymbol{\upmu}(\mathbf{x},\mathbf{t})\right|^{2}d\mathbf{x} = \boldsymbol{\upmu}(\mathbf{x},\mathbf{t}) \cdot \boldsymbol{\upmu}^{\*}(\mathbf{x},\mathbf{t}),\tag{15}$$

where *ψ* ð Þ *x*,*t* is the complex conjugate function. Following Eq. (14), we can rewrite:

$$
\Psi^\*\left(\mathbf{x}, \mathbf{t}\right) = \Psi^\*\left(\mathbf{x}\right) \mathbf{e}^{\mathbf{i}\left(\frac{\mathbf{x}}{\hbar}\right)\mathbf{t}}\tag{16}
$$

Finally, we can develop the density of the probability function using Eq. (14), and Eq. (16), which is independent of time.

$$\left|\boldsymbol{\Psi}(\boldsymbol{\mathfrak{x}},\boldsymbol{t})\right|^{2}d\boldsymbol{\mathfrak{x}}=\boldsymbol{\mathfrak{y}}(\boldsymbol{\mathfrak{x}})\cdot\boldsymbol{\mathfrak{y}}^{\*}\left(\boldsymbol{\mathfrak{x}}\right)=\left|\boldsymbol{\mathfrak{y}}(\boldsymbol{\mathfrak{x}})\right|^{2}\tag{17}$$

The main difference between classical and quantum mechanics is that in classical mechanics, the position of a particle can be determined precisely, whereas in quantum mechanics the position of a particle is related in terms of probability [1, 2].

*Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum… DOI: http://dx.doi.org/10.5772/intechopen.93258*

#### *2.2.2 Wave function behaviour: finite square well, infinite square well, and tunnelling behaviour*

In quantum mechanics, finite square well is an important invention to explain the particle wave function behaviour in the crystal. It is a further development of the infinite potential well, in which particle is confined in the square well. The finite potential well, there is a probability to find the particle outside the box. The idea in quantum mechanics is not like the classical idea, where if the total energy of the particle is less than the potential energy barrier of the walls it is not possible to find the particle outside the box. Alternatively, in quantum mechanics, there is a probability of the particle existing outside the box even if the particle energy is not enough by comparing the potential energy barrier of the walls [1, 8, 9].

We apply here the time independent Schrödinger equation for the case of an electron in free space. Consider the potential function *V x*ð Þ will be constant and energy must have the condition *E > V x*ð Þ. For analysis, we assume that the potential function *V x*ð Þ¼ **0** for the region II inside the box, as shown in **Figure 3**, and then the time-independent wave equation can be written as from Eq. (13) as

$$\frac{\partial^2 \boldsymbol{\Psi}(\mathbf{x})}{\partial \mathbf{x}^2} + \frac{2mE}{\hbar^2} \boldsymbol{\Psi}(\mathbf{x}) = \mathbf{0} \tag{18}$$

Letting *<sup>k</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi **<sup>2</sup>***mE* <sup>p</sup> *<sup>=</sup>*<sup>ℏ</sup> or *<sup>E</sup>* <sup>¼</sup> <sup>ℏ</sup>**<sup>2</sup>** *k***2** *=***2***m*, then Eq. (18) leads to

$$\frac{\partial^2 \Psi(\mathbf{x})}{\partial \mathbf{x}^2} = -k^2 \Psi(\mathbf{x}) \tag{19}$$

After solving the Eq. (19) using differential equation, the general solution becomes

$$
\psi(\mathbf{x}) = \mathbf{A}\sin\left(k\mathbf{x}\right) + \mathbf{B}\cos\left(k\mathbf{x}\right),
$$

where *A* and *B* are complex numbers, and *k* is any real number.

Now, for the region I and region III, outside the box, where the potential assumed to be constant, *V x*ð Þ¼ *V***0**, and Eq. (13) becomes

$$\frac{\partial^2 \Psi(\mathbf{x})}{\partial \mathbf{x}^2} + \frac{2m}{\hbar^2} (E - \mathbf{V}(\mathbf{x})) \,\Psi(\mathbf{x}) = \mathbf{0}$$

$$-\frac{\hbar^2}{2m} \frac{\delta^2 \Psi(\mathbf{x})}{\delta \mathbf{x}^2} = (E - \mathbf{V}\_0) \Psi(\mathbf{x})\tag{20}$$

#### **Figure 3.**

*Potential function of the finite potential well for different regions along the x-direction, with three discrete energy levels and corresponding wave function.*

*i*ℏ *ψ*ð Þ *x*

equation can be written as

we get

*Quantum Mechanics*

be written as

rewrite:

probability [1, 2].

**196**

*<sup>∂</sup>ϕ*ð Þ*<sup>t</sup>*

*<sup>i</sup>*<sup>ℏ</sup> **<sup>1</sup>** *ϕ*ð Þ*t*

*<sup>∂</sup><sup>t</sup>* ¼ � <sup>ℏ</sup>**<sup>2</sup>**

*<sup>∂</sup>ϕ*ð Þ*<sup>t</sup>*

� <sup>ℏ</sup>**<sup>2</sup> 2***m*

*<sup>∂</sup>***<sup>2</sup>***ψ*ð Þ *<sup>x</sup> ∂x***<sup>2</sup>** þ

According to the Max Born, the function j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>*,*<sup>t</sup>* <sup>2</sup>

as a probability density function.

and Eq. (16), which is independent of time.

**1** *ψ*ð Þ *x*

**2***m*

position or time independent function and the time dependent function,

**2***m*

*<sup>∂</sup><sup>t</sup>* ¼ � <sup>ℏ</sup>**<sup>2</sup>**

*ϕ*ð Þ*t ∂***2** *ψ*ð Þ *x*

Now, if we divide both sides of the Eq. (10) by the wave function, *ψ*ð*x*Þ *ϕ*ð Þ*t* ,

**1** *ψ*ð Þ *x*

After solving the Eq. (11) (using differential equation), the solution of the above

*<sup>∂</sup>***<sup>2</sup>***ψ*ð Þ *<sup>x</sup>*

*<sup>ϕ</sup>*ð Þ¼ *<sup>t</sup> <sup>e</sup>*�*<sup>i</sup> <sup>E</sup>*ð Þ<sup>ℏ</sup> *<sup>t</sup>* (12)

<sup>ℏ</sup>**<sup>2</sup>** ð Þ *<sup>E</sup>* � *V x*ð Þ *<sup>ψ</sup>*ð Þ¼ *<sup>x</sup>* **<sup>0</sup>** (13)

*dx* is the probability of finding

<sup>2</sup> (17)

*<sup>ψ</sup>*ð Þ¼ *<sup>x</sup>*,*<sup>t</sup> <sup>ψ</sup>*ð Þ *<sup>x</sup> <sup>ϕ</sup>*ð Þ¼ *<sup>t</sup> <sup>ψ</sup>*ð Þ *<sup>x</sup> <sup>e</sup>*�*<sup>i</sup> <sup>E</sup>*ð Þ<sup>ℏ</sup> *<sup>t</sup>* (14)

j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>*,*<sup>t</sup>* <sup>2</sup> *dx* <sup>¼</sup> *<sup>ψ</sup>*ð Þ� *<sup>x</sup>*,*<sup>t</sup> <sup>ψ</sup>* <sup>∗</sup> ð Þ *<sup>x</sup>*,*<sup>t</sup>* , (15)

*<sup>ψ</sup>* <sup>∗</sup> ð Þ¼ *<sup>x</sup>*,*<sup>t</sup> <sup>ψ</sup>* <sup>∗</sup> ð Þ *<sup>x</sup> <sup>e</sup><sup>i</sup> <sup>E</sup>*ð Þ<sup>ℏ</sup> *<sup>t</sup>* (16)

**2***m*

We notice the solution of Eq. (12) is the classical exponential form of a sinusoidal wave. As we see from Eq. (11) on the left hand side with function of time, which is equal to the constant of total energy of the particle, and the right hand side is a function of the position *x* only. After simplification of the above equation the time-independent portion of the Schrödinger wave equation can now

> *∂*2 *ψ*ð Þ *x*

Now, the total wave function can be written as in the form of product of the

the particle between *<sup>x</sup>* and *<sup>x</sup>* <sup>þ</sup> *dx* at a given time, we can express also j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>*,*<sup>t</sup>* <sup>2</sup> *dx*

where *ψ* ð Þ *x*,*t* is the complex conjugate function. Following Eq. (14), we can

Finally, we can develop the density of the probability function using Eq. (14),

j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>*,*<sup>t</sup>* <sup>2</sup> *dx* <sup>¼</sup> *<sup>ψ</sup>*ð Þ� *<sup>x</sup> <sup>ψ</sup>* <sup>∗</sup> ð Þ¼ *<sup>x</sup>* j j *<sup>ψ</sup>*ð Þ *<sup>x</sup>*

The main difference between classical and quantum mechanics is that in classical mechanics, the position of a particle can be determined precisely, whereas in quantum mechanics the position of a particle is related in terms of

*<sup>∂</sup>x***<sup>2</sup>** <sup>þ</sup> *V x*ð Þ¼ *<sup>E</sup>*

*<sup>∂</sup>x***<sup>2</sup>** <sup>þ</sup> *V x*ð Þ *<sup>ψ</sup>*ð*x*<sup>Þ</sup> *<sup>ϕ</sup>*ð Þ*<sup>t</sup>* (10)

*<sup>∂</sup>x***<sup>2</sup>** <sup>þ</sup> *V x*ð Þ (11)

We will get here two possible solutions, depending on energies, where *E* is smaller than *V***0**, that means the particle is bound the potential and *E* is greater than *V***0**, that means the particle is moving in free space, which is represented by travelling wave (shown in **Figure 3**).

The potential *V x*ð Þ as a function of the position is shown in **Figure 4**. The particle is assumed to exist in region II so the particle is contained within a finite region of space. The time-independent Schrödinger wave equation can be written as

$$\frac{\partial^2 \boldsymbol{\Psi}(\mathbf{x})}{\partial \mathbf{x}^2} + \frac{2m}{\hbar^2} (E - V(\mathbf{x})) \, \boldsymbol{\Psi}(\mathbf{x}) = \mathbf{0},\tag{21}$$

semiconductor device classically. As we know the Gauss's law is **∇** � *D* ¼ *ρ*, where **∇** is the divergence operator, **D** is the electric displacement law (*D* ¼ *εE*), *ρ* is the free charge density, *ε* is the permittivity of the medium, and **E** is the electric field (*E* ¼ �**∇V**). Maxwell's four equations describe the electric and magnetic fields arising from distributions of electric charge and currents, and how those fields

*Quantum tunnelling: The wavefunctions through the potential barrier, a significant tunnelling effect can be seen*

*Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum…*

<sup>∇</sup> � *<sup>E</sup>* <sup>¼</sup> *<sup>ρ</sup>*

<sup>∇</sup> � *<sup>E</sup>* ¼ � *<sup>δ</sup><sup>B</sup>*

∇ � *B* ¼ *μ***0***j* þ *μ***0***ϵ***<sup>0</sup>**

We can substitute the value of electric displacement in the basic equation of

we suppose that there is no magnetic field, then Eq. (24) can be rewritten as ∇ � *E* ¼ 0, The electric field as the gradient of a scalar function **V**, is called electrostatic potential. Thus we can write *E* ¼ �∇**V**, and the minus sign is chosen so that **V** is introduced as the potential energy per unit charge. Finally, we can develop the derivation of Poisson's equation using Eq. (22), and Eq. (24), which leads to

<sup>∇</sup> � *<sup>E</sup>* <sup>¼</sup> <sup>∇</sup> � �ð Þ¼ <sup>∇</sup>**<sup>V</sup>** *<sup>ρ</sup>*

<sup>∇</sup>**2V** ¼ � *<sup>ρ</sup>*

In this section, a connection between the bandstructure and quantum confinement effects with device characteristics in nano-scale devices is established. Three different devices are presented: a 25 nm gate length Si (Silicon) MOSFET (Metal Oxide Semiconductor Field Effect Transistor), a 32 nm Silicon-on-insulator (SOI) MOSFET and a 15 nm implant free (IF) In0.3Ga0.7As (Indium Gallium Arsenide) channel MOSFET. We use a 1D Poisson-Schrödinger solver across the middle of the

**2.3 Contribution of Schrödinger and Poisson equation in nano-particles**

*ϵ***0**

*ϵ***0**

*δt*

*ϵ***0**

*δE*

*ϵ***0**

, (22)

, (24)

*<sup>δ</sup><sup>t</sup>* (25)

, so called Eq. (22). In electrostatic,

(26)

∇ � *B* ¼ **0** (23)

change in time. Maxwell's equations described as follows [2]:

**Figure 5.**

**199**

*in three different regions.*

*DOI: http://dx.doi.org/10.5772/intechopen.93258*

Gauss's law, which can be rewritten as <sup>∇</sup> � *<sup>E</sup>* <sup>¼</sup> *<sup>ρ</sup>*

where *E* is the total energy of the particle. If *E* is finite, the wave function must be zero, or *ψ*ð Þ¼ *x* 0, in both regions I and III. A particle cannot penetrate these infinite potential barriers, so the probability of finding the particle in regions I and III is zero.

In the quantum mechanics, the particle in a box (also known as the infinite potential well) describes a particle free to move in a small space surrounded by impenetrable barriers (shown in **Figure 4**). In classical systems, for example, a particle trapped inside a large box can move at any speed within the box and it is no more likely to be found at one position than another. However, when the well becomes very narrow (on the scale of a few nanometers), quantum effects become important. The particle may only occupy certain positive energy levels [1, 9, 10].

The energy of the incident particle (*E > V*) in region I and transmitted particle (*E > V*) in region III through the potential barrier (*E < V*) in region II, where the tunnelled particle is the same but the probability amplitude is decreased. There is a finite probability that a particle impinging a potential barrier will penetrate the barrier and will appear in region III (shown in **Figure 5**). This quantum mechanical tunnelling phenomenon can be applied to semiconductor devices.

Quantum tunnelling is the quantum mechanical phenomenon where a subatomic particle's probability disappears from one side of a potential barrier and appears on the other side without any probability appearing inside the well. Quantum tunnelling is not predicted by the laws of classical mechanics where surmounting a potential barrier requires enough potential energy [9, 10].

#### *2.2.3 Maxwell's equations: Poisson equation*

To develop the Poisson equation we need to describe the famous Maxwell's equations in their differential form. In mathematics, Poisson's equation is a partial differential equation, which describe the potential field caused by a given charge distribution [2, 8]. Our goal is to find the density of the electron in the crystal of the

#### **Figure 4.**

*Potential function of the infinite potential well for different regions along the x-direction, with three discrete energy levels and corresponding wave function in the box or potential well.*

*Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum… DOI: http://dx.doi.org/10.5772/intechopen.93258*

**Figure 5.**

We will get here two possible solutions, depending on energies, where *E* is smaller than *V***0**, that means the particle is bound the potential and *E* is greater than

The potential *V x*ð Þ as a function of the position is shown in **Figure 4**. The particle is assumed to exist in region II so the particle is contained within a finite region of space. The time-independent Schrödinger wave equation can be written as

where *E* is the total energy of the particle. If *E* is finite, the wave function must be zero, or *ψ*ð Þ¼ *x* 0, in both regions I and III. A particle cannot penetrate these infinite potential barriers, so the probability of finding the particle in regions I and

In the quantum mechanics, the particle in a box (also known as the infinite potential well) describes a particle free to move in a small space surrounded by impenetrable barriers (shown in **Figure 4**). In classical systems, for example, a particle trapped inside a large box can move at any speed within the box and it is no more likely to be found at one position than another. However, when the well becomes very narrow (on the scale of a few nanometers), quantum effects become important. The particle may only occupy certain positive energy levels [1, 9, 10]. The energy of the incident particle (*E > V*) in region I and transmitted particle (*E > V*) in region III through the potential barrier (*E < V*) in region II, where the tunnelled particle is the same but the probability amplitude is decreased. There is a finite probability that a particle impinging a potential barrier will penetrate the barrier and will appear in region III (shown in **Figure 5**). This quantum mechanical

<sup>ℏ</sup><sup>2</sup> ð Þ *<sup>E</sup>* � *V x*ð Þ *<sup>ψ</sup>*ð Þ¼ *<sup>x</sup>* 0, (21)

*V***0**, that means the particle is moving in free space, which is represented by

2*m*

tunnelling phenomenon can be applied to semiconductor devices.

*2.2.3 Maxwell's equations: Poisson equation*

Quantum tunnelling is the quantum mechanical phenomenon where a subatomic particle's probability disappears from one side of a potential barrier and appears on the other side without any probability appearing inside the well. Quantum tunnelling is not predicted by the laws of classical mechanics where surmounting a potential barrier requires enough potential energy [9, 10].

To develop the Poisson equation we need to describe the famous Maxwell's equations in their differential form. In mathematics, Poisson's equation is a partial differential equation, which describe the potential field caused by a given charge distribution [2, 8]. Our goal is to find the density of the electron in the crystal of the

*Potential function of the infinite potential well for different regions along the x-direction, with three discrete*

*energy levels and corresponding wave function in the box or potential well.*

travelling wave (shown in **Figure 3**).

*Quantum Mechanics*

III is zero.

**Figure 4.**

**198**

*∂*2 *ψ*ð Þ *x ∂x*<sup>2</sup> þ

*Quantum tunnelling: The wavefunctions through the potential barrier, a significant tunnelling effect can be seen in three different regions.*

semiconductor device classically. As we know the Gauss's law is **∇** � *D* ¼ *ρ*, where **∇** is the divergence operator, **D** is the electric displacement law (*D* ¼ *εE*), *ρ* is the free charge density, *ε* is the permittivity of the medium, and **E** is the electric field (*E* ¼ �**∇V**). Maxwell's four equations describe the electric and magnetic fields arising from distributions of electric charge and currents, and how those fields change in time. Maxwell's equations described as follows [2]:

$$
\nabla \cdot \mathbf{E} = \frac{\rho}{\mathbf{e\_0}},
\tag{22}
$$

$$\nabla \cdot \mathbf{B} = \mathbf{0} \tag{23}$$

$$
\nabla \times \mathbf{E} = -\frac{\delta \mathbf{B}}{\delta \mathbf{t}},
\tag{24}
$$

$$
\nabla \times \mathbf{B} = \mu\_0 \mathbf{j} + \mu\_0 \varepsilon\_0 \frac{\delta E}{\delta t} \tag{25}
$$

We can substitute the value of electric displacement in the basic equation of Gauss's law, which can be rewritten as <sup>∇</sup> � *<sup>E</sup>* <sup>¼</sup> *<sup>ρ</sup> ϵ***0** , so called Eq. (22). In electrostatic, we suppose that there is no magnetic field, then Eq. (24) can be rewritten as ∇ � *E* ¼ 0, The electric field as the gradient of a scalar function **V**, is called electrostatic potential. Thus we can write *E* ¼ �∇**V**, and the minus sign is chosen so that **V** is introduced as the potential energy per unit charge. Finally, we can develop the derivation of Poisson's equation using Eq. (22), and Eq. (24), which leads to

$$
\nabla \cdot \mathbf{E} = \nabla \cdot (-\nabla \mathbf{V}) = \frac{\rho}{\varepsilon\_0}
$$

$$
\nabla^2 \mathbf{V} = -\frac{\rho}{\varepsilon\_0} \tag{26}
$$

#### **2.3 Contribution of Schrödinger and Poisson equation in nano-particles**

In this section, a connection between the bandstructure and quantum confinement effects with device characteristics in nano-scale devices is established. Three different devices are presented: a 25 nm gate length Si (Silicon) MOSFET (Metal Oxide Semiconductor Field Effect Transistor), a 32 nm Silicon-on-insulator (SOI) MOSFET and a 15 nm implant free (IF) In0.3Ga0.7As (Indium Gallium Arsenide) channel MOSFET. We use a 1D Poisson-Schrödinger solver across the middle of the gate along the channel of the devices. The goal is to obtain the calculations of an energy of bound states and associated carrier wavefunctions which are carried out self consistently with electrostatic potential. The obtained wavefunctions are then used to calculate a carrier density which allows to obtain a sheet density across the structure at given bias. We have chosen the SOI MOSFET for comparison because it is considered for low power applications. The SOI technology is developing now into the commercial area and is included in the ITRS. The SOI based MOSFETs have a silicon channel made of a narrow layer of less than 10 nm grown on a relatively thick SiO2 layer. Such strongly confined device channel creates an ultra-thin body (UTB) which provides enhanced carrier transport and, therefore, this transistor architecture is better to be referred as UTB SOI. The FD (fully depleted) SOI MOSFET has superior electrical characteristics and a threshold control from a bottom gate compared to the bulk CMOS device, which are described as follows [11, 12]:

1.decrease in a power dissipation and faster speed due to reduced junction area,

where *ψ* is the wave function, *E* is the energy eigenvalue, *V x*ð Þ is the potential energy assumed to be independent of time, ℏ is Planck's constant divided by 2π, *m* � *m x*ð Þ is the effective mass of an electron which is a position dependent, *H* represents the Hamiltonian operator associated with the sum of the kinetic and

*Schematics of conduction band structure of silicon in <100> oriented narrow channels [assumed (100) plane*

*Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum…*

*for the Si-SiO2 interface]. The energy levels are also shown in the silicon quantum well.*

The potential distribution *ϕ*ð Þ *x* in the semiconductor can be determined from a

*ε*0

*ρ* ¼ *ND*ð Þ� *x NA*ð Þþ *x p x*ð Þ� *n x*ð Þ*:* (30)

*ε*0

*ε*0

*V x*ð Þ¼ *qϕ*ð Þþ *x ΔEC*ð Þ *x* , (33)

(29)

, (31)

(32)

*d d x*ð Þ *<sup>ϕ</sup>*ð Þ¼ *<sup>x</sup>* �*ρ*ð Þ *<sup>x</sup>*

*d x*ð Þ *<sup>ϕ</sup>*ð Þ¼ *<sup>x</sup>* �*q N*½ � *<sup>D</sup>*ð Þ� *<sup>x</sup> NA*ð Þþ *<sup>x</sup> p x*ð Þ� *n x*ð Þ

*d*

where *ϕ* is the electrostatic potential, *ε<sup>S</sup>* is the semiconductor dielectric constant, *ε*<sup>0</sup> is the permittivity of free space. In the static behaviour, *ND* and *NA* are called the ionised donor and acceptor concentrations and, in the case of dynamic behaviour, *n* and *p* are known as electron and hole density distributions. When dealing with *n*type majority carriers semiconductor devices, we can ignore the holes contribution due to their slow movement compared to the electron dynamics. Then only the electrons and donors are considered. As a result the above Eq. (31) can be written as

*d x*ð Þ *<sup>ϕ</sup>*ð Þ¼ *<sup>x</sup>* �*q N*½ � *<sup>D</sup>*ð Þ� *<sup>x</sup> n x*ð Þ

The potential energy *V* in the Hamiltonian is related to the electrostatic potential

where *ΔEC* is the pseudopotential energy due to the band offset at the heterointerface. The wavefunction *ψ*ð Þ *x* in Eq. (27) and the electron density *n x*ð Þ in

potential energies of the system.

*DOI: http://dx.doi.org/10.5772/intechopen.93258*

**Figure 6.**

The charge density *ρ* is given by

*d*

*d d x*ð Þ *<sup>ε</sup>S*ð Þ *<sup>x</sup>*

Eq. (25) can be written as

*d d x*ð Þ *<sup>ε</sup>S*ð Þ *<sup>x</sup>*

*ϕ* as follows [15]:

Eq. (32) are related by

**201**

solution of the 1 D Poisson Eq. (26), which is given by

*d d x*ð Þ *<sup>ε</sup>S*ð Þ *<sup>x</sup>*


From a point of simplicity, we will first consider the UTB SOI transistor architecture because it is quite illustrative for quantum-mechanical calculations of a confined structure. We will consider a semiconductor material with a small energy gap sandwiched between energy barriers from a material with a larger energy gap. In this way, a quantum well is formed between the barriers which introduce a potential well with discrete energy levels, where particles are confined in one dimension and move free in other two directions as shown in **Figure 6** [13].

We will now focus exclusively on the calculation of quantum states related to electrons. The calculation of quantum states related to holes or any other particles or quasi-particles are equivalent. In the calculations, we will determine the conduction band profile, electron density, energy levels (eigenstates), wavefunctions (eigenfunctions) and electron sheet density in the semiconductor device structure under external potential. In this case, both Schrödinger and Poisson equations have to be solved self-consistently. The one-dimensional, time independent Schrödinger's wave equation for a particle in a potential distribution is a second order ordinary differential equation, which is given by [14].

$$\frac{\partial^2 \boldsymbol{\nu}(\mathbf{x})}{\partial \mathbf{x}^2} + \frac{2m}{\hbar^2} (E - V(\mathbf{x})) \, \boldsymbol{\nu}(\mathbf{x}) = \mathbf{0}$$

$$\left(-\frac{\hbar^2}{2} \frac{\delta}{\delta \boldsymbol{\kappa}} \frac{\mathbf{1}}{m(\boldsymbol{\kappa})} + V(\boldsymbol{\kappa})\right) \boldsymbol{\nu}(\mathbf{x}) = E \boldsymbol{\nu}(\mathbf{x})\tag{27}$$

$$\mathbf{H} \boldsymbol{\nu}(\mathbf{x}) = E \boldsymbol{\nu}(\mathbf{x}),$$

where

$$H = -\frac{\hbar^2}{2} \frac{\delta}{\delta \mathbf{x}} \frac{\mathbf{1}}{m(\mathbf{x})} + V(\mathbf{x}),\tag{28}$$

*Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum… DOI: http://dx.doi.org/10.5772/intechopen.93258*

**Figure 6.**

gate along the channel of the devices. The goal is to obtain the calculations of an energy of bound states and associated carrier wavefunctions which are carried out self consistently with electrostatic potential. The obtained wavefunctions are then used to calculate a carrier density which allows to obtain a sheet density across the structure at given bias. We have chosen the SOI MOSFET for comparison because it is considered for low power applications. The SOI technology is developing now into the commercial area and is included in the ITRS. The SOI based MOSFETs have a silicon channel made of a narrow layer of less than 10 nm grown on a relatively thick SiO2 layer. Such strongly confined device channel creates an ultra-thin body (UTB) which provides enhanced carrier transport and, therefore, this transistor architecture is better to be referred as UTB SOI. The FD (fully depleted) SOI MOSFET has superior electrical characteristics and a threshold control from a bottom gate compared to the bulk CMOS device, which are described as follows [11, 12]:

1.decrease in a power dissipation and faster speed due to reduced junction area,

From a point of simplicity, we will first consider the UTB SOI transistor architecture because it is quite illustrative for quantum-mechanical calculations of a confined structure. We will consider a semiconductor material with a small energy gap sandwiched between energy barriers from a material with a larger energy gap. In this way, a quantum well is formed between the barriers which introduce a potential well with discrete energy levels, where particles are confined in one dimension and move free in other two directions as shown in **Figure 6** [13].

We will now focus exclusively on the calculation of quantum states related to electrons. The calculation of quantum states related to holes or any other particles or quasi-particles are equivalent. In the calculations, we will determine the conduction

(eigenfunctions) and electron sheet density in the semiconductor device structure under external potential. In this case, both Schrödinger and Poisson equations have

Schrödinger's wave equation for a particle in a potential distribution is a second

<sup>ℏ</sup><sup>2</sup> ð Þ *<sup>E</sup>* � *V x*ð Þ *<sup>ψ</sup>*ð Þ¼ *<sup>x</sup>* <sup>0</sup>

*ψ*ð Þ¼ *x Eψ*ð Þ *x* (27)

*m x*ð Þ <sup>þ</sup> *V x*ð Þ, (28)

band profile, electron density, energy levels (eigenstates), wavefunctions

to be solved self-consistently. The one-dimensional, time independent

2*m*

1

*<sup>H</sup>* ¼ � <sup>ℏ</sup><sup>2</sup> 2 *δ δx*

*m x*ð Þ <sup>þ</sup> *V x*ð Þ

*Hψ*ð Þ¼ *x Eψ*ð Þ *x* ,

1

order ordinary differential equation, which is given by [14].

*∂*2 *ψ*ð Þ *x ∂x*<sup>2</sup> þ

� ℏ2 2 *δ δx*

where

**200**

5. reduced short-channel effects and an excellent latchup immunity.

2. steep subthreshold slope,

*Quantum Mechanics*

3.negligible floating body effects,

4.increased channel mobility,

*Schematics of conduction band structure of silicon in <100> oriented narrow channels [assumed (100) plane for the Si-SiO2 interface]. The energy levels are also shown in the silicon quantum well.*

where *ψ* is the wave function, *E* is the energy eigenvalue, *V x*ð Þ is the potential energy assumed to be independent of time, ℏ is Planck's constant divided by 2π, *m* � *m x*ð Þ is the effective mass of an electron which is a position dependent, *H* represents the Hamiltonian operator associated with the sum of the kinetic and potential energies of the system.

The potential distribution *ϕ*ð Þ *x* in the semiconductor can be determined from a solution of the 1 D Poisson Eq. (26), which is given by

$$\frac{d}{d(\infty)} \left( \epsilon\_S(\infty) \frac{d}{d(\infty)} \right) \phi(\infty) = \frac{-\rho(\infty)}{\varepsilon\_0} \tag{29}$$

The charge density *ρ* is given by

$$
\rho = N\_D(\mathbf{x}) - N\_A(\mathbf{x}) + p(\mathbf{x}) - n(\mathbf{x}).\tag{30}
$$

Eq. (25) can be written as

$$\frac{d}{d(\infty)}\left(\varepsilon\_{\rm S}(\infty)\frac{d}{d(\infty)}\right)\phi(\infty) = \frac{-q[N\_D(\infty) - N\_A(\infty) + p(\infty) - n(\infty)]}{\varepsilon\_0},\tag{31}$$

where *ϕ* is the electrostatic potential, *ε<sup>S</sup>* is the semiconductor dielectric constant, *ε*<sup>0</sup> is the permittivity of free space. In the static behaviour, *ND* and *NA* are called the ionised donor and acceptor concentrations and, in the case of dynamic behaviour, *n* and *p* are known as electron and hole density distributions. When dealing with *n*type majority carriers semiconductor devices, we can ignore the holes contribution due to their slow movement compared to the electron dynamics. Then only the electrons and donors are considered. As a result the above Eq. (31) can be written as

$$\frac{d}{d(\infty)} \left( \varepsilon\_S(\infty) \frac{d}{d(\infty)} \right) \phi(\infty) = \frac{-q[N\_D(\infty) - n(\infty)]}{\varepsilon\_0} \tag{32}$$

The potential energy *V* in the Hamiltonian is related to the electrostatic potential *ϕ* as follows [15]:

$$V(\mathbf{x}) = q\phi(\mathbf{x}) + \Delta E\_C(\mathbf{x}),\tag{33}$$

where *ΔEC* is the pseudopotential energy due to the band offset at the heterointerface. The wavefunction *ψ*ð Þ *x* in Eq. (27) and the electron density *n x*ð Þ in Eq. (32) are related by

$$m(\mathbf{x}) = \sum\_{k=1}^{M} \boldsymbol{\nu}\_k^\*(\mathbf{x}) \boldsymbol{\nu}\_k(\mathbf{x}) \boldsymbol{n}\_k,\tag{34}$$

where the summation runs over all the subbands, *M* is the number of bound states, and *nk* is the electron occupation for each state. The electron occupation of a state *k* is given by the Fermi-Dirac distribution:

$$m\_k = \frac{m}{\pi\hbar^2} \int\_{E\_k}^{\infty} \frac{1}{1 + \exp\left(E - E\_F/k\_B T\right)} dE,\tag{35}$$

mobile holes into the substrate. Therefore, the semiconductor is depleted of mobile carriers and a negative charge occurs at the interface because the fixed ionised

*Ideal metal-oxide-semiconductor (MOS) structure with p-type silicon substrate in a flat band condition.*

*Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum…*

We will investigate a Si MOS structure at a cross-section in the middle of a gate of the 25 nm gate length Si MOSFET. The structure shown in **Figure 8(a)** has a p-type silicon substrate, oxynitride (ON) gate oxide with a thickness of 1.6 nm, a

The conduction band profile in a MOS structure of bulk silicon, biased at gate voltage of VG = 1.0 V is shown in **Figure 8(b)**. The ground state energy rises from the conduction band edge as shown in **Figure 8(b)**. This phenomenon is called surface quantization by applied higher gate voltage. The surface quantization is often expressed by a triangular well approximation and the potential near the interface has almost a triangular shape because the potential barrier of SiO2 is relatively high in silicon MOS structure [13]. **Figure 8(b)** shows also the classically calculated electron density which will peak at the interface and predicts a much larger electron density and higher energy level when compared to the lower gate voltage [14, 16]. The quantum-mechanically calculated electron density is smaller and a displacement of the charge from the interface occurs when compared to the

The investigated 32 nm gate length silicon-on-insulator (SOI) MOSFET is grown on a silicon (Si) substrate. The SOI structure has a layer of silicon dioxide (SiO2)

*(a) A schematic metal-oxide-semiconductor (MOS) structure for the 25 nm gate length MOSFET with p-type silicon substrate, (b) conduction band, electron density (classical and quantum-mechanical), energy level, Fermi energy level and wavefunction under an applied bias of VG = 1.0 V across the channel for the MOS*

acceptor atoms are in fixed positions [16].

*DOI: http://dx.doi.org/10.5772/intechopen.93258*

dielectric constant of *εON* = 7 and a metal gate.

*2.4.2 Silicon-on-insulator (SOI) MOS structure*

*structure of the 25 nm gate length Si MOSFET, where T = 300 K.*

classical calculation.

**Figure 8.**

**203**

**Figure 7.**

where *Ek* is the eigenenergy, is the *EF* Fermi energy, and *kBT* is the thermal energy. An iteration procedure is employed to obtain self-consistent solutions for Eqs. (27) and (32). Starting with a trial potential ð Þ *x* , the wave functions, and their corresponding eigenenergies, *Ek* are used to calculate the electron density distribution *nx* using Eqs. (34) and (35).

#### **2.4 Simulation results: wavefunctions behaviour of particles in the semiconductor devices: 1D Poisson-Schrödinger solver**

The previous method of solving the Schrödinger-Poisson equations (see Section 2.3) has been applied to calculate the conduction band profile, electron concentration, energy levels (eigenstates) and wavefunctions (eigenfunctions) in a cross-section placed in middle of the gate of Metal-Oxide-Semiconductor (MOS) structure.

#### *2.4.1 Si MOS structure*

The MOS structure consists of a Metal-Oxide-Semiconductor capacitor, which is in the heart of the MOSFET. **Figure 7** shows the ideal MOS structure for p-type silicon in the flat band condition. The MOS structure is called the flat-band condition if the two following conditions are met [16]:


The energy bands in the semiconductor near the oxide-semiconductor interface bend as a voltage is applied across the MOS capacitor [13]. We will assume three different bias voltages. One is below the threshold voltage, VT, the second is just above the threshold voltage, and third one is at an on-current condition. The threshold voltage is defined as the applied gate voltage required to create the inversion layer charge and is one of the important parameters of MOSFETs. For enhancement mode, n-type MOS structure, the accumulation is for VG < 0, the depletion for VT > VG > 0, the inversion for VG � VT and the strong inversion for VG >> 0 [14].

An accumulation layer of holes occurs in the oxide-semiconductor junction typically for negative voltages when the negative charge on the gate attracts holes from the substrate to the oxide-semiconductor interface. The induced space charge region is created for positive voltages. The positive charge on the gate pushes the

*Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum… DOI: http://dx.doi.org/10.5772/intechopen.93258*

#### **Figure 7.**

*n x*ð Þ¼ <sup>X</sup> *M*

∞ð

*Ek*

**2.4 Simulation results: wavefunctions behaviour of particles in the semiconductor devices: 1D Poisson-Schrödinger solver**

state *k* is given by the Fermi-Dirac distribution:

tion if the two following conditions are met [16]:

tion *nx* using Eqs. (34) and (35).

*Quantum Mechanics*

*2.4.1 Si MOS structure*

the Si-SiO2 interface.

VG >> 0 [14].

**202**

*nk* <sup>¼</sup> *<sup>m</sup> π*ℏ<sup>2</sup> *k*¼**1**

*ψ* <sup>∗</sup>

where the summation runs over all the subbands, *M* is the number of bound states, and *nk* is the electron occupation for each state. The electron occupation of a

where *Ek* is the eigenenergy, is the *EF* Fermi energy, and *kBT* is the thermal energy. An iteration procedure is employed to obtain self-consistent solutions for Eqs. (27) and (32). Starting with a trial potential ð Þ *x* , the wave functions, and their corresponding eigenenergies, *Ek* are used to calculate the electron density distribu-

The previous method of solving the Schrödinger-Poisson equations (see Section 2.3) has been applied to calculate the conduction band profile, electron concentration, energy levels (eigenstates) and wavefunctions (eigenfunctions) in a cross-section placed in middle of the gate of Metal-Oxide-Semiconductor (MOS) structure.

The MOS structure consists of a Metal-Oxide-Semiconductor capacitor, which is in the heart of the MOSFET. **Figure 7** shows the ideal MOS structure for p-type silicon in the flat band condition. The MOS structure is called the flat-band condi-

1.The work function of metal and silicon are equal, which implies that in all the materials, all energy levels in both the silicon and oxide are flat. When there is no applied voltage between the metal and silicon, their Fermi levels line up.

2.There exists no charge, the electric field is zero everywhere in the oxide and at

The energy bands in the semiconductor near the oxide-semiconductor interface bend as a voltage is applied across the MOS capacitor [13]. We will assume three different bias voltages. One is below the threshold voltage, VT, the second is just above the threshold voltage, and third one is at an on-current condition. The threshold voltage is defined as the applied gate voltage required to create the inversion layer charge and is one of the important parameters of MOSFETs. For enhancement mode, n-type MOS structure, the accumulation is for VG < 0, the depletion for VT > VG > 0, the inversion for VG � VT and the strong inversion for

An accumulation layer of holes occurs in the oxide-semiconductor junction typically for negative voltages when the negative charge on the gate attracts holes from the substrate to the oxide-semiconductor interface. The induced space charge region is created for positive voltages. The positive charge on the gate pushes the

1

*<sup>k</sup>* ð Þ *x ψk*ð Þ *x nk*, (34)

<sup>1</sup> <sup>þ</sup> *exp E*ð Þ � *EF=kBT dE*, (35)

*Ideal metal-oxide-semiconductor (MOS) structure with p-type silicon substrate in a flat band condition.*

mobile holes into the substrate. Therefore, the semiconductor is depleted of mobile carriers and a negative charge occurs at the interface because the fixed ionised acceptor atoms are in fixed positions [16].

We will investigate a Si MOS structure at a cross-section in the middle of a gate of the 25 nm gate length Si MOSFET. The structure shown in **Figure 8(a)** has a p-type silicon substrate, oxynitride (ON) gate oxide with a thickness of 1.6 nm, a dielectric constant of *εON* = 7 and a metal gate.

The conduction band profile in a MOS structure of bulk silicon, biased at gate voltage of VG = 1.0 V is shown in **Figure 8(b)**. The ground state energy rises from the conduction band edge as shown in **Figure 8(b)**. This phenomenon is called surface quantization by applied higher gate voltage. The surface quantization is often expressed by a triangular well approximation and the potential near the interface has almost a triangular shape because the potential barrier of SiO2 is relatively high in silicon MOS structure [13]. **Figure 8(b)** shows also the classically calculated electron density which will peak at the interface and predicts a much larger electron density and higher energy level when compared to the lower gate voltage [14, 16]. The quantum-mechanically calculated electron density is smaller and a displacement of the charge from the interface occurs when compared to the classical calculation.

#### *2.4.2 Silicon-on-insulator (SOI) MOS structure*

The investigated 32 nm gate length silicon-on-insulator (SOI) MOSFET is grown on a silicon (Si) substrate. The SOI structure has a layer of silicon dioxide (SiO2)

#### **Figure 8.**

*(a) A schematic metal-oxide-semiconductor (MOS) structure for the 25 nm gate length MOSFET with p-type silicon substrate, (b) conduction band, electron density (classical and quantum-mechanical), energy level, Fermi energy level and wavefunction under an applied bias of VG = 1.0 V across the channel for the MOS structure of the 25 nm gate length Si MOSFET, where T = 300 K.*

with a thickness of 20 nm, which is called buried oxide (BOX) and is fabricated on a Si substrate, and a silicon body (which creates a device channel) with a thickness of 8 nm. A Hafnium Oxide (HfO2) layer is deposited above the silicon body as a gate oxide with a thickness of 1.19 nm and a dielectric constant of *εHf O*<sup>2</sup> ¼ 20 and a top metal contact referred to as a gate as shown in **Figure 9(a)**. The metal gate will be able to bend the semiconductor bands with the application of a gate potential [13].

We have used, again, the 1D self-consistent solution of the Poisson-Schrödinger equation to obtain the conduction band profile, energy levels, wavefunctions and

. The Al0.3Ga0.7As layer at

The III-V MOSFET consists of In0.3Ga0.7As channel with thickness of 5 nm, high-*ε* dielectric layer of Gadolinium Gallium Oxide (GdGaO) as a gate dielectric with a thickness of 1.5 nm and whose dielectric constant is *εGGO* ¼ 20. The In0.3Ga0.7As channel is located between an Al0.3Ga0.7As layer with a thickness of 1.5 nm and an Al0.3Ga0.7As layer of a thickness of 3 nm. The δ-doping layer is placed

*Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum…*

the bottom of the structure is grown as a thick buffer layer of 50 nm as shown in

**Figure 10(b)** shows the conduction band, five discrete energy levels and electron concentration (classical and quantum mechanical) across the channel for the

At VG = 1.0 V, we obtain three discrete energy levels in the quantum well with corresponding wavefunction for these energy levels shown in **Figures 11(a)** in SOI MOSFET. We summarise that in future technology the bulk MOSFET will be replaced by an ultra-thin-body (UTB) silicon-on-insulator (SOI) on the basis of better electrostatistic integrity, low channel doping to get high mobility, high

*(a) Cross-section of the 15 nm gate length In0.3Ga0.7As channel MOS structure with a high- ε dielectric layer which is located below the metal gate, and (b) conduction band, electron density and discrete energy levels under an applied bias of VG = 1.0 V, across the channel for a MOS structure of the 15 nm gate length*

*(a) Electron wave functions under an applied bias of VG = 1.0 V across the channel for a MOS structure of the 32 nm gate length SOI MOSFET. (b) The wavefunctions under an applied bias of VG = 1.0 V, across the*

*channel for a MOS structure of the 15 nm gate length In0.3Ga0.7As MOSFET.*

electron density in a confined body of this heterostructure MOSFET.

**Figure 10(a)**. The whole device is grown on a GaAs substrate [13, 14].

below the channel with a concentration of 7 � <sup>10</sup><sup>12</sup> cm�<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.93258*

15 nm gate length In0.3Ga0.7As MOSFET biased at VG = 1.0 V.

dielectric material to prevent gate leakage and metal gate [17–19].

**Figure 10.**

**Figure 11.**

**205**

*In0.3Ga0.7As MOSFET.*

We have investigated the specified 32 nm gate length SOI MOSFET using, again, a self-consistent solution of 1D Schrödinger and Poisson equations. **Figure 9(b)** shows the electron conduction band and density profiles, which are obtained along a slice taken through the middle of a SOI MOS structure, from the surface to the substrate biased at VG = 1.0 V at room temperature. In this structure, the potential energy creates a square quantum well, because the potential difference between the front interface and the back interface is small and the potential barriers are very high. Electrons are therefore confined in the ultra-thin Si body, which is sandwiched between the gate oxide and the BOX. The electron energy in the perpendicular direction is quantized and the energy of the ground state rises [14, 16] when compared to the conduction band. We can find two discrete energy levels in the quantum well. The classically calculated electron density will again peak at the interface of oxide and semiconductor. The quantum-mechanically calculated electron density will peak away from the oxide-semiconductor interface due to displacement of the charge from the interface [13].

#### *2.4.3 MOS structure for an InGaAs channel transistor*

We have selected an In0.3Ga0.7As channel because of its optimal electron mobility and low effective mass. We investigate the effect of a confined channel in the implant free (IF) In0.3Ga0.7As channel MOSFET with a gate length of 15 nm aimed for the future sub-22 nm Si technology. The IF MOSFET is derived from a HEMT structure which has

1.an oxide layer to prevent gate tunnelling,

2.a δ-doping layer placed below the channel. This placement allows the metal gate to maintain a good control of carrier transport in the channel, and

3.an ultra-thin body channel to the heterostructure used in a transistor design.

#### **Figure 9.**

*(a) A schematic for silicon on insulator (SOI) structure with the 32 nm gate length. (b) Electron density (classical and quantum-mechanical) distribution, conduction band and energy levels under an applied bias of VG = 1.0 V.*

*Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum… DOI: http://dx.doi.org/10.5772/intechopen.93258*

We have used, again, the 1D self-consistent solution of the Poisson-Schrödinger equation to obtain the conduction band profile, energy levels, wavefunctions and electron density in a confined body of this heterostructure MOSFET.

The III-V MOSFET consists of In0.3Ga0.7As channel with thickness of 5 nm, high-*ε* dielectric layer of Gadolinium Gallium Oxide (GdGaO) as a gate dielectric with a thickness of 1.5 nm and whose dielectric constant is *εGGO* ¼ 20. The In0.3Ga0.7As channel is located between an Al0.3Ga0.7As layer with a thickness of 1.5 nm and an Al0.3Ga0.7As layer of a thickness of 3 nm. The δ-doping layer is placed below the channel with a concentration of 7 � <sup>10</sup><sup>12</sup> cm�<sup>2</sup> . The Al0.3Ga0.7As layer at the bottom of the structure is grown as a thick buffer layer of 50 nm as shown in **Figure 10(a)**. The whole device is grown on a GaAs substrate [13, 14].

**Figure 10(b)** shows the conduction band, five discrete energy levels and electron concentration (classical and quantum mechanical) across the channel for the 15 nm gate length In0.3Ga0.7As MOSFET biased at VG = 1.0 V.

At VG = 1.0 V, we obtain three discrete energy levels in the quantum well with corresponding wavefunction for these energy levels shown in **Figures 11(a)** in SOI MOSFET. We summarise that in future technology the bulk MOSFET will be replaced by an ultra-thin-body (UTB) silicon-on-insulator (SOI) on the basis of better electrostatistic integrity, low channel doping to get high mobility, high dielectric material to prevent gate leakage and metal gate [17–19].

#### **Figure 10.**

with a thickness of 20 nm, which is called buried oxide (BOX) and is fabricated on a Si substrate, and a silicon body (which creates a device channel) with a thickness of 8 nm. A Hafnium Oxide (HfO2) layer is deposited above the silicon body as a gate oxide with a thickness of 1.19 nm and a dielectric constant of *εHf O*<sup>2</sup> ¼ 20 and a top metal contact referred to as a gate as shown in **Figure 9(a)**. The metal gate will be able to bend the semiconductor bands with the application of a gate potential [13]. We have investigated the specified 32 nm gate length SOI MOSFET using, again, a self-consistent solution of 1D Schrödinger and Poisson equations. **Figure 9(b)** shows the electron conduction band and density profiles, which are obtained along a slice taken through the middle of a SOI MOS structure, from the surface to the substrate biased at VG = 1.0 V at room temperature. In this structure, the potential energy creates a square quantum well, because the potential difference between the front interface and the back interface is small and the potential barriers are very

high. Electrons are therefore confined in the ultra-thin Si body, which is sandwiched between the gate oxide and the BOX. The electron energy in the perpendicular direction is quantized and the energy of the ground state rises [14, 16] when compared to the conduction band. We can find two discrete energy levels in the quantum well. The classically calculated electron density will again peak at the interface of oxide and semiconductor. The quantum-mechanically calculated electron density will peak away from the oxide-semiconductor interface due

We have selected an In0.3Ga0.7As channel because of its optimal electron mobility and low effective mass. We investigate the effect of a confined channel in the implant free (IF) In0.3Ga0.7As channel MOSFET with a gate length of 15 nm aimed for the future sub-22 nm Si technology. The IF MOSFET is derived from a HEMT

2.a δ-doping layer placed below the channel. This placement allows the metal gate to maintain a good control of carrier transport in the channel, and

3.an ultra-thin body channel to the heterostructure used in a transistor design.

*(a) A schematic for silicon on insulator (SOI) structure with the 32 nm gate length. (b) Electron density (classical and quantum-mechanical) distribution, conduction band and energy levels under an applied bias of*

to displacement of the charge from the interface [13].

*2.4.3 MOS structure for an InGaAs channel transistor*

1.an oxide layer to prevent gate tunnelling,

structure which has

*Quantum Mechanics*

**Figure 9.**

*VG = 1.0 V.*

**204**

*(a) Cross-section of the 15 nm gate length In0.3Ga0.7As channel MOS structure with a high- ε dielectric layer which is located below the metal gate, and (b) conduction band, electron density and discrete energy levels under an applied bias of VG = 1.0 V, across the channel for a MOS structure of the 15 nm gate length In0.3Ga0.7As MOSFET.*

#### **Figure 11.**

*(a) Electron wave functions under an applied bias of VG = 1.0 V across the channel for a MOS structure of the 32 nm gate length SOI MOSFET. (b) The wavefunctions under an applied bias of VG = 1.0 V, across the channel for a MOS structure of the 15 nm gate length In0.3Ga0.7As MOSFET.*

**Figure 10(b)** shows the conduction band and electron concentration for the 15 nm gate length In0.3Ga0.7As MOSFET biased at VG = 1.0 V. Five discrete energy levels can be observed at this high bias with the corresponding wave functions in the quantum well shown in **Figure 11(b)** [20].

### **3. Conclusions**

We have been carried out using a self-consistent solution of 1D Poisson-Schrödinger equation to determine conduction band profiles, electron density, energy levels (eigenstates) and wavefunctions (eigenfunctions) in the Si, SOI and InGaAs MOS structures under external potential. We have afterwards simulated the electron sheet density as a function of the applied gate bias and made a comparison among the three device structures, the 25 nm gate length bulk Si, 32 nm UTB SOI, and 15 nm gate length InGaAs MOSFETs.

We have investigated the effect of electron confinement in nanoscaled transistor channels using 1D simulation through cross-sections of the devices. These investigations have been carried out using a self-consistent solution of 1D Poisson-Schrödinger equation to determine conduction band profiles, electron density, energy levels (eigenstates) and wavefunctions (eigenfunctions) in the Si, SOI and In0.3Ga0.7As MOS structures under external potential. We have afterwards simulated the electron sheet density as a function of the applied gate bias and made a comparison among the three device structures, the 25 nm gate length bulk Si MOSFET, the 32 nm UTB SOI Si MOSFET, and the 15 nm gate length IF In0.3Ga0.7As MOSFET [20].

I can envisage that my future work could be related to the investigations of the new physical phenomena present in the UTB MOSFET architectures. As explained previously, the planar and non-planar UTB device architectures are preferred solutions for future technology nodes because the conventional bulk MOSFETs suffer from a poor electrostatic behaviour when scaled to sub-22 nm gate lengths exhibiting unsatisfactory short channel effects. These short channel effects can be summarised as follows:

1. reduced carrier mobility at high channel doping, hampering the device performance,

**Author details**

London, United Kingdom

**207**

Aynul Islam1,2 and Anika Tasnim Aynul<sup>3</sup>

provided the original work is properly cited.

1 Bangor College, Bangor University, United Kingdom

\*Address all correspondence to: anika\_tasnim@live.co.uk

2 Central South University Forestry and Technology, Hunan, China

\*

*Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum…*

*DOI: http://dx.doi.org/10.5772/intechopen.93258*

3 Department of Physics and Astronomy, University College of London (UCL),

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,


The UTB MOSFET architectures do not show such severe short channel effects because they have superior electrostatic integrity thanks to the electron confinement of their channel region.

*Analysis of Quantum Confinement and Carrier Transport of Nano-Transistor in Quantum… DOI: http://dx.doi.org/10.5772/intechopen.93258*

### **Author details**

**Figure 10(b)** shows the conduction band and electron concentration for the 15 nm gate length In0.3Ga0.7As MOSFET biased at VG = 1.0 V. Five discrete energy levels can be observed at this high bias with the corresponding wave functions in the

We have been carried out using a self-consistent solution of 1D Poisson-Schrödinger equation to determine conduction band profiles, electron density, energy levels (eigenstates) and wavefunctions (eigenfunctions) in the Si, SOI and InGaAs MOS structures under external potential. We have afterwards simulated the electron sheet density as a function of the applied gate bias and made a comparison among the three device structures, the 25 nm gate length bulk Si, 32 nm UTB SOI,

We have investigated the effect of electron confinement in nanoscaled transistor channels using 1D simulation through cross-sections of the devices. These investigations have been carried out using a self-consistent solution of 1D Poisson-Schrödinger equation to determine conduction band profiles, electron density, energy levels (eigenstates) and wavefunctions (eigenfunctions) in the Si, SOI and In0.3Ga0.7As MOS structures under external potential. We have afterwards simulated the electron sheet density as a function of the applied gate bias and made a comparison among the three device structures, the 25 nm gate length bulk Si MOSFET, the 32 nm UTB SOI Si MOSFET, and the 15 nm gate length IF

I can envisage that my future work could be related to the investigations of the new physical phenomena present in the UTB MOSFET architectures. As explained previously, the planar and non-planar UTB device architectures are preferred solutions for future technology nodes because the conventional bulk MOSFETs suffer from a poor electrostatic behaviour when scaled to sub-22 nm gate lengths exhibiting unsatisfactory short channel effects. These short channel effects can be

1. reduced carrier mobility at high channel doping, hampering the device

3.and high gate tunnelling current and poor electrostatic control despite

The UTB MOSFET architectures do not show such severe short channel effects because they have superior electrostatic integrity thanks to the electron confine-

quantum well shown in **Figure 11(b)** [20].

and 15 nm gate length InGaAs MOSFETs.

In0.3Ga0.7As MOSFET [20].

summarised as follows:

performance,

ment of their channel region.

**206**

2.band-to-band drain leakage current [19],

employment of metal/high-o gate stacks, etc. [1, 3]. ˛

**3. Conclusions**

*Quantum Mechanics*

Aynul Islam1,2 and Anika Tasnim Aynul<sup>3</sup> \*

1 Bangor College, Bangor University, United Kingdom

2 Central South University Forestry and Technology, Hunan, China

3 Department of Physics and Astronomy, University College of London (UCL), London, United Kingdom

\*Address all correspondence to: anika\_tasnim@live.co.uk

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Donald Neamen A. Semiconductor Physics and Devices, Chapter 2. 3rd ed. New Delhi/New York: University of New Mexico; 2007

[2] Rae AM, Napolitano J. Quantum Mechanics, Chapter 3 and 5. 6th ed. London/New York: CRC Press/Taylor and Francis Group; 1986

[3] Ridley BK. Quantum Processes in Semiconductors. London: Oxford; 1982

[4] Frank DJ, Dennard R, Nowak E, Solomon P, Taur Y, Wong H-S. Proceedings of the IEEE. 2001;**89**: 259-288

[5] Winstead B, Ravaioli U. IEEE Transactions on Electron Devices. 2003; **50**(2):440-446

[6] Lundstrom M. Fundamentals of Carrier Transport. 2nd ed. Cambridge, UK: Cambridge University Press; 2000

[7] Kazutaka T. Numerical Simulation of Submicron Semiconductor Devices, Chapter 2. New York: Artech House, Inc; 1993. p. 102

[8] Jacoboni C, Lugli P. The Monte Carlo Method for Semiconductor Device Simulation. Vienna, Austria: Springer-Verlag; 1989. p. 114

[9] Griffiths David J. Introduction to Quantum Mechanics. 2nd ed. Edinburgh Gate, Harlow: Prentice Hall; 2004. ISBN 978-0-13-111892-8

[10] Physicist Erwin Schrödinger's Google doodle marks quantum mechanics work*.* The Guardian. 13 August 2013 [Accessed: 25 August 2013]

[11] Schrödinger E. An undulatory theory of the mechanics of atoms and molecules. Physical Review. 1926;**28**(6): 10491070

[12] Laloe F. Do We Really Understand Quantum Mechanics. New York: Cambridge University Press; 2012. ISBN: 978-1-107-02501-1

**Chapter 11**

*Aghaddin Mamedov*

**Abstract**

**1. Introduction**

**209**

Development of Supersymmetric

Background/Local Gauge Field

Coupling of Electromagnetism

with the Nucleon's Background

Space-Time Frame: The Physics

A new reformulated gauge field theory comprising discrete super symmetry matrixes **U (1) = SU (2) + SO (3)** has been developed which explains why all the elementary particles appear in three families with very similar structures. The three families' performance is the product of discrete conservation of energy—momentum eigenvalue **Es = 1/2Ea** within space–time frame which appears to be the genetic code of new physics. A new supersymmetric gauge field theory of photon was developed, which describes fundamental conservation laws through invariant translation of the discrete symmetries of nature. A new gauge theory describes all the fundamental laws through isomorphism of the discrete space–time **SU (2)** frame and energy-momentum **SO (3)** symmetry group. Coupling of space and time phases of energy conservation generates the background gauge field, which in conjugation with the local gauge field mediates discrete performance of three fractional proton-neutron families of baryon structure. The presented theory requires to have a new look to our understanding of symmetry and conservation laws.

**Keywords:** supersymmetric theory, discrete double gauge field, discrete space–time symmetry, strong interactions, matter–antimatter symmetry

From the beginning, I would like to show that there is no matter–antimatter asymmetry in nature and the matter–antimatter asymmetry would eliminate existence of our world in cyclic mode, moving to the randomness. Matter–antimatter, holding discrete supersymmetric genetic code **2Es = Ea**, appear in different phases of energy conservation with the change of frequency. The phenomenon called

Theory of Nucleon Based on

beyond the Standard Model

[13] Aynul I, Kalna K. Nano-Transistor Scaling and their Characteristics Using Monte Carlo. Moldova, UK: LAP LAMBERT Academic Publishing; 2018. p. 60. ISBN-13: 978-613-9-94747-8. Available from: https://www.lap-pub lishing.com/

[14] Aynul I, Kalna K. Analysis of electron transport in the nano-scaled Si, SOI and III -V MOSFETs: Si/SiO2 interface charges and quantum mechanical effects. In: IOP Conf. Series: Materials Science and Engineering. Vol. 504. UK: IOP Publishing; 2019. p. 012021. DOI: 10.1088/1757-899X/ 504/1/012021

[15] Shankar R. Principles of Quantum Mechanics. 2nd ed. New York/London: Kluwer Academic/Plenum Publishers; 1943. ISBN: 978-0-306-44790-7

[16] Aynul I. "Monte Carlo Device Modelling of Electron Transport in Nanoscale Transistors" Doctor of Philosophy. Wales, United Kingdom: College of Engineering, Swansea University Swansea SA2 8PP; 2012

[17] Jacoboni C, Lugli P. The Monte Carlo Method for Semiconductor Device Simulation. Vienna, Austria: Springer-Verlag; 1989. p. 90

[18] Oda S, Ferry DK. Silicon Nanoelectronics. London/New York: Technology and Engineering; 2006. pp. 89-95

[19] Frank DJ, Laux SE, Fischetti MV. Monte Carlo simulation of a 30 nm dualgate MOSFET: How short can Si go. In: Technical Digest - International Electron Devices Meet; 1992. pp. 553-556

[20] Sekigawa T, Hayashi Y. Solid-State Electronics. 1984;**27**:827-828

#### **Chapter 11**

**References**

*Quantum Mechanics*

1982

259-288

**50**(2):440-446

Inc; 1993. p. 102

Verlag; 1989. p. 114

978-0-13-111892-8

10491070

**208**

New Mexico; 2007

and Francis Group; 1986

[1] Donald Neamen A. Semiconductor Physics and Devices, Chapter 2. 3rd ed. New Delhi/New York: University of

[12] Laloe F. Do We Really Understand Quantum Mechanics. New York: Cambridge University Press; 2012.

[13] Aynul I, Kalna K. Nano-Transistor Scaling and their Characteristics Using Monte Carlo. Moldova, UK: LAP LAMBERT Academic Publishing; 2018. p. 60. ISBN-13: 978-613-9-94747-8. Available from: https://www.lap-pub

[14] Aynul I, Kalna K. Analysis of electron transport in the nano-scaled Si, SOI and III -V MOSFETs: Si/SiO2 interface charges and quantum

mechanical effects. In: IOP Conf. Series: Materials Science and Engineering. Vol. 504. UK: IOP Publishing; 2019. p. 012021. DOI: 10.1088/1757-899X/

[15] Shankar R. Principles of Quantum Mechanics. 2nd ed. New York/London: Kluwer Academic/Plenum Publishers; 1943. ISBN: 978-0-306-44790-7

[16] Aynul I. "Monte Carlo Device Modelling of Electron Transport in Nanoscale Transistors" Doctor of Philosophy. Wales, United Kingdom: College of Engineering, Swansea University Swansea SA2 8PP; 2012

[17] Jacoboni C, Lugli P. The Monte Carlo Method for Semiconductor Device Simulation. Vienna, Austria: Springer-

[18] Oda S, Ferry DK. Silicon Nanoelectronics. London/New York: Technology and Engineering; 2006. pp. 89-95

[19] Frank DJ, Laux SE, Fischetti MV. Monte Carlo simulation of a 30 nm dualgate MOSFET: How short can Si go. In: Technical Digest - International Electron

[20] Sekigawa T, Hayashi Y. Solid-State

Devices Meet; 1992. pp. 553-556

Electronics. 1984;**27**:827-828

Verlag; 1989. p. 90

ISBN: 978-1-107-02501-1

lishing.com/

504/1/012021

[2] Rae AM, Napolitano J. Quantum Mechanics, Chapter 3 and 5. 6th ed. London/New York: CRC Press/Taylor

[3] Ridley BK. Quantum Processes in Semiconductors. London: Oxford;

[4] Frank DJ, Dennard R, Nowak E, Solomon P, Taur Y, Wong H-S. Proceedings of the IEEE. 2001;**89**:

[5] Winstead B, Ravaioli U. IEEE

[6] Lundstrom M. Fundamentals of Carrier Transport. 2nd ed. Cambridge, UK: Cambridge University Press; 2000

Transactions on Electron Devices. 2003;

[7] Kazutaka T. Numerical Simulation of Submicron Semiconductor Devices, Chapter 2. New York: Artech House,

[8] Jacoboni C, Lugli P. The Monte Carlo Method for Semiconductor Device Simulation. Vienna, Austria: Springer-

[9] Griffiths David J. Introduction to Quantum Mechanics. 2nd ed. Edinburgh Gate, Harlow: Prentice Hall; 2004. ISBN

[10] Physicist Erwin Schrödinger's Google doodle marks quantum mechanics work*.* The Guardian. 13 August 2013 [Accessed: 25 August 2013]

[11] Schrödinger E. An undulatory theory of the mechanics of atoms and molecules. Physical Review. 1926;**28**(6): Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based on Coupling of Electromagnetism with the Nucleon's Background Space-Time Frame: The Physics beyond the Standard Model

*Aghaddin Mamedov*

#### **Abstract**

A new reformulated gauge field theory comprising discrete super symmetry matrixes **U (1) = SU (2) + SO (3)** has been developed which explains why all the elementary particles appear in three families with very similar structures. The three families' performance is the product of discrete conservation of energy—momentum eigenvalue **Es = 1/2Ea** within space–time frame which appears to be the genetic code of new physics. A new supersymmetric gauge field theory of photon was developed, which describes fundamental conservation laws through invariant translation of the discrete symmetries of nature. A new gauge theory describes all the fundamental laws through isomorphism of the discrete space–time **SU (2)** frame and energy-momentum **SO (3)** symmetry group. Coupling of space and time phases of energy conservation generates the background gauge field, which in conjugation with the local gauge field mediates discrete performance of three fractional proton-neutron families of baryon structure. The presented theory requires to have a new look to our understanding of symmetry and conservation laws.

**Keywords:** supersymmetric theory, discrete double gauge field, discrete space–time symmetry, strong interactions, matter–antimatter symmetry

#### **1. Introduction**

From the beginning, I would like to show that there is no matter–antimatter asymmetry in nature and the matter–antimatter asymmetry would eliminate existence of our world in cyclic mode, moving to the randomness. Matter–antimatter, holding discrete supersymmetric genetic code **2Es = Ea**, appear in different phases of energy conservation with the change of frequency. The phenomenon called

symmetry breaking is the discrete supersymmetric invariant translation of background gauge symmetry to the local gauge symmetry with the invariant inverse.

intervals and linearity in differential equations might be the reason for appearance of uncertainty problems of quantum mechanics and infinity in classic physics formulations. Similarly Nobel Laureate Hoof't showed [8] that "when we send distances and time intervals to zero we do assume that the philosophy of differential

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based…*

In the present paper, we will discuss a new reformulated gauge theory, combin-

ing "twin brothers" of background/local gauge fields, conjugated with the new energy-momentum rotational symmetry group and associated with the discrete space–time symmetry. We will describe fundamental change of physical laws when we shift from the theories based on continuous energy conservation law to the discrete energy-momentum invariant translation, carried within the discrete space– time frame. Simply, we will present an invariant action-response exchange parity, relative to the response, which drastically changes features of classic, relativistic, and quantum theories. *The paper will describe why all the elementary particles come in three families with very similar structure. The three families' performance is the product of energy conservation within energy-momentum exchange interaction, which became the genetic code of our existence and new physics. We believe that our theory of supersymme-*

*try is the rebirth of gauge field theory, which does not need application of*

**2.1 The illness of linear differential equations for description of energy**

First, I would like to discuss shortly literature information [9] to show that Newton's traditional differential equations, describing change of an event in abstract space and time, do not provide exact solution. It describes change of an event in abstract time by a smooth continuous vector field, which has local

diffeomorphism, preserving only limited property of an event. Diffeomorphism is a function in smooth manifold, which describes differentiation where the original position is lost. The present mathematical knowledge does not provide any solution on how to eliminate the diffeomorphism problem of differentiation to save the initial property of a function. That is why all theories, describing symmetry, have to use renormalization, which drives them, similar to quantum uncertainty, to the

Presently there is no mathematical solution on how to get nonlinear differential equation, which may describe change of dynamical local position from point to point on the discrete space–time, where space and time may change their

dimensions relative to the energy-momentum flux to the space–time frame. This approach is very useful for a wide class of mathematical problems because it solves boundary value problems (which is in reality initial value problems) of differential

Traditional differentiation describes change of interval of one variable in relation to other, for example, change of space in relation of time—**dS/dt.** We suggest a new mathematical theory for differentiation of a function without loss of the origin/ local dynamical structure within multiple variables, such as space, time, and energy. Presentation of differentiation through coupling of intervals of change of multiple variables with their superposition origin produces deterministic outcome regardless

The theory, which we suggest, comprises the principle that any interaction, to hold symmetry, after change in the space–time frame, should look the same as its

**2. The genetic code of origin as the superposition**

equations works."

*DOI: http://dx.doi.org/10.5772/intechopen.93087*

*renormalization.*

**conservation**

local approximate symmetry.

equations.

**211**

of scale of interactions.

Nature does not distinguish a difference between the laws, describing different scale events, and selects very simple principle, which holds symmetry with the perfect conservation laws. The main problem of classic, relativistic, and quantum mechanics theories is the application of mathematical models, which describe a break of *continuous symmetry*, *associated with the continuous energy conservation*. Due to the consumption of energy in dynamical processes with the discrete energy portions, application of continuous functions, such as Lagrangian and Hamiltonian, for differentiation of change leads to the runaway of the energy solutions to infinity. Artificial renormalization of Lagrangian/Hamiltonian dynamical equations leads to the approximate symmetry; therefore we cannot use these linear differential equations to get correct fundamental laws of nature. Due to these problems, some authors, for example, Weinberg, suggest [1] that nature is approximately simple and Yang-Mill symmetry naturally should produce approximate symmetry. Weinberg sand Glashow suggested that [1, 2] nuclear interactions have spontaneous symmetry breaking that is why these interactions may produce only approximate symmetry.

The theories, describing spontaneous breaking of continuous symmetry, with application of renormalization approach, do not provide proper mathematics of energy conservation, associated with the symmetry. Presently there is no theory, which may describe conservation of action during "change" of an event at small space and time intervals. An action is the product of energy consumption, and due to the discrete consumption of energy, the outcome product of an action has to produce discrete action formulation. However, Lagrangian continuous action principle (Hamiltonian as well) does not hold this requirement.

Our opinion is that the fundamental laws of nature cannot appear in differential formulations in correct way, if these equations do not include dynamical superposition origin and describe interactions through change of continuous function. Without involvement of the original position to the differential equation and using the continuous function, we cannot conserve energy at the origin and remove renormalization groups, adding to the physical theories.

Feynman showed [3] that you could describe an event in the Hamiltonian in the form of differential equation, which describes how the function changes in term of operator. We may provide our comment that such a task is not realizable with the Hamiltonian, because it is a continuous function and does not involve dynamical initial position. The action is the discrete space–time function; therefore, the continuous outcome of a discrete action without relation to dynamic local position is uncertain.

Feynman [4] applied renormalized Lagrangian action to quantum mechanics but even in Feynman's renormalized Lagrangian action is not conserved. The main problem of Lagrangian action and Feynman approach is the application of linear continuous action-response relation.

Therefore, we need an entirely new theory to describe reversible fundamental laws of nature, combining discrete conservation of energy with the boundarymapped space–time, which would combine all kind of fields and forces within discrete background symmetry. Application of discrete energy conservation law and discrete symmetry as the product of this law may change drastically our present knowledge on the nature of forces and their roles in fundamental interactions. It is possible that nature and performance of forces, merging at the background symmetry, will be different, and reversibility of dynamical laws at discrete symmetry may change completely the role and classification of forces.

In our previous studies [5–7], we showed that wrong description of dynamical laws through simple continuous displacement in space–time structure using only

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based… DOI: http://dx.doi.org/10.5772/intechopen.93087*

intervals and linearity in differential equations might be the reason for appearance of uncertainty problems of quantum mechanics and infinity in classic physics formulations. Similarly Nobel Laureate Hoof't showed [8] that "when we send distances and time intervals to zero we do assume that the philosophy of differential equations works."

In the present paper, we will discuss a new reformulated gauge theory, combining "twin brothers" of background/local gauge fields, conjugated with the new energy-momentum rotational symmetry group and associated with the discrete space–time symmetry. We will describe fundamental change of physical laws when we shift from the theories based on continuous energy conservation law to the discrete energy-momentum invariant translation, carried within the discrete space– time frame. Simply, we will present an invariant action-response exchange parity, relative to the response, which drastically changes features of classic, relativistic, and quantum theories. *The paper will describe why all the elementary particles come in three families with very similar structure. The three families' performance is the product of energy conservation within energy-momentum exchange interaction, which became the genetic code of our existence and new physics. We believe that our theory of supersymmetry is the rebirth of gauge field theory, which does not need application of renormalization.*

#### **2. The genetic code of origin as the superposition**

#### **2.1 The illness of linear differential equations for description of energy conservation**

First, I would like to discuss shortly literature information [9] to show that Newton's traditional differential equations, describing change of an event in abstract space and time, do not provide exact solution. It describes change of an event in abstract time by a smooth continuous vector field, which has local diffeomorphism, preserving only limited property of an event. Diffeomorphism is a function in smooth manifold, which describes differentiation where the original position is lost. The present mathematical knowledge does not provide any solution on how to eliminate the diffeomorphism problem of differentiation to save the initial property of a function. That is why all theories, describing symmetry, have to use renormalization, which drives them, similar to quantum uncertainty, to the local approximate symmetry.

Presently there is no mathematical solution on how to get nonlinear differential equation, which may describe change of dynamical local position from point to point on the discrete space–time, where space and time may change their dimensions relative to the energy-momentum flux to the space–time frame. This approach is very useful for a wide class of mathematical problems because it solves boundary value problems (which is in reality initial value problems) of differential equations.

Traditional differentiation describes change of interval of one variable in relation to other, for example, change of space in relation of time—**dS/dt.** We suggest a new mathematical theory for differentiation of a function without loss of the origin/ local dynamical structure within multiple variables, such as space, time, and energy. Presentation of differentiation through coupling of intervals of change of multiple variables with their superposition origin produces deterministic outcome regardless of scale of interactions.

The theory, which we suggest, comprises the principle that any interaction, to hold symmetry, after change in the space–time frame, should look the same as its

symmetry breaking is the discrete supersymmetric invariant translation of background gauge symmetry to the local gauge symmetry with the invariant inverse. Nature does not distinguish a difference between the laws, describing different scale events, and selects very simple principle, which holds symmetry with the perfect conservation laws. The main problem of classic, relativistic, and quantum mechanics theories is the application of mathematical models, which describe a break of *continuous symmetry*, *associated with the continuous energy conservation*. Due to the consumption of energy in dynamical processes with the discrete energy portions, application of continuous functions, such as Lagrangian and Hamiltonian, for differentiation of change leads to the runaway of the energy solutions to infinity. Artificial

renormalization of Lagrangian/Hamiltonian dynamical equations leads to the

why these interactions may produce only approximate symmetry.

ciple (Hamiltonian as well) does not hold this requirement.

renormalization groups, adding to the physical theories.

change completely the role and classification of forces.

uncertain.

*Quantum Mechanics*

**210**

continuous action-response relation.

approximate symmetry; therefore we cannot use these linear differential equations to get correct fundamental laws of nature. Due to these problems, some authors, for example, Weinberg, suggest [1] that nature is approximately simple and Yang-Mill symmetry naturally should produce approximate symmetry. Weinberg sand Glashow suggested that [1, 2] nuclear interactions have spontaneous symmetry breaking that is

The theories, describing spontaneous breaking of continuous symmetry, with application of renormalization approach, do not provide proper mathematics of energy conservation, associated with the symmetry. Presently there is no theory, which may describe conservation of action during "change" of an event at small space and time intervals. An action is the product of energy consumption, and due to the discrete consumption of energy, the outcome product of an action has to produce discrete action formulation. However, Lagrangian continuous action prin-

Our opinion is that the fundamental laws of nature cannot appear in differential formulations in correct way, if these equations do not include dynamical superposition origin and describe interactions through change of continuous function. Without involvement of the original position to the differential equation and using the continuous function, we cannot conserve energy at the origin and remove

Feynman showed [3] that you could describe an event in the Hamiltonian in the form of differential equation, which describes how the function changes in term of operator. We may provide our comment that such a task is not realizable with the Hamiltonian, because it is a continuous function and does not involve dynamical initial position. The action is the discrete space–time function; therefore, the continuous outcome of a discrete action without relation to dynamic local position is

Feynman [4] applied renormalized Lagrangian action to quantum mechanics but even in Feynman's renormalized Lagrangian action is not conserved. The main problem of Lagrangian action and Feynman approach is the application of linear

Therefore, we need an entirely new theory to describe reversible fundamental laws of nature, combining discrete conservation of energy with the boundarymapped space–time, which would combine all kind of fields and forces within discrete background symmetry. Application of discrete energy conservation law and discrete symmetry as the product of this law may change drastically our present knowledge on the nature of forces and their roles in fundamental interactions. It is possible that nature and performance of forces, merging at the background symmetry, will be different, and reversibility of dynamical laws at discrete symmetry may

In our previous studies [5–7], we showed that wrong description of dynamical laws through simple continuous displacement in space–time structure using only

background superposition/dynamical local origin of energy-momentum content and space–time frame. The basic statement of such a concept is very simple: "particles may hold their "non-charged" state of rest only in discrete mode." Such an approach is the modification of Aristotle's concept [10] that "natural state of a body to be at rest" which does not present rest in discrete mode, therefore does not hold conservation of energy.

Eq. (1) after differentiation may display Maxwell equations in an alternative way. If a system after change looks the same in discrete mode in opposite phase, the

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based…*

The positive and negative solutions of dynamic supersymmetry outcomes of a

We can assume that the positive solution of Eq. (3) presents the discrete symmetric function in space–time phase in the local gauge field, while the negative sign is an antisymmetric solution of the symmetric function of an event in an opposite energetic phase of the background gauge field. The positive and negative solutions of Eq. (3) appear as a discrete change of the symmetric function from one phase to another phase, which is a shift of energy conservation from space–time phase (holding by ordinary matter) to the energy phase. These phases as background/local gauge fields discretely transform to each other, leading conservation of energy and

symmetry in discrete mode within opposite energy and space–time phases. The classic physics Eq. (3) in some sense is similar with Schrödinger's wave

**d**

which presents the original function but does not undergo any changes.

dynamic local states of variables, there will be no need of application of

**dS dt** ¼ �

**3. Development of a new mathematical theory for differentiation**

intervals and small displacement in space by improvement of mathematical

Hoof't showed [8] that it is possible to eliminate the bad effect of small time

The problem of Schrödinger's Eq. (4) is that it describes change of wave function only in one phase, which is time. The space phase representative is Hilbert space,

If classic physics could describe the symmetry and energy conservation law within the space–time frame with conjugation of space and time intervals with the

Schrodinger's wave function (4), which uses probability approach. The wave function of quantum mechanics with the local states of space and time coordinates could have deterministic classic equation to describe the exact symmetry of Nature. The deterministic equation of background space–time symmetry after the change of an

*S***1**

where the left side of the equation describes uniform change of space and time coordinates of an event, while the right side presents the original local space–time frame. The positive sign describes outcome of the ordinary matter phase, while the negative sign shows the outcome of antimatter. The statements of Eqs. (1)–(3) and (5), without Dirac's relativistic quantum mechanics, naturally predict existence of antiparticles to hold discrete conservation of energy within different states.

**F**´ð Þ¼� **s**,**t F s**ð Þ ,**t** (2)

F´ s, t ð Þ¼�F s, t ð Þ (3)

**dt <sup>ψ</sup>** ¼ �**iH<sup>ψ</sup>** (4)

**t1** (5)

equation of symmetry has a negative phase solution:

discrete event together will have a form:

*DOI: http://dx.doi.org/10.5772/intechopen.93087*

event in discrete mode may look the same:

function:

**of change**

**213**

The nature of rest is well described by Nobel Laureate Anderson [11]. By Anderson opinion, a system at stationary state of rest could not stay long and stationary state can be only equal superposition and its inverse. By his opinion, only superposition and inverse mixture may describe the absence of dipole moment. Unfortunately, Anderson did not put his statement into mathematical formulation. However, Anderson's "equal mixture" is equivalent to the equal numbers of matter–antimatter, which, as we will show later, needs modification.

We found out that we could solve the gap in "Anderson's equal mixtures "and describe stable steady-state performance of matter if we will apply vector type of discrete exchange interactions between two symmetric states, which can bring the system to superposition in discrete mode and hold discrete CP invariance of strong interactions. The superposition displays the genetic particle, while displacement from the superposition appears as the antiparticle of the superposition. It is easy to show that the origin of this principle is the conjugation of discrete conservation of energy with the discrete space–time, which is in hold for any fields/particles regardless of scale and completeness.

#### **2.2 The basic statement of symmetry**

While symmetries are conjugated with the corresponding conservation laws, we will start our analysis from principles of energy conservation. Distribution of energy in a medium requires certain space locality and time duration. The portion of energy, consumed for displacement of space, appears as the potential ingredient, while the time portion of the total energy presents kinetic energy. We may present the potential and kinetic ingredients of energy in the form of conjugated space and time portions. The suggested approach is different from Lagrangian or Hamiltonian, because these functions present continuous conservation of certain abstract amounts.

The Lagrangian or Hamiltonian functions, as Feynman stated [12], describe an abstract mathematical principle, which involves a certain numerical quantity, which has to be conserved. These formulations present some abstract number, which does not change, and after the change, we should have the same number.

Due to the conjugation with the conservation of energy, an event symmetry after the change should look the same. Therefore, a mathematical formulation of symmetry should show that (a) we have the same number of energy after change of an event and (b) an event looks the same as origin. We will describe how we can get such a mathematical formulation.

The exchange interaction of superposition (initial neutral state) with its displacement may produce two outcomes: (a) the symmetry of particles is continuous, such as the outcome of change looks the same as origin continuously, but breaks down spontaneously; (b) the outcome of change looks the same in discrete mode, with invariant translation without violation of symmetry. Later we will show that the outcome of interactions after change may look the same if the fundamental laws describe conservation of energy only in discrete mode.

Mathematically this statement, in general, may have the following form:

$$\mathbf{F}'(\mathbf{s}, \mathbf{t}) = \mathbf{F}\left(\mathbf{s}, \mathbf{t}\right) \tag{1}$$

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based… DOI: http://dx.doi.org/10.5772/intechopen.93087*

Eq. (1) after differentiation may display Maxwell equations in an alternative way. If a system after change looks the same in discrete mode in opposite phase, the equation of symmetry has a negative phase solution:

$$\mathbf{F}'(\mathbf{s}, \mathbf{t}) = -\mathbf{F}(\mathbf{s}, \mathbf{t}) \tag{2}$$

The positive and negative solutions of dynamic supersymmetry outcomes of a discrete event together will have a form:

$$\mathbf{F}'(\mathbf{s}, \mathbf{t}) = \pm \mathbf{F}(\mathbf{s}, \mathbf{t}) \tag{3}$$

We can assume that the positive solution of Eq. (3) presents the discrete symmetric function in space–time phase in the local gauge field, while the negative sign is an antisymmetric solution of the symmetric function of an event in an opposite energetic phase of the background gauge field. The positive and negative solutions of Eq. (3) appear as a discrete change of the symmetric function from one phase to another phase, which is a shift of energy conservation from space–time phase (holding by ordinary matter) to the energy phase. These phases as background/local gauge fields discretely transform to each other, leading conservation of energy and symmetry in discrete mode within opposite energy and space–time phases.

The classic physics Eq. (3) in some sense is similar with Schrödinger's wave function:

$$\frac{d}{dt}\Psi = -\mathbf{i}\mathbf{H}\Psi\tag{4}$$

The problem of Schrödinger's Eq. (4) is that it describes change of wave function only in one phase, which is time. The space phase representative is Hilbert space, which presents the original function but does not undergo any changes.

If classic physics could describe the symmetry and energy conservation law within the space–time frame with conjugation of space and time intervals with the dynamic local states of variables, there will be no need of application of Schrodinger's wave function (4), which uses probability approach. The wave function of quantum mechanics with the local states of space and time coordinates could have deterministic classic equation to describe the exact symmetry of Nature. The deterministic equation of background space–time symmetry after the change of an event in discrete mode may look the same:

$$\frac{d\mathbf{S}}{dt} = \pm \frac{\mathbf{S1}}{t\mathbf{1}}\tag{5}$$

where the left side of the equation describes uniform change of space and time coordinates of an event, while the right side presents the original local space–time frame. The positive sign describes outcome of the ordinary matter phase, while the negative sign shows the outcome of antimatter. The statements of Eqs. (1)–(3) and (5), without Dirac's relativistic quantum mechanics, naturally predict existence of antiparticles to hold discrete conservation of energy within different states.

#### **3. Development of a new mathematical theory for differentiation of change**

Hoof't showed [8] that it is possible to eliminate the bad effect of small time intervals and small displacement in space by improvement of mathematical

background superposition/dynamical local origin of energy-momentum content and space–time frame. The basic statement of such a concept is very simple: "particles may hold their "non-charged" state of rest only in discrete mode." Such an approach is the modification of Aristotle's concept [10] that "natural state of a body to be at rest" which does not present rest in discrete mode, therefore does not hold

The nature of rest is well described by Nobel Laureate Anderson [11]. By Anderson opinion, a system at stationary state of rest could not stay long and stationary state can be only equal superposition and its inverse. By his opinion, only superposition and

We found out that we could solve the gap in "Anderson's equal mixtures "and describe stable steady-state performance of matter if we will apply vector type of discrete exchange interactions between two symmetric states, which can bring the system to superposition in discrete mode and hold discrete CP invariance of strong interactions. The superposition displays the genetic particle, while displacement from the superposition appears as the antiparticle of the superposition. It is easy to show that the origin of this principle is the conjugation of discrete conservation of energy with the discrete space–time, which is in hold for any fields/particles

While symmetries are conjugated with the corresponding conservation laws, we will start our analysis from principles of energy conservation. Distribution of energy in a medium requires certain space locality and time duration. The portion of energy, consumed for displacement of space, appears as the potential ingredient, while the time portion of the total energy presents kinetic energy. We may present the potential and kinetic ingredients of energy in the form of conjugated space and time portions. The suggested approach is different from Lagrangian or Hamiltonian, because these functions present continuous conservation of certain abstract

The Lagrangian or Hamiltonian functions, as Feynman stated [12], describe an

The exchange interaction of superposition (initial neutral state) with its displacement may produce two outcomes: (a) the symmetry of particles is continuous, such as the outcome of change looks the same as origin continuously, but breaks down spontaneously; (b) the outcome of change looks the same in discrete mode, with invariant translation without violation of symmetry. Later we will show that the outcome of interactions after change may look the same if the fundamental laws

Mathematically this statement, in general, may have the following form:

**F**´ð Þ¼ **s**,**t F s**ð Þ ,**t** (1)

abstract mathematical principle, which involves a certain numerical quantity, which has to be conserved. These formulations present some abstract number, which does not change, and after the change, we should have the same number. Due to the conjugation with the conservation of energy, an event symmetry after the change should look the same. Therefore, a mathematical formulation of symmetry should show that (a) we have the same number of energy after change of an event and (b) an event looks the same as origin. We will describe how we can get

inverse mixture may describe the absence of dipole moment. Unfortunately, Anderson did not put his statement into mathematical formulation. However, Anderson's "equal mixture" is equivalent to the equal numbers of matter–antimatter,

which, as we will show later, needs modification.

regardless of scale and completeness.

**2.2 The basic statement of symmetry**

such a mathematical formulation.

describe conservation of energy only in discrete mode.

amounts.

**212**

conservation of energy.

*Quantum Mechanics*

formulation of small-scale transformation, for example, by renormalization group. However, renormalization tool leads to the approximate symmetry and renormalized artificial outcome of a natural event. The other way, which he suggested, is to find a new, improved theory.

state. This is the main problem of physical laws, applying the renormalization group

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based…*

Lagrange and Hamilton suggested conservation of energy in the form of linear

The specific feature of our approach is that energy, distributed within space and time portions, appears in the form of non-separable energy-momentum exchange entities. Energy in one phase appears as the consumed charged part in the space– time frame and in another phase appears as itself, comprising color ingredients of neutral photon-antiphoton pair. On this basis, we may present energy and momentum in two forms: (a) energy-momentum exists in the form of electrically charged matter–antimatter pairs, and (b) energy-momentum exists in the form of color charge pairs, where every part is an own particle of the other part. The condition of energy and momentum in forms (a) and (b) are completely different. However, the color charged bosonic pairs, which appear as "the neutral twin brothers" in the form of Majorana particles, are the superposition where it has a trend to move. In space– time phase, energy appears as Dirac's particles. It seems obvious that, at superposi-

**4. Energy-momentum: (a) charged antiparticle-particle pair and**

differential equations as well. The main concern of these equations is that the position coordinates and velocity components are independent variables and deriv-

atives of the Lagrangian with respect to the variables taken separately.

tion of color charge "neutral twin brothers," all the ingredients of energymomentum, as internal products, will exist in the form of twin particles.

Now we may apply this mathematical tool for characterization of any type of change, particularly ingredients of space–time. The parameters Δ**S**/**S**<sup>1</sup> and Δ**t**/**t**<sup>1</sup> have no unity and are unit-less parameters, which makes easy to compare them as the equivalent entities. Using Wheeler's [13] statement that the equation of special relativity **E = mc2** allows to transfer space and time equivalently to each other, we may show problems of such a statement. For this purpose, we may analyze the relationship between energy and mass portions without application of Lorentz transformations.

> *m***1** *E***1** ;

Eq. (6) describes change of energy-mass equivalence with the effect of initial condition (we may call rest mass and rest energy) in the form of "non-Lorentz transformation." By literature information [14], the exact value of Lorentz factor at velocity close to speed of light is **2.00**. If we use numeral value **γ = 2.00**, as an exact Lorentz factor [14], at uniform speed of light **c<sup>2</sup> = 1**, Eq. (6) produces condition.

for energy mass invariant translation. The energy and mass invariance (7) appears as the product of discrete exchange of energy-momentum relation and produces half-integer-integer spin interactions of mass and integer spin carrier particles.

Based on Planck's discrete energy radiation and empirical principle of energy conservation, we can formulate a nonempirical mathematical expression of energy.

**E1**

**m1** <sup>¼</sup> *<sup>c</sup>***2**,Δ<sup>E</sup> <sup>¼</sup> γΔm*c***<sup>2</sup>** (6)

Δm ¼ **1***=***2ΔE** (7)

to remove uncertainty of initial position.

*DOI: http://dx.doi.org/10.5772/intechopen.93087*

<sup>γ</sup> <sup>¼</sup> *<sup>Δ</sup>***<sup>E</sup> E1** : *Δm* **m1**

**5. Electromagnetic energy as the origin of space-time**

**5.1 Alternative model of space-time structure**

**215**

<sup>¼</sup> *<sup>Δ</sup>***<sup>E</sup>** *<sup>Δ</sup>***<sup>m</sup>** *:*

**(b) neutral twin particles**

To find new theories, we need to eliminate two problems: nonindependence feature of uncertain space displacement and time intervals in combined space–time unit and linearity of the change. We cannot get any help from special relativity (SR) and Minkowski's space–time to eliminate independent features of space and time intervals because they do not involve local origin and connect opposite time interval with the three space intervals into a nonsymmetric four-momentum frame (3:1), which involves abstract intervals of neutral space and time variables without their local positions.

We cannot use principles of general relativity (GR) theory as well because general relativity does not provide boundary-mapped reversible dynamical law due to its continuous space–time frame. GR does not have a background, which is the reason that GR's geometric, continuous space–time structure at small-scale interactions cannot find origin and runs away to the infinity. Wheeler's suggestion [13] on "space tells mass how to move, mass tells to space-time how to curve" does not produce a complete concept in a sense that it produces uncertainty because GR's space–time cannot tell to mass the path and boundary to move and mass cannot tell space–time boundary where to stop.

First, we will look how the features of dynamics change if we gradually reduce time interval **Δt**, moving from the high scale to the small-scale event, as was done by Hoof't [8]. However, we will analyze not an interval as Hoof't did, but a function **Δf**/**f1,** which as a mathematical operator may give information about change of a function in relation to its dynamical local origin. This function is a sufficient entity for the identification of change. The non-unitary function **Δf**/**f1** shows quantum behavior and with the fractional feature (portion) produces the outcomes with the integer numbers **(f2/f1–1**). The mathematical operator in the form of **Δf**/**f1** portion describes the fraction of the change in relation to its dynamical origin. Similarly, the operator Δ**S**/**S**<sup>1</sup> describes displacement of space with the applied force in relation to its origin, while the operator Δ**t**/**t**<sup>1</sup> describes the fluctuation of time about instant of action. The functions Δ**S**/**S**<sup>1</sup> and Δ**t**/**t**<sup>1</sup> describe the entanglement of the displacement with the initial superposition as the genetic code of the event. The relation of change around its origin **Δs/S**<sup>1</sup> generates a spherical space, while relation of time interval to instant of time produces a round time structure. Therefore, there is no preferred inertial system and mathematical model, which may display an event better than its initial superposition state.

In planet-scale events, reduction of the distance twice, as was shown by Hoof't [8], does not affect significantly the linearity of change. The parameter Δ**S**/**S**<sup>1</sup> also describes a similar effect of the change to the linearity. However, if we reduce interval of time twice in a small-scale event, using Δ**t**/**t**<sup>1</sup> function, we will be able to describe catastrophic effect of the change to the linearity of the motion.

The relation of intervals of time and displacement to the origin creates entanglement of the final and initial states of coordinates. The origin of an event in this case "tells the body how to move and the final state of a motion gets the information where to stop." However, the entanglement of interval of change with the origin leads to the deterministic nature of the dynamical event within a certain boundary, and it is the only way for elimination of the infinity problem of small-scale interactions. The effect of initial/local coordinate of time and space of a body appears as an action of initial energy contents (such as inertial mass, inertial energy) of a body to the change of pathway. Presently all the physical laws use only independent intervals to describe the change of an event without relation of change to the initial local *Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based… DOI: http://dx.doi.org/10.5772/intechopen.93087*

state. This is the main problem of physical laws, applying the renormalization group to remove uncertainty of initial position.

#### **4. Energy-momentum: (a) charged antiparticle-particle pair and (b) neutral twin particles**

Lagrange and Hamilton suggested conservation of energy in the form of linear differential equations as well. The main concern of these equations is that the position coordinates and velocity components are independent variables and derivatives of the Lagrangian with respect to the variables taken separately.

The specific feature of our approach is that energy, distributed within space and time portions, appears in the form of non-separable energy-momentum exchange entities. Energy in one phase appears as the consumed charged part in the space– time frame and in another phase appears as itself, comprising color ingredients of neutral photon-antiphoton pair. On this basis, we may present energy and momentum in two forms: (a) energy-momentum exists in the form of electrically charged matter–antimatter pairs, and (b) energy-momentum exists in the form of color charge pairs, where every part is an own particle of the other part. The condition of energy and momentum in forms (a) and (b) are completely different. However, the color charged bosonic pairs, which appear as "the neutral twin brothers" in the form of Majorana particles, are the superposition where it has a trend to move. In space– time phase, energy appears as Dirac's particles. It seems obvious that, at superposition of color charge "neutral twin brothers," all the ingredients of energymomentum, as internal products, will exist in the form of twin particles.

Now we may apply this mathematical tool for characterization of any type of change, particularly ingredients of space–time. The parameters Δ**S**/**S**<sup>1</sup> and Δ**t**/**t**<sup>1</sup> have no unity and are unit-less parameters, which makes easy to compare them as the equivalent entities. Using Wheeler's [13] statement that the equation of special relativity **E = mc2** allows to transfer space and time equivalently to each other, we may show problems of such a statement. For this purpose, we may analyze the relationship between energy and mass portions without application of Lorentz transformations.

$$\gamma = \frac{\Delta \mathbf{E}}{\mathbf{E1}} \mathbf{:} \frac{\Delta m}{\mathbf{m}\_1} = \frac{\Delta \mathbf{E}}{\Delta \mathbf{m}} \frac{m\mathbf{1}}{E\mathbf{1}} \mathbf{:} \frac{\mathbf{E1}}{m\mathbf{1}} = \mathbf{c2}, \Delta \mathbf{E} = \gamma \Delta \mathbf{m} \mathbf{c2} \tag{6}$$

Eq. (6) describes change of energy-mass equivalence with the effect of initial condition (we may call rest mass and rest energy) in the form of "non-Lorentz transformation." By literature information [14], the exact value of Lorentz factor at velocity close to speed of light is **2.00**. If we use numeral value **γ = 2.00**, as an exact Lorentz factor [14], at uniform speed of light **c<sup>2</sup> = 1**, Eq. (6) produces condition.

$$
\Delta \mathbf{m} = \mathbf{1}/2\Delta \mathbf{E} \tag{7}
$$

for energy mass invariant translation. The energy and mass invariance (7) appears as the product of discrete exchange of energy-momentum relation and produces half-integer-integer spin interactions of mass and integer spin carrier particles.

#### **5. Electromagnetic energy as the origin of space-time**

#### **5.1 Alternative model of space-time structure**

Based on Planck's discrete energy radiation and empirical principle of energy conservation, we can formulate a nonempirical mathematical expression of energy.

formulation of small-scale transformation, for example, by renormalization group.

To find new theories, we need to eliminate two problems: nonindependence feature of uncertain space displacement and time intervals in combined space–time unit and linearity of the change. We cannot get any help from special relativity (SR) and Minkowski's space–time to eliminate independent features of space and time intervals because they do not involve local origin and connect opposite time interval with the three space intervals into a nonsymmetric four-momentum frame (3:1), which involves abstract intervals of neutral space and time variables without their

We cannot use principles of general relativity (GR) theory as well because general relativity does not provide boundary-mapped reversible dynamical law due to its continuous space–time frame. GR does not have a background, which is the reason that GR's geometric, continuous space–time structure at small-scale interactions cannot find origin and runs away to the infinity. Wheeler's suggestion [13] on "space tells mass how to move, mass tells to space-time how to curve" does not produce a complete concept in a sense that it produces uncertainty because GR's space–time cannot tell to mass the path and boundary to move and mass cannot tell

First, we will look how the features of dynamics change if we gradually reduce time interval **Δt**, moving from the high scale to the small-scale event, as was done by Hoof't [8]. However, we will analyze not an interval as Hoof't did, but a function **Δf**/**f1,** which as a mathematical operator may give information about change of a function in relation to its dynamical local origin. This function is a sufficient entity for the identification of change. The non-unitary function **Δf**/**f1** shows quantum behavior and with the fractional feature (portion) produces the outcomes with the integer numbers **(f2/f1–1**). The mathematical operator in the form of **Δf**/**f1** portion describes the fraction of the change in relation to its dynamical origin. Similarly, the operator Δ**S**/**S**<sup>1</sup> describes displacement of space with the applied force in relation to its origin, while the operator Δ**t**/**t**<sup>1</sup> describes the fluctuation of time about instant of action. The functions Δ**S**/**S**<sup>1</sup> and Δ**t**/**t**<sup>1</sup> describe the entanglement of the displacement with the initial superposition as the genetic code of the event. The relation of change around its origin **Δs/S**<sup>1</sup> generates a spherical space, while relation of time interval to instant of time produces a round time structure. Therefore, there is no preferred inertial system and mathematical model,

which may display an event better than its initial superposition state.

describe catastrophic effect of the change to the linearity of the motion.

In planet-scale events, reduction of the distance twice, as was shown by Hoof't [8], does not affect significantly the linearity of change. The parameter Δ**S**/**S**<sup>1</sup> also describes a similar effect of the change to the linearity. However, if we reduce interval of time twice in a small-scale event, using Δ**t**/**t**<sup>1</sup> function, we will be able to

The relation of intervals of time and displacement to the origin creates entanglement of the final and initial states of coordinates. The origin of an event in this case "tells the body how to move and the final state of a motion gets the information where to stop." However, the entanglement of interval of change with the origin leads to the deterministic nature of the dynamical event within a certain boundary, and it is the only way for elimination of the infinity problem of small-scale interactions. The effect of initial/local coordinate of time and space of a body appears as an action of initial energy contents (such as inertial mass, inertial energy) of a body to the change of pathway. Presently all the physical laws use only independent intervals to describe the change of an event without relation of change to the initial local

However, renormalization tool leads to the approximate symmetry and renormalized artificial outcome of a natural event. The other way, which he

suggested, is to find a new, improved theory.

space–time boundary where to stop.

local positions.

*Quantum Mechanics*

**214**

The basic principle of energy conservation states that "Energy can only be transferred from one form to another." Transformation of energy from one form to another requires boundary within the space–time frame, carrying conservation of energy through space and time portions. While conservation laws associated with the time and space frame symmetries, we may consider that equally distributed space and time portions of energy hold simultaneous conservation of energy and momentum within symmetric frame. On this bass, the energy portions, equally distributed in space or time phases, both cover the half of the total available energy: **Es = Et = 1/2Ea.** This equation is the equivalent expression of Eq. (7). Similarly, the total energy comprises the mixture of energy portions, equally distributed within two parts of the space–time frame: 2**Es = Ea**.

Based on these simple equations, we may construct mathematical model of energy conservation, which has to combine energy-momentum conservations within the space–time frame. Conjugation of energy-momentum conservations within exchange interaction, which appears in discrete mode, generates principles of discrete symmetry. In this sense, special relativity's energy-mass relation **E = mc<sup>2</sup>** does not hold invariant discrete energy-mass exchange relation and cannot describe discrete symmetry of energy-mass relation, localized within the discrete space–time frame*:*

Based on such an approach, we may present space–time as a frame, which comprises cross product of space portion as materialization of energy and cross product of time portion, which at decay of space–time returns an energy to the origin:

$$\mathrm{Es}\frac{\mathrm{d}\mathbf{S}}{\mathrm{S}\_{1}} - (\mathrm{Ea} - \mathrm{Es})\frac{\mathrm{d}\mathbf{t}\_{1}}{\mathrm{t}\_{1}} = \mathbf{0} \tag{8}$$

The right side of Eq. (9) describes energy-momentum exchange interaction, relative to the original momentum of superposition, which generates shift of energy conservation from the space–time frame to the original energy phase. *The particle of space–time moves through the electric field, and electric field in reverse order propagates through the space–time field of matter.* Eq. (10) describes the cross product wave function where the local space–time wave **S1/t1** is carried by the flux of energymomentum wave (**Ea/Es-1**) which changes wavelength and wave amplitude of space–time by S1 and t1. Conservation of energy-momentum is associated with the symmetry in time and space; therefore symmetry has to be the cross product of

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based…*

Model (10) has two important features on coupling of particle with the field. The eigenvector of model (9) connects force, field, and particle together: (a) the mathematical operator describes change of symmetry generating electromagnetic field in relation to initial momentum of a particle, and (b) it presents change of space–time position of a "non-Aristotelian" particle in relation to the symmetry-

The superposition's genetic code in the form of dynamical space–time (**S**1/**t**1) unit has discrete coupling with the electromagnetic field (10). Model (10) in general form describes relation of energy portions, distributed within space–time field, which generates discrete vector space as a product of discrete energy-momentum relation. The suggested approach is different from Sudarshan and Marshak's **V-A** theory of weak force [15], while without discrete eigenfunction, producing integer spin particles you cannot reverse a particle to the background gauge field. However, Hilbert space of quantum mechanics and **V-A** theory do not carry such a performance. *The other feature of Eq. (9) is reciprocal isomorphic discrete symmetry of space– time and energy-momentum exchange interaction, which became the outcome products of each other, forming the supersymmetric gauge equation.* Such an approach allows com-

The background gauge field's force carrier **Ea** holds the symmetry of **Es** matter ingredients of eigenfunction **(Ea Es)/Es** in the space–time frame of local gauge field. When the symmetry generator is turned off (**Ea = 0)**, the **Es** through coupling of local and background particles return to the background gauge field in the form of neutral pairs of gauge field. Based on model (10), which combines space–time with the electromagnetism (energy-momentum conservation), the origin of space– time appears to be the background gauge field energy, which generates the basic unit of matter space–time frame and holds its conservation within conjugation of

We found out that simplifying strong interactions to the linear exchange of photons or meson within continuous symmetry is the reason for appearance of problems of particle physics theories. Particularly, Yukawa's meson theory of strong interactions, describing linear exchange of mesons, and **V-A** theory of CP violation

Model (9) has a philosophical meaning: we do not present time as itself, which as an entity is different from space. We present a certain entity in time phase and this entity is the energy. That is why time has no independent existence from space energy portions and is not an abstract parameter, which may flow independently. The same philosophy is relative to the space as well. We did not present an event in abstract three-dimensional space or within four-momentum frame of SR; we describe the vector space, which changes dimension and direction in accordance with the flux of energy and momentum to this space. Such a space of space–time

bining all the conservation laws within these symmetric interactions.

**5.2 The space and time particles of the space-time frame**

space–time and energy-momentum relation.

*DOI: http://dx.doi.org/10.5772/intechopen.93087*

generating field.

background/local gauge phases.

are examples of such theories.

**217**

The first part of Eq. (8) presents the portion of consumed energy (**Es**) in space phase with the positive sign, while the second ingredient of the equation shows the remaining energy portions within the time ingredient of total energy with the negative sign. Model (8) gives the following equations:

$$\frac{\frac{\text{dS}}{\text{S}\_1}}{\frac{\text{dt}}{\text{t}\_1}} = \frac{\text{E}\_\text{a} - \text{E}\_\text{s}}{\text{E}\_\text{s}} \tag{9}$$

$$\frac{d\mathbf{S}}{dt} = \frac{\mathbf{S\_1}}{\mathbf{t\_1}} \left(\frac{\mathbf{E\_a}}{\mathbf{E\_s}} - \mathbf{1}\right) \tag{10}$$

$$
\lambda = \frac{\mathbf{E\_a}}{\mathbf{E\_s}} - \mathbf{1} \tag{11}
$$

$$\text{Let } \lambda = \mathbf{1}, \mathbf{E}\mathbf{s} = \mathbf{1}/2\mathbf{E}\mathbf{a} \tag{12}$$

**S**<sup>1</sup> and **t**<sup>1</sup> are the space and time variables, corresponding to the origin/dynamic local boundary, and **Ea–Es** and **Es** are the energy portions, distributed in space and time within energy-momentum exchange interaction at conditions corresponding to the background/dynamical local boundaries of **S**<sup>1</sup> and **t**1. The background superposition as the gauge field holds the hidden initial space and time variables, which carry invariant translation of energy and corresponding symmetry from one form of energy to another and inverse. The local dynamical gauge position is the mathematical operator, which translates energy in the form of force from the local matter phase to energy phase. The **Ea** electromagnetic energy of model (9) is the symmetry generator of local gauge field, while **Es** appears as the local momentum ingredient of energy of the background gauge field.

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based… DOI: http://dx.doi.org/10.5772/intechopen.93087*

The right side of Eq. (9) describes energy-momentum exchange interaction, relative to the original momentum of superposition, which generates shift of energy conservation from the space–time frame to the original energy phase. *The particle of space–time moves through the electric field, and electric field in reverse order propagates through the space–time field of matter.* Eq. (10) describes the cross product wave function where the local space–time wave **S1/t1** is carried by the flux of energymomentum wave (**Ea/Es-1**) which changes wavelength and wave amplitude of space–time by S1 and t1. Conservation of energy-momentum is associated with the symmetry in time and space; therefore symmetry has to be the cross product of space–time and energy-momentum relation.

Model (10) has two important features on coupling of particle with the field. The eigenvector of model (9) connects force, field, and particle together: (a) the mathematical operator describes change of symmetry generating electromagnetic field in relation to initial momentum of a particle, and (b) it presents change of space–time position of a "non-Aristotelian" particle in relation to the symmetrygenerating field.

The superposition's genetic code in the form of dynamical space–time (**S**1/**t**1) unit has discrete coupling with the electromagnetic field (10). Model (10) in general form describes relation of energy portions, distributed within space–time field, which generates discrete vector space as a product of discrete energy-momentum relation. The suggested approach is different from Sudarshan and Marshak's **V-A** theory of weak force [15], while without discrete eigenfunction, producing integer spin particles you cannot reverse a particle to the background gauge field. However, Hilbert space of quantum mechanics and **V-A** theory do not carry such a performance. *The other feature of Eq. (9) is reciprocal isomorphic discrete symmetry of space– time and energy-momentum exchange interaction, which became the outcome products of each other, forming the supersymmetric gauge equation.* Such an approach allows combining all the conservation laws within these symmetric interactions.

The background gauge field's force carrier **Ea** holds the symmetry of **Es** matter ingredients of eigenfunction **(Ea Es)/Es** in the space–time frame of local gauge field. When the symmetry generator is turned off (**Ea = 0)**, the **Es** through coupling of local and background particles return to the background gauge field in the form of neutral pairs of gauge field. Based on model (10), which combines space–time with the electromagnetism (energy-momentum conservation), the origin of space– time appears to be the background gauge field energy, which generates the basic unit of matter space–time frame and holds its conservation within conjugation of background/local gauge phases.

We found out that simplifying strong interactions to the linear exchange of photons or meson within continuous symmetry is the reason for appearance of problems of particle physics theories. Particularly, Yukawa's meson theory of strong interactions, describing linear exchange of mesons, and **V-A** theory of CP violation are examples of such theories.

#### **5.2 The space and time particles of the space-time frame**

Model (9) has a philosophical meaning: we do not present time as itself, which as an entity is different from space. We present a certain entity in time phase and this entity is the energy. That is why time has no independent existence from space energy portions and is not an abstract parameter, which may flow independently. The same philosophy is relative to the space as well. We did not present an event in abstract three-dimensional space or within four-momentum frame of SR; we describe the vector space, which changes dimension and direction in accordance with the flux of energy and momentum to this space. Such a space of space–time

The basic principle of energy conservation states that "Energy can only be transferred from one form to another." Transformation of energy from one form to another requires boundary within the space–time frame, carrying conservation of energy through space and time portions. While conservation laws associated with the time and space frame symmetries, we may consider that equally distributed space and time portions of energy hold simultaneous conservation of energy and momentum within symmetric frame. On this bass, the energy portions, equally distributed in space or time phases, both cover the half of the total available energy: **Es = Et = 1/2Ea.** This equation is the equivalent expression of Eq. (7). Similarly, the total energy comprises the mixture of energy portions, equally distributed within

Based on these simple equations, we may construct mathematical model of energy conservation, which has to combine energy-momentum conservations within the space–time frame. Conjugation of energy-momentum conservations within exchange interaction, which appears in discrete mode, generates principles of discrete symmetry. In this sense, special relativity's energy-mass relation **E = mc<sup>2</sup>** does not hold invariant discrete energy-mass exchange relation and cannot describe discrete symmetry of energy-mass relation, localized within the discrete space–time

Based on such an approach, we may present space–time as a frame, which comprises cross product of space portion as materialization of energy and cross product of

� ð Þ **Ea** � **Es dt1**

The first part of Eq. (8) presents the portion of consumed energy (**Es**) in space phase with the positive sign, while the second ingredient of the equation shows the remaining energy portions within the time ingredient of total energy with the

> <sup>¼</sup> **Ea** � **Es Es**

> > **Ea Es** � **1**

**S**<sup>1</sup> and **t**<sup>1</sup> are the space and time variables, corresponding to the origin/dynamic local boundary, and **Ea–Es** and **Es** are the energy portions, distributed in space and time within energy-momentum exchange interaction at conditions corresponding to the background/dynamical local boundaries of **S**<sup>1</sup> and **t**1. The background superposition as the gauge field holds the hidden initial space and time variables, which carry invariant translation of energy and corresponding symmetry from one form of energy to another and inverse. The local dynamical gauge position is the mathematical operator, which translates energy in the form of force from the local matter phase to energy phase. The **Ea** electromagnetic energy of model (9) is the symmetry generator of local gauge field, while **Es** appears as the local momentum ingredient

**<sup>λ</sup>** <sup>¼</sup> **Ea Es** **t1**

¼ **0** (8)

� **1** (11)

at **λ** ¼ **1**, **Es** ¼ **1***=***2Ea** (12)

(9)

(10)

time portion, which at decay of space–time returns an energy to the origin:

**dS S1 dt t1**

**dS dt** <sup>¼</sup> **S1 t1**

Es **dS S1**

negative sign. Model (8) gives the following equations:

of energy of the background gauge field.

**216**

two parts of the space–time frame: 2**Es = Ea**.

frame*:*

*Quantum Mechanics*

may change from three-dimensional frame to two- or round dimensional space– time. Based on the frequency of energy-momentum flux, space–time at the small scale moves to the round dimension, which is not possible by Hamiltonian's or Lagrangian's only time-dependent linear equations.

elementary space–time frame. The condition **Es = 1/2Ea** of model (10) generates the

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based…*

� **dt t1**

When symmetry generator electromagnetic energy is turned off (**Ea = 0)**, we will get decay of space–time and shift of energy from local gauge space–time frame

Eqs. (13) and (14) are the alternative presentations of Eq. (5). The portions of energy, carried by space and time identities **dS/S1** and **dt/t1**, play a role of the quantum operator of annihilation or creation, through coupling with the energy-

The concept of supersymmetry, which we suggest in (9), describes the conjugated symmetry, which involves simultaneous symmetry of space–time frame and

In accordance with our concept, a change of particle's displacement around their

In accordance with our theory, the spin is the conserved vector quantity, produced from conservation of energy within discrete energy-momentum exchange relation, which generates for ingredients of this interaction's spin numbers (12). The space and time portions of energy in exchange interaction (12) appear as interaction of fields, which produces the ingredients of this interaction in the form of fermions

The quantum physics' presentation of spin, as a cross product of vector position

momentum conservation in a proper way. The quantum mechanic's specification of spin is a very abstract concept because the point particle is not a particle, which does not have a space–time frame of matter and therefore cannot produce half spin identity in the form of fermion. We suggest the identification of angular momentum as a product of the particle's space–time position vector and energy-momentum exchange interaction (10), which produces not the pseudo-vector but the local space vector. This vector generates a deterministic pathway of a particle's dynamics. In such a model, the dynamic local position became the deterministic position vector. Therefore, we may identify fermions and bosons only as the products of space–time frame. Due to these features, quantum mechanics cannot explain unusual feature of baryon frame where two identical quarks in proton or neutron

with the momentum, does not produce quantity, which may carry energy-

superposition generates the conserved quantity called spin, which in quantum physics has identification, as the angular momentum. By quantum physics, the spin number for a point particle is the product of pseudo-vector position (relative to

The condition **Es = 1/2Ea** became the energy-momentum genetic code of particle-antiparticle interactions in the discrete space–time frame. The genetic code of Eq. (13) in the space–time frame generates a three-jet performance of ingredients

energy-momentum exchange interactions, carrying both in discrete mode.

**5.3 The theory of spin as the product of discrete energy-momentum**

some unknown origin) and its momentum vector **r** � **p** [16].

frame do not obey the Pauli rules of quantum statistics.

¼ **0** (13)

¼ **0** (14)

**dS S1**

**dS S1** þ **dt t1**

invariant translations in a space–time frame:

*DOI: http://dx.doi.org/10.5772/intechopen.93087*

to the energy phase of the background gauge field:

momentum exchange interactions.

of energy-mass exchange interaction.

**exchange interaction**

and bosons.

**219**

Replacement of intervals by combination of the origin with the displacement in the form of portions is the new algebraic expression of dimension, which as the mathematical operation carries *natural renormalization* of the change to the initial origin. Relation of the change to the initial origin generates **S2/S1–1** quantum operator, producing outcomes by the integer numbers.

Based on such a phenomenon, any particle of space–time field or antiparticle of energy field has no independent existence. The condition **Ea** > **0** is the displacement from the superposition (field excitation) with the generation of three-dimensional space, which produces a local field and its particle (electrically charged mass). When an energy flux to space–time discontinued (**Ea = 0**), the distinction between field and particle disappears, and the superposition and field merge. At **Ea = 0**, the negative energy matrix produces **U (1)** symmetry group of gauge field, similar to Maxwell's theory of electromagnetism. Model (9) can be applied for any interaction of **(Ea Es)** as a field and **Es** as a particle.

The other specific feature of our theory is that the relation of the change to its origin generates an original code as its own reference frame of an event. The state of origin of the space or time particle became their own antiparticle. When a particle does not change **(Ea = 0)**, its position in space, merging with its antiparticle, generates neutral particle of discrete rest. Conjugation and merging of two states (fields) became the main principle of discrete symmetry and conservation of energy within a certain boundary.

Our theory uses the background gauge field as the only possible reference frame, where all the interactions take their origin. The relation of an event or the particle's space–time frame to the background gauge field became the obvious concept, while the background gauge field is the source of interactions and mediates the space– time frame of a particle within the local gauge field. The background state is the source of symmetry generating electromagnetic force-gravitation exchange interaction in the local gauge field, which has to deliver energy back to its origin. *Gravitation appears as the short-range force, which holds discrete performance of electromagnetic force and generates stable existence of a nucleon in discrete mode.*

The inertial frame of reference in classic physics and special relativity is the same and states "the body with net zero force does not accelerate and such a body is at rest or moves at a constant velocity." Based on our theory, this statement is not completely true. When the net force flux to the space–time frame is zero **(Ea = 0)**, particles move to the reference background gauge field which holds discrete performance but not constant rest. We explain SR's time delay statement differently. By SR, "the clock of a moving body will tick slower than clock that is in rest in his inertial frame of reference." SR states that if the particle's speed approaches the speed of light, the massless particles that travel with speed of light is unaffected by passage of time.

First, the massless particles cannot have free travel due to the requirement of energy conservation within boundary. Based on model (10), time instant **t**<sup>1</sup> is proportional to eigenvalue (12), and with the reduction of this value, the time instant and clock will tick slower than the background state.

The relation of change to initial origin eliminates unity which allows describing energy portions in space and time phases within any symmetric dimensions which may change from linearity of planet-scale event (string like dimension) to round dimension of baryon-scale interactions. The minimum portion of quanta, produced from the nonlinear energy-momentum exchange vector interaction, generates an

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based… DOI: http://dx.doi.org/10.5772/intechopen.93087*

elementary space–time frame. The condition **Es = 1/2Ea** of model (10) generates the invariant translations in a space–time frame:

$$\frac{\mathbf{dS}}{\mathbf{S\_1}} - \frac{\mathbf{dt}}{\mathbf{t\_1}} = \mathbf{0} \tag{13}$$

When symmetry generator electromagnetic energy is turned off (**Ea = 0)**, we will get decay of space–time and shift of energy from local gauge space–time frame to the energy phase of the background gauge field:

$$\frac{\mathbf{dS}}{\mathbf{S\_1}} + \frac{\mathbf{dt}}{\mathbf{t\_1}} = \mathbf{0} \tag{14}$$

Eqs. (13) and (14) are the alternative presentations of Eq. (5). The portions of energy, carried by space and time identities **dS/S1** and **dt/t1**, play a role of the quantum operator of annihilation or creation, through coupling with the energymomentum exchange interactions.

The condition **Es = 1/2Ea** became the energy-momentum genetic code of particle-antiparticle interactions in the discrete space–time frame. The genetic code of Eq. (13) in the space–time frame generates a three-jet performance of ingredients of energy-mass exchange interaction.

The concept of supersymmetry, which we suggest in (9), describes the conjugated symmetry, which involves simultaneous symmetry of space–time frame and energy-momentum exchange interactions, carrying both in discrete mode.

#### **5.3 The theory of spin as the product of discrete energy-momentum exchange interaction**

In accordance with our concept, a change of particle's displacement around their superposition generates the conserved quantity called spin, which in quantum physics has identification, as the angular momentum. By quantum physics, the spin number for a point particle is the product of pseudo-vector position (relative to some unknown origin) and its momentum vector **r** � **p** [16].

In accordance with our theory, the spin is the conserved vector quantity, produced from conservation of energy within discrete energy-momentum exchange relation, which generates for ingredients of this interaction's spin numbers (12). The space and time portions of energy in exchange interaction (12) appear as interaction of fields, which produces the ingredients of this interaction in the form of fermions and bosons.

The quantum physics' presentation of spin, as a cross product of vector position with the momentum, does not produce quantity, which may carry energymomentum conservation in a proper way. The quantum mechanic's specification of spin is a very abstract concept because the point particle is not a particle, which does not have a space–time frame of matter and therefore cannot produce half spin identity in the form of fermion. We suggest the identification of angular momentum as a product of the particle's space–time position vector and energy-momentum exchange interaction (10), which produces not the pseudo-vector but the local space vector. This vector generates a deterministic pathway of a particle's dynamics. In such a model, the dynamic local position became the deterministic position vector. Therefore, we may identify fermions and bosons only as the products of space–time frame. Due to these features, quantum mechanics cannot explain unusual feature of baryon frame where two identical quarks in proton or neutron frame do not obey the Pauli rules of quantum statistics.

may change from three-dimensional frame to two- or round dimensional space– time. Based on the frequency of energy-momentum flux, space–time at the small scale moves to the round dimension, which is not possible by Hamiltonian's or

Replacement of intervals by combination of the origin with the displacement in the form of portions is the new algebraic expression of dimension, which as the mathematical operation carries *natural renormalization* of the change to the initial origin. Relation of the change to the initial origin generates **S2/S1–1** quantum

Based on such a phenomenon, any particle of space–time field or antiparticle of energy field has no independent existence. The condition **Ea** > **0** is the displacement from the superposition (field excitation) with the generation of three-dimensional space, which produces a local field and its particle (electrically charged mass). When an energy flux to space–time discontinued (**Ea = 0**), the distinction between field and particle disappears, and the superposition and field merge. At **Ea = 0**, the negative energy matrix produces **U (1)** symmetry group of gauge field, similar to Maxwell's theory of electromagnetism. Model (9) can be applied for any interaction

The other specific feature of our theory is that the relation of the change to its origin generates an original code as its own reference frame of an event. The state of origin of the space or time particle became their own antiparticle. When a particle does not change **(Ea = 0)**, its position in space, merging with its antiparticle, generates neutral particle of discrete rest. Conjugation and merging of two states (fields) became the main principle of discrete symmetry and conservation of energy

Our theory uses the background gauge field as the only possible reference frame, where all the interactions take their origin. The relation of an event or the particle's space–time frame to the background gauge field became the obvious concept, while the background gauge field is the source of interactions and mediates the space– time frame of a particle within the local gauge field. The background state is the source of symmetry generating electromagnetic force-gravitation exchange interaction in the local gauge field, which has to deliver energy back to its origin. *Gravitation appears as the short-range force, which holds discrete performance of elec-*

*tromagnetic force and generates stable existence of a nucleon in discrete mode.*

The inertial frame of reference in classic physics and special relativity is the same and states "the body with net zero force does not accelerate and such a body is at rest or moves at a constant velocity." Based on our theory, this statement is not completely true. When the net force flux to the space–time frame is zero **(Ea = 0)**, particles move to the reference background gauge field which holds discrete performance but not constant rest. We explain SR's time delay statement differently. By SR, "the clock of a moving body will tick slower than clock that is in rest in his inertial frame of reference." SR states that if the particle's speed approaches the speed of light, the massless particles that travel with speed of light is unaffected by

First, the massless particles cannot have free travel due to the requirement of energy conservation within boundary. Based on model (10), time instant **t**<sup>1</sup> is proportional to eigenvalue (12), and with the reduction of this value, the time

The relation of change to initial origin eliminates unity which allows describing energy portions in space and time phases within any symmetric dimensions which may change from linearity of planet-scale event (string like dimension) to round dimension of baryon-scale interactions. The minimum portion of quanta, produced from the nonlinear energy-momentum exchange vector interaction, generates an

instant and clock will tick slower than the background state.

Lagrangian's only time-dependent linear equations.

operator, producing outcomes by the integer numbers.

of **(Ea Es)** as a field and **Es** as a particle.

within a certain boundary.

*Quantum Mechanics*

passage of time.

**218**

The genetic code of supersymmetry **Es = 1/2Ea** explains this paradox. The antisymmetric wave function (13) holds the invariance of baryon performance through discrete symmetry, carried within background and local gauge fields. From supersymmetric genetic code.

It is necessary to note a very important feature of translation of ingredients of Eq. (14) when the ingredients of this equation doubled. The half-integer fermions of this equation became integer carrier particles, while integer carrier particles became double integer carrier particles. Therefore, in energy phase the performance

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based…*

Based on model (10), depending on the energy flux to the space–time frame, the helicity of the ingredients of the space–time frame changes. In accordance with Eq. (14) when electromagnetic interactions turned off (**Ea = 0**), the difference between space and time phase particles disappears, and all the particles behave as integer number particles of the background gauge field. In this case, the chirality and handiness of particles gets the same left-handed direction. When local symmetry generator force is not available (**Ea = 0**), the momentum of matter space–time phase (10) transforms to the energy of the gauge field and gets a negative sign. Simply, "the ingredients of energy return to itself." In this case, Dirac's neutrinos transform to the neutral Majorana neutrinos having left helicity to the background gauge field where bosonic particles involve gamma rays, neutral fermions pair, and

To understand the nature of particles and forces, we have to analyze the decay mechanism of produced pion families, where the W vector bosons were intermedi-

π� ! **μ**� þ **νμ**

<sup>π</sup>þπ�π**<sup>0</sup>** ! **<sup>2</sup><sup>γ</sup>** <sup>þ</sup> **<sup>μ</sup>**þ*=***μ**� <sup>þ</sup> **νμ***=***νμ**

The decay of the local gauge field to hold vector conservation produces a new vector, which comprises generation of intermediate W vector bosons from decay of π pions. The **2Es = Ea** code of the background gauge field requires equal numbers of electron and neutrino family pairs which is realized by the equal branching ratios of the decay of intermediate W vector bosons. Eq. **(**19**)** describes decay condition in

The product stream composition generates composite of neutral particles, which exists in annihilation mode in the background gauge field to hold equation 2Es = Ea:

In the discrete energy conservation mode, the particles cannot hold annihilation process for a long time. Coupling of gamma rays with the neutral particles leads to

The intermediate step in the symmetric translation from fermions to gauge field is the transformation of proton-antiproton pair to neutron-antineutron Majoranatype particles, which decompose to kaon family mesons. Due to the existence of three fractional proton-antiprotons, comprising other flavors of quarks, the decomposition of neutron-antineutron pair produces three kaon-type mesons. The decay products of other unidentified two kaons, which we may call **Kaon2** and

**2γ** \$ �**μ**<sup>þ</sup>*=***μ**� þ **νμ***=***νμ**

the generation of charge and electromagnetic force of local gauge field:

� \$ **μ**�*=***νμ**

**Kaon3**, can be described similarly by Eqs. (16)–(19).

The produced ingredients of the decay form the balance equation:

π<sup>þ</sup> ! **μ**<sup>þ</sup> þ **νμ** (16)

<sup>π</sup>**<sup>0</sup>** ! **<sup>2</sup><sup>γ</sup>** (18)

� (17)

� (19)

� (20)

� <sup>þ</sup> **<sup>μ</sup>**<sup>þ</sup>*=***νμ** <sup>þ</sup> **Ea electromagnetic force** (21)

of forces holding space–time frame changes in the opposite order.

*DOI: http://dx.doi.org/10.5772/intechopen.93087*

neutral neutrinos.

ate ingredients:

average for muon family leptons.

**2γ** þ **μ**<sup>þ</sup>*=***μ**� þ **νμ***<sup>=</sup>* **νμ**

**221**

**Es = 1/2Ea** (12) follows why quark ingredients should have **2/3** and **1/3** fractional charges. From three portions of energy (charges), only two portions describe one type of charge, and the other one portion describes another charge, holding the requirement of discrete **Es = 1/2Ea** symmetry.

#### **6. The invariant translations within fermion-boson pairs**

By Wilczek's [17] opinion of getting symmetry and maintaining the balance of conserved quantum numbers, the extra particles should exist by an equal number of antiparticles. However, our theory predicts that invariant translation of ingredients of energy-momentum exchange interaction should not involve equal numbers of particle-antiparticle pairs but has to follow the condition of Eq. (12) **Es = 1/2Ea**. We think that the concept of equal numbers appeared from wrong identification of a particle as a point-like particle, which cannot produce identity for fermion. In accordance with the invariant translation (12), from one charged fermion, we can produce only half-neutral boson. Therefore, based on invariant particle-antiparticle translation (12), to get a neutral bosonic particle, we have to double the number of particles to produce a neutral boson:

$$\mathbf{2} \ (\mathbf{E}\mathbf{s} = \mathbf{1}/2\mathbf{E}\mathbf{a}) \to (\mathbf{2}\mathbf{E}\mathbf{s}^{\mathsf{t}} = \mathbf{E}\mathbf{a}^{\mathsf{t}}) \tag{15}$$

This operation is similar to quantum mechanic's doubling of wave function. However, in our case it is due to combining of energy portions, distributed in space and time phases to get full portion of energy at the origin in the form of boson. Elimination of dipole moment requires removal of charges in the space–time frame of matter, which requires decay of the space–time frame of ordinary matter and restoration of energy at the origin. However, the fractional charges of nucleons of baryon structure do not allow separation of quarks with elimination of charges. To eliminate this restriction, virtual particles with the fractional charges of baryon structure undergo coupling to pion families, which, as intermediate bosons, carry easy decay with production of neutral particles of the background gauge field.

The π-mesons generation through coupling of proton-antiproton or quarkantiquark pairs during decay of space–time frame was proven by experiments, carried out in Berkeley Center where it was observed that formation of neutral field, which could be accounted for neutral π-mesons, created by collisions of high-energy protons. In addition, it was shown that the neutral mesons decayed into two mesons with the lifetime of the order of **10**�**<sup>13</sup> s** or less [18].

The produced π-mesons family became intermediate spin zero bosons due to the decay of the space–time frame of matter, while the spin number is the product of energy-momentum exchange interaction within the space–time frame of ordinary matter. On this basis, the coupling particles get the performance of the neutral particles of gauge field. The doubling of particles (14) at **Ea = 0** reverses the performance of the forces due to the transition of energy conservation from space– time phase to energy phase with the change of sign (13).

The shift of symmetry from space–time frame local field to the background gauge field symmetry leads to the disappearance of spin and generation of gauge field particles due to the coupling of initial and local momentum in the form **(**�**Es/Es**) of Eq. (10).

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based… DOI: http://dx.doi.org/10.5772/intechopen.93087*

It is necessary to note a very important feature of translation of ingredients of Eq. (14) when the ingredients of this equation doubled. The half-integer fermions of this equation became integer carrier particles, while integer carrier particles became double integer carrier particles. Therefore, in energy phase the performance of forces holding space–time frame changes in the opposite order.

Based on model (10), depending on the energy flux to the space–time frame, the helicity of the ingredients of the space–time frame changes. In accordance with Eq. (14) when electromagnetic interactions turned off (**Ea = 0**), the difference between space and time phase particles disappears, and all the particles behave as integer number particles of the background gauge field. In this case, the chirality and handiness of particles gets the same left-handed direction. When local symmetry generator force is not available (**Ea = 0**), the momentum of matter space–time phase (10) transforms to the energy of the gauge field and gets a negative sign. Simply, "the ingredients of energy return to itself." In this case, Dirac's neutrinos transform to the neutral Majorana neutrinos having left helicity to the background gauge field where bosonic particles involve gamma rays, neutral fermions pair, and neutral neutrinos.

To understand the nature of particles and forces, we have to analyze the decay mechanism of produced pion families, where the W vector bosons were intermediate ingredients:

$$
\mathfrak{n}^+ \to \mathfrak{\mu}^+ + \mathfrak{\nu}\_{\mathfrak{\mu}} \tag{16}
$$

$$
\mathfrak{n}^- \to \mathfrak{\mu}^- + \mathfrak{\nu}\_{\mathfrak{\mu}}^- \tag{17}
$$

$$
\pi^0 \to 2\gamma \tag{18}
$$

The produced ingredients of the decay form the balance equation:

$$
\pi^+ \pi^- \pi^0 \to 2\chi + \mathfrak{\mu}^+/\mathfrak{\mu}^- + \nu\_{\mathfrak{\mu}/} \nu\_{\mathfrak{\mu}}^- \tag{19}
$$

The decay of the local gauge field to hold vector conservation produces a new vector, which comprises generation of intermediate W vector bosons from decay of π pions. The **2Es = Ea** code of the background gauge field requires equal numbers of electron and neutrino family pairs which is realized by the equal branching ratios of the decay of intermediate W vector bosons. Eq. **(**19**)** describes decay condition in average for muon family leptons.

The product stream composition generates composite of neutral particles, which exists in annihilation mode in the background gauge field to hold equation 2Es = Ea:

$$2\mathfrak{y} \leftrightarrow -\mathfrak{u}^+/\mathfrak{u}^- + \nu\_{\mathfrak{u}/}\nu\_{\mathfrak{u}}{}^- \tag{20}$$

In the discrete energy conservation mode, the particles cannot hold annihilation process for a long time. Coupling of gamma rays with the neutral particles leads to the generation of charge and electromagnetic force of local gauge field:

$$\left(2\mathfrak{\cdot} + \mathfrak{\mu}^{+}/\mathfrak{\mu}^{-} + \mathfrak{\nu}\_{\mathfrak{\mu}/}\mathfrak{\nu}\_{\mathfrak{\mu}}^{-} \leftrightarrow \mathfrak{\mu}^{-}/\mathfrak{\nu}\_{\mathfrak{\mu}}^{-} + \mathfrak{\mu}^{+}/\mathfrak{\nu}\_{\mathfrak{\mu}} + \text{Eq } \left(\mathbf{electromagnetic} \mathbf{figure}\right) \tag{21}\right)$$

The intermediate step in the symmetric translation from fermions to gauge field is the transformation of proton-antiproton pair to neutron-antineutron Majoranatype particles, which decompose to kaon family mesons. Due to the existence of three fractional proton-antiprotons, comprising other flavors of quarks, the decomposition of neutron-antineutron pair produces three kaon-type mesons. The decay products of other unidentified two kaons, which we may call **Kaon2** and **Kaon3**, can be described similarly by Eqs. (16)–(19).

The genetic code of supersymmetry **Es = 1/2Ea** explains this paradox. The antisymmetric wave function (13) holds the invariance of baryon performance through discrete symmetry, carried within background and local gauge fields. From

**Es = 1/2Ea** (12) follows why quark ingredients should have **2/3** and **1/3** fractional charges. From three portions of energy (charges), only two portions describe one type of charge, and the other one portion describes another charge, holding the

By Wilczek's [17] opinion of getting symmetry and maintaining the balance of conserved quantum numbers, the extra particles should exist by an equal number of antiparticles. However, our theory predicts that invariant translation of ingredients of energy-momentum exchange interaction should not involve equal numbers of particle-antiparticle pairs but has to follow the condition of Eq. (12) **Es = 1/2Ea**. We think that the concept of equal numbers appeared from wrong identification of a particle as a point-like particle, which cannot produce identity for fermion. In accordance with the invariant translation (12), from one charged fermion, we can produce only half-neutral boson. Therefore, based on invariant particle-antiparticle translation (12), to get a neutral bosonic particle, we have to double the number of

This operation is similar to quantum mechanic's doubling of wave function. However, in our case it is due to combining of energy portions, distributed in space and time phases to get full portion of energy at the origin in the form of boson. Elimination of dipole moment requires removal of charges in the space–time frame of matter, which requires decay of the space–time frame of ordinary matter and restoration of energy at the origin. However, the fractional charges of nucleons of baryon structure do not allow separation of quarks with elimination of charges. To eliminate this restriction, virtual particles with the fractional charges of baryon structure undergo coupling to pion families, which, as intermediate bosons, carry easy decay with production of neutral particles of the background gauge field. The π-mesons generation through coupling of proton-antiproton or quarkantiquark pairs during decay of space–time frame was proven by experiments, carried out in Berkeley Center where it was observed that formation of neutral field, which could be accounted for neutral π-mesons, created by collisions of high-energy protons. In addition, it was shown that the neutral mesons decayed into two mesons

The produced π-mesons family became intermediate spin zero bosons due to the decay of the space–time frame of matter, while the spin number is the product of energy-momentum exchange interaction within the space–time frame of ordinary matter. On this basis, the coupling particles get the performance of the neutral particles of gauge field. The doubling of particles (14) at **Ea = 0** reverses the performance of the forces due to the transition of energy conservation from space–

The shift of symmetry from space–time frame local field to the background gauge field symmetry leads to the disappearance of spin and generation of gauge field particles due to the coupling of initial and local momentum in the form

**2 Es** ð Þ! <sup>¼</sup> **<sup>1</sup>***=***2Ea 2Est** <sup>¼</sup> **Eat** ð Þ (15)

supersymmetric genetic code.

*Quantum Mechanics*

requirement of discrete **Es = 1/2Ea** symmetry.

particles to produce a neutral boson:

with the lifetime of the order of **10**�**<sup>13</sup> s** or less [18].

time phase to energy phase with the change of sign (13).

**(**�**Es/Es**) of Eq. (10).

**220**

**6. The invariant translations within fermion-boson pairs**

In accordance with the equation **Es = 1/2Ea**, for transformation of fermions to bosons (transformation of mass back to energy), we have to double the numbers of particles through coupling of space and time phases, carrying energy portions. In reverse order, for generation of fermions from bosons (generation of mass from energy), we have to separate space and time phases (12) to produce charge and reduce spin numbers of particles.

the electromagnetic field where the electron was placed in a positive-energy eigenstate to get decay into negative-energy eigenstates. However, such an

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based…*

without the application of relativistic quantum approach.

in the form of photons.

*DOI: http://dx.doi.org/10.5772/intechopen.93087*

of Dirac's electron "sea."

the supersymmetric feature of nature.

**223**

approach had a problem that the real electron would disappear by emitting energy

Based on our theory on discrete performance of nucleon ingredients, it is easy to show that generation of photons from fractional electron charges at coupling mode predicts existence of its antiparticle-positron. Model (9) presents electromagnetic interaction of space–time particle **(Es)**, particularly electron, with the electromagnetic field (**Ea)**, through deterministic energy-momentum exchange interaction

In the absence of electromagnetic field (**Es = 0)**, this interaction moves to the background energy field through merging of photon's fractional electric charges to particle-antiparticle pair **e/e** and **ν/ν** with the generation of a neutral current instead

For formulation of gauge field theory, instead of Dirac's relativistic approach, we

The basic principles of our supersymmetric theory replaced Dirac conditions through (a) and (b). The problem of Dirac's approach is that the space and time derivatives enter to the equation with the second order, which led to the loss of the original function and its first-order derivative. That is why Dirac's equation could not find the local position of an electron in motion in a deterministic way and used probability density. The second problem of Dirac equation is that he introduced to his equation relativistic energy-momentum relation in the form of linear space–time vector, similar to Sudarshan's V-A vector that could not produce integer spin carrier neutral particle field from half-integer spin carrying fermion particle. This was the reason for the theory to produce "electron sea." In accordance with the supersymmetric theory, fermion-showing performance as a particle in local space–time gauge phase became a field of neutral bosons of background gauge phase of energy. This is

In gauge energy phase, electron and positron coupling to **e/e** generates, together with the Majorana **ν /ν** neutrinos, a vacuum "sea" of neutral current. Therefore, the "Dirac sea of electrons" in reality is the gauge field of neutral particles. The energy of the field is finite and has a boundary within the space–time frame, existing through discrete shift between space–time and energy phases of energy conservation.

Particle physics does not provide any information why Dirac's neutrinos have to

transform to Majorana neutrinos. The exchange interaction (**Ea/Es** � **Es/Es)** of Eq. (10) determines the nature of neutrinos. When interaction with electromagnetic energy is off (**Ea = 0)**, the difference between **Es** in the denominator and nominator of Eq. (10) disappears. The **Es** of nominator presents local momentum, while the **Es** in the denominator describes initial momentum as genetic code of superposition. At (**Ea = 0)**, the momentum ingredients of expression **Es/Es** became equivalent and cancel each other with the disappearance of charges of quarks **(e**�**/ν**�**)/(e+/ν)** ! **(e/e)/(ν/ν)** and generation of Majorana neutrinos **(ν/ν**) and neutral **e/e** fermion pairs of gauge field in the form of bosons. In the energy phase, the gauge field Majorana particles became boson particles, and the chirality and the handiness get the same direction. *Transformation of fermions of local gauge field to the*

**9. Performance of Dirac's and Majorana neutrinos in model**

used classic principles: (a) formulation has to hold symmetric space and time derivatives, in relation to origin, and (b) energy-momentum exchange relation has to present the momentum and energy as the space and time parts of a space–time

vector instead of a four-momentum frame of Dirac's relativistic theory.

While our theory involves meson families as intermediate particles within halfinteger-integer particle transformations, we may compare our supersymmetry theory with the basic principles of Yukawa's meson theory. The background of Yukawa's theory is the spontaneous breaking of continuous symmetry [19] and involves interaction between scalar ϕ and a Dirac field *ψ*. Yukawa's vector in the form of pseudo-scalar field is the linear combination of nuclear force-electric dipole moment (**φφ0**) which is very similar to V-A theory [15]. Both theories cannot describe CP invariance of strong interactions, while the linear combination of vectors could not produce translation of interactions to the initial state.

In Yukawa model, meson is the force carrier, but in accordance with our theory, meson is the product of invariant translation from baryon frame and is the intermediate ingredient of the background gauge field where all the forces merged.

#### **7. The Yang-Mills theory and the mass gap of Yang-Mills theory**

The main feature of Yang-Mill theory [1, 20] is that to produce differentiable manifold it applies continuous elements of Lie group. Yang-Mills theory, using differentiable manifold of non-Abelian Lie group and continuous Lagrangian, tried to describe the behavior of elementary particles through the combination of electromagnetic and weak forces. The non-Abelian Lie group is opposite to discrete symmetry, while traditional differentiation is not applicable for discrete symmetry group. The Yang-Mills theory does not have mathematical formulation for proper matrix reduction to get the nonzero mass particles of the local gauge field.

The Yang-Mills theory does not explain why the weak force has continuous energy spectrum. Pauli [21] suggested the production of massless neutrino together with the electron to explain continuous spectrum, but this explanation was not valid because quantum field theories have no mechanism for translation of space–time fermions to the gauge field bosons, showing continuous energy spectrum. Yang and Mill had no choice and selected the only possible way—application of non-Abelian Lie group, having Lagrangian manifold. The other problem of Yang-Mills theory is the application of energy-momentum four-vector (R4 ), which leads to the V-A-type energy-momentum spectrum that produces a gap in energy between zero and some positive number.

However, the Yang-Mills theory is the only correct concept among all particle physics theories, used to describe strong interactions. The very important feature of the Yang-Mills theory is that it suggests simultaneous production of massless photons in addition to three massive bosons. Unfortunately, this excellent suggestion has no proper mathematics, which could describe invariant translation of gauge bosons to fermions of strong interactions.

#### **8. Dirac's relativistic quantum theory and problems of Dirac's "electron sea"**

Dirac [22] applied the relativistic theory to Schrodinger's equation to get relativistic wave function of electron motion. The problem of Dirac equation was negative energy solution. To solve this problem, Dirac assumed interaction of electron with

the electromagnetic field where the electron was placed in a positive-energy eigenstate to get decay into negative-energy eigenstates. However, such an approach had a problem that the real electron would disappear by emitting energy in the form of photons.

Based on our theory on discrete performance of nucleon ingredients, it is easy to show that generation of photons from fractional electron charges at coupling mode predicts existence of its antiparticle-positron. Model (9) presents electromagnetic interaction of space–time particle **(Es)**, particularly electron, with the electromagnetic field (**Ea)**, through deterministic energy-momentum exchange interaction without the application of relativistic quantum approach.

In the absence of electromagnetic field (**Es = 0)**, this interaction moves to the background energy field through merging of photon's fractional electric charges to particle-antiparticle pair **e/e** and **ν/ν** with the generation of a neutral current instead of Dirac's electron "sea."

For formulation of gauge field theory, instead of Dirac's relativistic approach, we used classic principles: (a) formulation has to hold symmetric space and time derivatives, in relation to origin, and (b) energy-momentum exchange relation has to present the momentum and energy as the space and time parts of a space–time vector instead of a four-momentum frame of Dirac's relativistic theory.

The basic principles of our supersymmetric theory replaced Dirac conditions through (a) and (b). The problem of Dirac's approach is that the space and time derivatives enter to the equation with the second order, which led to the loss of the original function and its first-order derivative. That is why Dirac's equation could not find the local position of an electron in motion in a deterministic way and used probability density. The second problem of Dirac equation is that he introduced to his equation relativistic energy-momentum relation in the form of linear space–time vector, similar to Sudarshan's V-A vector that could not produce integer spin carrier neutral particle field from half-integer spin carrying fermion particle. This was the reason for the theory to produce "electron sea." In accordance with the supersymmetric theory, fermion-showing performance as a particle in local space–time gauge phase became a field of neutral bosons of background gauge phase of energy. This is the supersymmetric feature of nature.

In gauge energy phase, electron and positron coupling to **e/e** generates, together with the Majorana **ν /ν** neutrinos, a vacuum "sea" of neutral current. Therefore, the "Dirac sea of electrons" in reality is the gauge field of neutral particles. The energy of the field is finite and has a boundary within the space–time frame, existing through discrete shift between space–time and energy phases of energy conservation.

#### **9. Performance of Dirac's and Majorana neutrinos in model**

Particle physics does not provide any information why Dirac's neutrinos have to transform to Majorana neutrinos. The exchange interaction (**Ea/Es** � **Es/Es)** of Eq. (10) determines the nature of neutrinos. When interaction with electromagnetic energy is off (**Ea = 0)**, the difference between **Es** in the denominator and nominator of Eq. (10) disappears. The **Es** of nominator presents local momentum, while the **Es** in the denominator describes initial momentum as genetic code of superposition. At (**Ea = 0)**, the momentum ingredients of expression **Es/Es** became equivalent and cancel each other with the disappearance of charges of quarks **(e**�**/ν**�**)/(e+/ν)** ! **(e/e)/(ν/ν)** and generation of Majorana neutrinos **(ν/ν**) and neutral **e/e** fermion pairs of gauge field in the form of bosons. In the energy phase, the gauge field Majorana particles became boson particles, and the chirality and the handiness get the same direction. *Transformation of fermions of local gauge field to the*

In accordance with the equation **Es = 1/2Ea**, for transformation of fermions to bosons (transformation of mass back to energy), we have to double the numbers of particles through coupling of space and time phases, carrying energy portions. In reverse order, for generation of fermions from bosons (generation of mass from energy), we have to separate space and time phases (12) to produce charge and

While our theory involves meson families as intermediate particles within halfinteger-integer particle transformations, we may compare our supersymmetry the-

In Yukawa model, meson is the force carrier, but in accordance with our theory, meson is the product of invariant translation from baryon frame and is the intermediate ingredient of the background gauge field where all the forces merged.

The main feature of Yang-Mill theory [1, 20] is that to produce differentiable manifold it applies continuous elements of Lie group. Yang-Mills theory, using differentiable manifold of non-Abelian Lie group and continuous Lagrangian, tried to describe the behavior of elementary particles through the combination of electromagnetic and weak forces. The non-Abelian Lie group is opposite to discrete symmetry, while traditional differentiation is not applicable for discrete symmetry group. The Yang-Mills theory does not have mathematical formulation for proper

The Yang-Mills theory does not explain why the weak force has continuous energy spectrum. Pauli [21] suggested the production of massless neutrino together with the electron to explain continuous spectrum, but this explanation was not valid because quantum field theories have no mechanism for translation of space–time fermions to the gauge field bosons, showing continuous energy spectrum. Yang and Mill had no choice and selected the only possible way—application of non-Abelian Lie group, having Lagrangian manifold. The other problem of Yang-Mills theory is the application of

), which leads to the V-A-type energy-momentum

ory with the basic principles of Yukawa's meson theory. The background of Yukawa's theory is the spontaneous breaking of continuous symmetry [19] and involves interaction between scalar ϕ and a Dirac field *ψ*. Yukawa's vector in the form of pseudo-scalar field is the linear combination of nuclear force-electric dipole moment (**φφ0**) which is very similar to V-A theory [15]. Both theories cannot describe CP invariance of strong interactions, while the linear combination of vectors could not produce translation of interactions to the initial state.

**7. The Yang-Mills theory and the mass gap of Yang-Mills theory**

matrix reduction to get the nonzero mass particles of the local gauge field.

spectrum that produces a gap in energy between zero and some positive number. However, the Yang-Mills theory is the only correct concept among all particle physics theories, used to describe strong interactions. The very important feature of the Yang-Mills theory is that it suggests simultaneous production of massless photons in addition to three massive bosons. Unfortunately, this excellent suggestion has no proper mathematics, which could describe invariant translation of gauge

**8. Dirac's relativistic quantum theory and problems of Dirac's**

Dirac [22] applied the relativistic theory to Schrodinger's equation to get relativistic wave function of electron motion. The problem of Dirac equation was negative energy solution. To solve this problem, Dirac assumed interaction of electron with

reduce spin numbers of particles.

*Quantum Mechanics*

energy-momentum four-vector (R4

bosons to fermions of strong interactions.

**"electron sea"**

**222**

#### *background gauge bosons as the intermediate particles is the necessary step for invariant translations of strong interactions.*

of neutrinos, and the produced particles remained spin ½ fermions. The neutrinos helicity was the main problem of all of the field theories which did not allow to describe fundamental laws of nature in a proper way. The question why nature has no right-handed neutrino produced an opinion [25] that "God decided that Nature should be left handed." Due to this problem, particle physics theories suggest that nature respects parity with regard to all the fundamental forces with the exception

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based…*

Based on our model (9), the generation of free neutrinos and antineutrinos takes place at cutoff electromagnetic interactions (**Ea = 0)**, which reverses the momentum to the background state. The problem is that Weyl's theory determines the spin only relative to the positive momentum vector, and

individual ½ spin carrier massless Weyl spinors violate conservation of parity. For this reason, Pauli specified Weyl spinors "unphysical" [21]. Weyl's theory could not combine neutrinos to hold parity conservation in the form pair of virtual Majorana

Weyl's theory cannot explain why production of his spinors in the unitary transformations takes place only in the presence of half angle. Hamilton rotation about some axis, in a similar way, connects half angle and the Pauli matrixes. The

Model (10) shows that at **Es = 1/2Ea**, the invariant translation within complex space–time coordinates are connected within tangent 45 which describes space– time symmetry **t1 ΔS=S1 Δt** in the form of coordinates **y=x** symmetry. This is the half-angle mystery of Weyl and Hamilton translations. The two-dimensional space– time frame in association with the **Es = 1/2 Ea** discrete energy-momentum symme-

Hamiltonian and momentum are the adjoint elements of the Lie Algebra group that generate linear transition in space and time. Model (10) presents the nonlinear energy-momentum exchange relation as the adjoint elements of three-dimensional **SO (3)** group and shows that Hamiltonian linear

transformation alone cannot do invariant translation. Due to the involvement of symmetry generator force **Ea**, the translation has to be with the change of

dimension. The invariant translation requires conjugation of invariant space–time frame **SU (2)** with the three-dimensional energy-mass exchange transformation through **SO (3)** group where change of space–time dimension is the driving force of

Under unitary transformations, one rotation (360°) does not bring the state of a body to the origin. One rotation brings **SU (2)** x **SO (3)** local gauge symmetry to **U (1)** matrix with simultaneous transformation of a three-dimensional particle frame to linear gauge field. Therefore, full translation of opposite phases holds condition: **SU (2) x SO (3)/U (1).** Doubling the spin numbers of quark ingredients through coupling of space and time portions of energy **Es = 1/2Ea** to **2Es = Ea** produces unstable pions which produce non-charged boson-like ingredients of the

The second transformation with the reversing of **U (1)** symmetry group brings the linear gauge field back to the space–time **(SU (2)** x **SO (3))** frame of three particles of baryon structure**.** Coupling of neutral **e/e**, **ν/ν**, and **ꝩ**/**ꝩ** ingredients in such a translation generates quarks of baryon structure. The invariance between bosons and fermions in the form of strong interactions is possible only in discrete

With the reversed momentum line (**Ea = 0**), the antineutrino changes its helicity and becomes the left-handed particle, which leads to the coupling of two neutrinos in the form of bosonic twin particles. The right-handed neutrino would block generation of **U (1)** field and its translation back to **SO (3)** matrix, that is why

of the weak interaction, which involves neutrinos.

*DOI: http://dx.doi.org/10.5772/intechopen.93087*

presence of half angle in both cases was unavoidable [25].

particles.

try carries this translation.

background gauge field **U (1).**

mode with the change of space–time dimensions.

translations.

**225**

The mixture of neutral electron-positron pairs and Majorana neutrinos generates the spin 2 neutral particles of graviton of spin zero gauge field, carrying gravitation force to the background vacuum with the velocity faster than electromagnetic force in any medium. Therefore, gravitation force appears in reverse order from electromagnetic energy through coupling of entangled space and time portions of energy to restore it at the background vacuum (**Ea = 0)**. It is not "spooky action at a distance" [23] but coupling of entangled non-separable portions of energy, existing in different forms.

The mass of Majorana neutrinos (Majorana mass) in the background gauge field is very low, but they became massive as Dirac neutrinos of baryon structure due to the entanglement with the charged electron family particles. Due to the decay of space–time phase at (**Ea = 0**), the particles of gauge field has the continuum spectrum. The **e/e** and **ν/ν** pars have no independent existence, but with the gamma rays, they form dark matter and energy content with ratio 33 and 66%, holding **Ea = 2Es** gauge field frame of energy conservation.

At vacuum expectation value takes place discrete shift of gauge field energy back to the local gauge field of space–time. Majorana neutrinos became again Dirac neutrinos with the generation of charge and massive particles of exchange interaction **(EaEs)/Es**.

#### **10. The problems of Weyl spinors and quantum field theory**

Quantum field theory suggests existence of massless half spin fermions and provides relativistic Weyl equation for description of massless half spin fermions. Due to the connection of Weyl's spinor to Dirac's theory of half spin electron, Weyl spinors describes Dirac fermions in the form of two ½ spin massless fermions. Quantum field theory does not explain physical nature of Weyl's massless spin ½ particles, while spin ½ fermions in Dirac structure are massive particles. Dirac's theory describes energymomentum relation as a continuous function that is why physical nature of predicted massless ½ spin fermions remains open. In accordance with our theory, neutrinos, existing in pair with the electron and positron, as Dirac fermions in quark's structure, in energy phase transform to integer spin "twin" Majorana particles. In mathematics, usually such an inversion has to meet requirements of spinors.

The spinor is the mathematical operation [24], which produces vector space by addition of vectors together or multiplication by numbers, called scalar. The vector addition or scalar multiple operation must satisfy requirements, called axioms. The real vector space presents a physical quantity such as force and multiplication of a force by a real multiplier which produces another force vector.

Spinor, as a vector, exhibits inversion, when a physical system constantly rotates through a full turn (360°). In the following chapters, we will explain scalar multiplication in strong interactions and will provide mathematical framework, carrying the inversion of particles from half spin to integer spin neutral particles with the simultaneous change in the nature of existing force.

#### **11. Development of new gauge field as the frame for discrete conservation of energy**

#### **11.1 General principles**

The idea of a gauge theory appeared from Weyl spinors, but Weyl's theory, as we mentioned, could not produce the gauge scalar field due to the helicity problem

#### *Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based… DOI: http://dx.doi.org/10.5772/intechopen.93087*

of neutrinos, and the produced particles remained spin ½ fermions. The neutrinos helicity was the main problem of all of the field theories which did not allow to describe fundamental laws of nature in a proper way. The question why nature has no right-handed neutrino produced an opinion [25] that "God decided that Nature should be left handed." Due to this problem, particle physics theories suggest that nature respects parity with regard to all the fundamental forces with the exception of the weak interaction, which involves neutrinos.

Based on our model (9), the generation of free neutrinos and antineutrinos takes place at cutoff electromagnetic interactions (**Ea = 0)**, which reverses the momentum to the background state. The problem is that Weyl's theory determines the spin only relative to the positive momentum vector, and individual ½ spin carrier massless Weyl spinors violate conservation of parity. For this reason, Pauli specified Weyl spinors "unphysical" [21]. Weyl's theory could not combine neutrinos to hold parity conservation in the form pair of virtual Majorana particles.

Weyl's theory cannot explain why production of his spinors in the unitary transformations takes place only in the presence of half angle. Hamilton rotation about some axis, in a similar way, connects half angle and the Pauli matrixes. The presence of half angle in both cases was unavoidable [25].

Model (10) shows that at **Es = 1/2Ea**, the invariant translation within complex space–time coordinates are connected within tangent 45 which describes space– time symmetry **t1 ΔS=S1 Δt** in the form of coordinates **y=x** symmetry. This is the half-angle mystery of Weyl and Hamilton translations. The two-dimensional space– time frame in association with the **Es = 1/2 Ea** discrete energy-momentum symmetry carries this translation.

Hamiltonian and momentum are the adjoint elements of the Lie Algebra group that generate linear transition in space and time. Model (10) presents the nonlinear energy-momentum exchange relation as the adjoint elements of three-dimensional **SO (3)** group and shows that Hamiltonian linear transformation alone cannot do invariant translation. Due to the involvement of symmetry generator force **Ea**, the translation has to be with the change of dimension. The invariant translation requires conjugation of invariant space–time frame **SU (2)** with the three-dimensional energy-mass exchange transformation through **SO (3)** group where change of space–time dimension is the driving force of translations.

Under unitary transformations, one rotation (360°) does not bring the state of a body to the origin. One rotation brings **SU (2)** x **SO (3)** local gauge symmetry to **U (1)** matrix with simultaneous transformation of a three-dimensional particle frame to linear gauge field. Therefore, full translation of opposite phases holds condition: **SU (2) x SO (3)/U (1).** Doubling the spin numbers of quark ingredients through coupling of space and time portions of energy **Es = 1/2Ea** to **2Es = Ea** produces unstable pions which produce non-charged boson-like ingredients of the background gauge field **U (1).**

The second transformation with the reversing of **U (1)** symmetry group brings the linear gauge field back to the space–time **(SU (2)** x **SO (3))** frame of three particles of baryon structure**.** Coupling of neutral **e/e**, **ν/ν**, and **ꝩ**/**ꝩ** ingredients in such a translation generates quarks of baryon structure. The invariance between bosons and fermions in the form of strong interactions is possible only in discrete mode with the change of space–time dimensions.

With the reversed momentum line (**Ea = 0**), the antineutrino changes its helicity and becomes the left-handed particle, which leads to the coupling of two neutrinos in the form of bosonic twin particles. The right-handed neutrino would block generation of **U (1)** field and its translation back to **SO (3)** matrix, that is why

*background gauge bosons as the intermediate particles is the necessary step for invariant*

The mixture of neutral electron-positron pairs and Majorana neutrinos generates the spin 2 neutral particles of graviton of spin zero gauge field, carrying gravitation force to the background vacuum with the velocity faster than electromagnetic force in any medium. Therefore, gravitation force appears in reverse order from electromagnetic energy through coupling of entangled space and time portions of energy to restore it at the background vacuum (**Ea = 0)**. It is not "spooky action at a distance" [23] but coupling of entangled non-separable portions of energy, existing in different forms. The mass of Majorana neutrinos (Majorana mass) in the background gauge field is very low, but they became massive as Dirac neutrinos of baryon structure due to the entanglement with the charged electron family particles. Due to the decay of space–time phase at (**Ea = 0**), the particles of gauge field has the continuum spectrum. The **e/e** and **ν/ν** pars have no independent existence, but with the gamma rays, they form dark matter and energy content with ratio 33 and 66%, holding

At vacuum expectation value takes place discrete shift of gauge field energy back to the local gauge field of space–time. Majorana neutrinos became again Dirac neutrinos with the generation of charge and massive particles of exchange interac-

Quantum field theory suggests existence of massless half spin fermions and provides relativistic Weyl equation for description of massless half spin fermions. Due to the connection of Weyl's spinor to Dirac's theory of half spin electron, Weyl spinors describes Dirac fermions in the form of two ½ spin massless fermions. Quantum field theory does not explain physical nature of Weyl's massless spin ½ particles, while spin ½ fermions in Dirac structure are massive particles. Dirac's theory describes energymomentum relation as a continuous function that is why physical nature of predicted massless ½ spin fermions remains open. In accordance with our theory, neutrinos, existing in pair with the electron and positron, as Dirac fermions in quark's structure, in energy phase transform to integer spin "twin" Majorana particles. In mathematics,

The spinor is the mathematical operation [24], which produces vector space by addition of vectors together or multiplication by numbers, called scalar. The vector addition or scalar multiple operation must satisfy requirements, called axioms. The real vector space presents a physical quantity such as force and multiplication of a

Spinor, as a vector, exhibits inversion, when a physical system constantly rotates through a full turn (360°). In the following chapters, we will explain scalar multiplication in strong interactions and will provide mathematical framework, carrying the inversion of particles from half spin to integer spin neutral particles with the

The idea of a gauge theory appeared from Weyl spinors, but Weyl's theory, as we mentioned, could not produce the gauge scalar field due to the helicity problem

**10. The problems of Weyl spinors and quantum field theory**

usually such an inversion has to meet requirements of spinors.

force by a real multiplier which produces another force vector.

**11. Development of new gauge field as the frame for discrete**

simultaneous change in the nature of existing force.

**conservation of energy**

**11.1 General principles**

**224**

*translations of strong interactions.*

*Quantum Mechanics*

tion **(EaEs)/Es**.

**Ea = 2Es** gauge field frame of energy conservation.

nature does not allow its existence. Weyl's theory due to the wrong helicity misses these translations.

The integer spin carrying particles, produced at zero electromagnetic interactions within baryon's space–time frame, can hold invariance of gauge field only in

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based…*

The symmetries of space–time (**Es = 1/2Ea)** and energy (2**Es = Ea)** phases do not have independent existence and only in conjugation carry discrete conservation of

We assume that **2Es = Ea** frame of the background gauge field involves a combination of elastic (Thomson effect) and inelastic scattering (Compton effect) where inelastic scattering gradually transforms to elastic scattering. At background vacuum expectation value takes place translation of the background gauge field energy to space–time frame of local gauge field by elastic scattering, which involves absorbing of gamma rays by the virtual matter bosons. This process shifts the continuous spectrum of longitudinal waves of the background field of bosons to the discrete spectrum of transference waves of charged matter particles. This is the

According to quantum mechanics, vacuum energy without renormalization mathematically is infinite. However, this statement is true only if the background's gauge field energy has no shift to the local gauge field of matter's space–time frame. During shift of the background gauge field's energy to the local space–time field, Majorana neutrinos transform to Dirac neutrinos with the transformation of color

In accordance with our theory, if field does not change, it cannot hold energy conservation and symmetry within reversible dynamic translations. Energy can exist only through propagation in space–time frame, and in reverse order, space– time is the matter product of energy distribution. On this basis, conjugated existence of background energy and local gauge matter fields is the necessary condition

The energy-momentum exchange relation of the model (12) in the form of eigenvector generates exchange of particle with the field. The energy-momentum exchange relation of eigenvector (12) describes the relation of two fields, such as electromagnetic-gravitation fields, which carry invariant translation to each other. Electromagnetic force in the form of **Ea** can be a vector field and at the same time photon particle. At **Ea = 0**, the electromagnetic force disappears as a field/particle

Model (10) describes local gauge field **S1/t1**, which carries energy at each point of space–time. Local gauge field carries electromagnetic force in space–time frame in the form of energy-momentum content and strength of electromagnetic field determined by its coupling with the local space–time field. Model (10) combines all types of interactions and translates them to each other through energy-momentum exchange interaction. In this case, background/local gauge field of articles appear as

The vector space, as specified in mathematics, moves through plane wave, which is field. By the requirement of vector space [24], the field where the vector has to move requires existence of two equivalent field functions that determine the field value. These functions involve two parameters, which are time and displacement along the direction. In accordance with our model (13), the symmetry of energy portions, distributed evenly within space and time phases, generates two

process, which eliminates generation of ultraviolent divergences.

**11.3 Mechanism of conjugation of background and local gauge fields**

charges to the electric charges of quarks.

*DOI: http://dx.doi.org/10.5772/intechopen.93087*

and transforms to gauge field of boson ingredients.

for conservation of energy.

the "two worlds of particles."

**227**

ð Þ **e** � *=***ν**� *=*ð Þ\$ **e** þ *=***ν** ð Þ **e***=***e** *=*ð Þþ **ν***=***ν ꝩ***=***ꝩ** (23)

discrete mode:

energy.

The other problem of quantum mechanics is that it eliminates participation of neutrinos in strong interactions due to the absence of charges. However, our theory shows that neutrinos are the necessary ingredients for generation of strong force, while coupling with the neutral **e/e** pairs generates formation of charges through their reproduced right–left helicity in **SO (3)** group.

Without conjugation of **SU (2)** and **SO (3**) symmetry groups, the description of chiral symmetry is not possible. At **Ea = 0**, Dirac particles transform to Majorana neutral massless particles which eliminates the difference between handiness and chirality.

#### **11.2 Mathematical framework of discrete gauge field**

Gauge, in common sense [26] is a measurement of a relative position of a system with reference to another abstract system to determine boundary of measurement. The gauge theory has no mathematical framework relative to the proper reference frame for measurement of change. In this aspect, the gauge theory has the same reference frame problem of classic physics.

Based on literature [26], the gauge symmetry has specification, as "is a lack of change when some field being applied." The meaning of this statement is that the measurable quantity after the change looks the same. Linearly differentiable Lagrangian of non-Abelian algebra due to the absence of the space–time frame cannot provide a mathematical formulation on how the gauge field after the change may look the same.

The theory, which we apply, provides a mathematical framework to the gauge theory to measure a quantity, relative to its initial superposition state. We suggest that change of energy and momentum in discrete mode generates the dynamical operator of gauge field, which describes the measurement in relation to the initial origin with the integer numbers of energy portions.

When the superposition of a gauge field after displacement within space, time, and energy looks the same, the invariant transformation produces invariance for all the inner ingredients of the change **ΔS=S1**, **Δt=t1**, and **Ea –Es = Es** with the realization of condition **1=1** (9).

The genetic code of exchange interactions **Es = 1/2Ea** keeps the discrete symmetry of force carrier and electrically charged ingredients of space–time at different spin numbers (12). In the energy phase (13) of gauge field, the genetic code **Es = 1/2Ea** undergoes multiplication by scalar **2** to **Ea = 2Es** which holds the discrete symmetry within color charge ingredients of gauge field, leaving spin untouched. When the electromagnetic interactions is off (**Ea = 0)**, coupling of space and time portions of energy generates transformation of half spin matter fermions to integer bosons ν**/ν + e/e + ꝩ/ꝩ** family of background vacuum:

$$\frac{\mathbf{dS}}{\mathbf{S\_1}} = \frac{\mathbf{dt}}{\mathbf{t\_1}} \left( - \frac{\mathbf{E\_s}}{\mathbf{E\_s}} \right) \tag{22}$$

When electromagnetic energy is off (Ea = 0), merging of space and time portions of energy generates left side helicity for all the non-charged ingredients of gauge field of background vacuum. At this condition, all the half-integer fermions merging with their own antiparticles form neutral integer spin carrying bosonic particles. Eq. (22) is the equation of vacuum, where space and time portions of energy merging generates a one-dimensional space–time frame of vacuum.

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based… DOI: http://dx.doi.org/10.5772/intechopen.93087*

The integer spin carrying particles, produced at zero electromagnetic interactions within baryon's space–time frame, can hold invariance of gauge field only in discrete mode:

$$(\mathbf{e} - / \boldsymbol{\nu} -)/(\mathbf{e} + / \boldsymbol{\nu}) \leftrightarrow (\mathbf{e}/\mathbf{e})/(\boldsymbol{\nu}/\boldsymbol{\nu}) + \mathbf{p}/\mathbf{p} \tag{23}$$

The symmetries of space–time (**Es = 1/2Ea)** and energy (2**Es = Ea)** phases do not have independent existence and only in conjugation carry discrete conservation of energy.

We assume that **2Es = Ea** frame of the background gauge field involves a combination of elastic (Thomson effect) and inelastic scattering (Compton effect) where inelastic scattering gradually transforms to elastic scattering. At background vacuum expectation value takes place translation of the background gauge field energy to space–time frame of local gauge field by elastic scattering, which involves absorbing of gamma rays by the virtual matter bosons. This process shifts the continuous spectrum of longitudinal waves of the background field of bosons to the discrete spectrum of transference waves of charged matter particles. This is the process, which eliminates generation of ultraviolent divergences.

According to quantum mechanics, vacuum energy without renormalization mathematically is infinite. However, this statement is true only if the background's gauge field energy has no shift to the local gauge field of matter's space–time frame. During shift of the background gauge field's energy to the local space–time field, Majorana neutrinos transform to Dirac neutrinos with the transformation of color charges to the electric charges of quarks.

#### **11.3 Mechanism of conjugation of background and local gauge fields**

In accordance with our theory, if field does not change, it cannot hold energy conservation and symmetry within reversible dynamic translations. Energy can exist only through propagation in space–time frame, and in reverse order, space– time is the matter product of energy distribution. On this basis, conjugated existence of background energy and local gauge matter fields is the necessary condition for conservation of energy.

The energy-momentum exchange relation of the model (12) in the form of eigenvector generates exchange of particle with the field. The energy-momentum exchange relation of eigenvector (12) describes the relation of two fields, such as electromagnetic-gravitation fields, which carry invariant translation to each other. Electromagnetic force in the form of **Ea** can be a vector field and at the same time photon particle. At **Ea = 0**, the electromagnetic force disappears as a field/particle and transforms to gauge field of boson ingredients.

Model (10) describes local gauge field **S1/t1**, which carries energy at each point of space–time. Local gauge field carries electromagnetic force in space–time frame in the form of energy-momentum content and strength of electromagnetic field determined by its coupling with the local space–time field. Model (10) combines all types of interactions and translates them to each other through energy-momentum exchange interaction. In this case, background/local gauge field of articles appear as the "two worlds of particles."

The vector space, as specified in mathematics, moves through plane wave, which is field. By the requirement of vector space [24], the field where the vector has to move requires existence of two equivalent field functions that determine the field value. These functions involve two parameters, which are time and displacement along the direction. In accordance with our model (13), the symmetry of energy portions, distributed evenly within space and time phases, generates two

nature does not allow its existence. Weyl's theory due to the wrong helicity misses

their reproduced right–left helicity in **SO (3)** group.

**11.2 Mathematical framework of discrete gauge field**

reference frame problem of classic physics.

origin with the integer numbers of energy portions.

bosons ν**/ν + e/e + ꝩ/ꝩ** family of background vacuum:

realization of condition **1=1** (9).

The other problem of quantum mechanics is that it eliminates participation of neutrinos in strong interactions due to the absence of charges. However, our theory shows that neutrinos are the necessary ingredients for generation of strong force, while coupling with the neutral **e/e** pairs generates formation of charges through

Without conjugation of **SU (2)** and **SO (3**) symmetry groups, the description of chiral symmetry is not possible. At **Ea = 0**, Dirac particles transform to Majorana neutral massless particles which eliminates the difference between handiness and

Gauge, in common sense [26] is a measurement of a relative position of a system with reference to another abstract system to determine boundary of measurement. The gauge theory has no mathematical framework relative to the proper reference frame for measurement of change. In this aspect, the gauge theory has the same

Based on literature [26], the gauge symmetry has specification, as "is a lack of change when some field being applied." The meaning of this statement is that the measurable quantity after the change looks the same. Linearly differentiable Lagrangian of non-Abelian algebra due to the absence of the space–time frame cannot provide a mathematical formulation on how the gauge field after the change

The theory, which we apply, provides a mathematical framework to the gauge theory to measure a quantity, relative to its initial superposition state. We suggest that change of energy and momentum in discrete mode generates the dynamical operator of gauge field, which describes the measurement in relation to the initial

When the superposition of a gauge field after displacement within space, time, and energy looks the same, the invariant transformation produces invariance for all the inner ingredients of the change **ΔS=S1**, **Δt=t1**, and **Ea –Es = Es** with the

The genetic code of exchange interactions **Es = 1/2Ea** keeps the discrete symmetry of force carrier and electrically charged ingredients of space–time at different

**Es = 1/2Ea** undergoes multiplication by scalar **2** to **Ea = 2Es** which holds the discrete symmetry within color charge ingredients of gauge field, leaving spin untouched. When the electromagnetic interactions is off (**Ea = 0)**, coupling of space and time portions of energy generates transformation of half spin matter fermions to integer

spin numbers (12). In the energy phase (13) of gauge field, the genetic code

**dS S1**

<sup>¼</sup> **dt t1**

When electromagnetic energy is off (Ea = 0), merging of space and time portions of energy generates left side helicity for all the non-charged ingredients of gauge field of background vacuum. At this condition, all the half-integer fermions merging with their own antiparticles form neutral integer spin carrying bosonic particles. Eq. (22) is the equation of vacuum, where space and time portions of energy merging generates a one-dimensional space–time frame of vacuum.

� **Es Es** 

(22)

these translations.

*Quantum Mechanics*

chirality.

may look the same.

**226**

field functions of negative displacement by the conjugation of space and time variables:

$$\mathbf{s\_1}\Delta\mathbf{t} = -\mathbf{t\_1}\Delta\mathbf{s} \tag{24}$$

**11.4 Invariant translation of symmetries within background/local gauge fields**

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based…*

Background gauge/local gauge fields exist in the form of field-anti-field pair. Due to the energy-momentum non-commutation, the local gauge field is the non-Abelian, while background gauge is Abelian field. Generation of non-Abelian local gauge field from background Abelian's gauge field is not spontaneously symmetry breaking. The local gauge field at **Ea = 0** of model (9) merges with the background gauge field, as particle-antiparticle pair. In this case, the difference between particle and field disappears. The background and local gauge fields are connected through **SO (3)** rotational matrix which carries invariant translation of particle to field. Quantum mechanics mediates physical quantity by the square of the wave function, but **SO (3)** group of model (9) mediates physical quantity of the background gauge field by coupling of space–time portions of energy, carried in the form of matter–

Model (9) suggests that CP symmetry of strong interactions is in hold only through cross product of **SU (2) x SO (3)** symmetry groups within two transformations: one is charge cancelation translation, and the second is parity transformation. Yang-Mills [24, 28] attempted to apply gauge theory to the strong interactions through elevating of global symmetry to local gauge symmetry, but this attempt produced symmetry breaking. Without conjugation of **SU (2)** matrix of space–time frame and energy-momentum exchange interactions **SO (3)** (9), the background gauge field **U (1**) cannot carry invariant translation of mass to the opposite phase of

By Glashow's opinion [2] electromagnetism is mediated not only by photons; it

In our theory electromagnetism and gravitation are unified within **SU (2) x SO (3)** symmetry of local gauge field which involves unification of charges as the internal products of baryon's space–time frame. The genetic code of baryon particles **Es = 1/2 Ea** holds all the internal conservation laws: baryon conservation, isospin conservation, hypercharge conservation, and boson-fermion spin invariant translation. It is known that the hypercharge of **SU (3)** symmetry is one of two quantum numbers of the hadrons and alongside with isospin *I***<sup>3</sup>** follows the formula: **Q=J3 + 1/2Y**. For multiples

According to model (9), at local gauge field, the hypercharge current coupling is the condition **Es = 1/2 Ea** which describes local space–time symmetry at **J3 = 0**. At the background gauge field, the hypercharge conservation **Ea = 2Es** describes multiple bosons, similar to **J = 2Q.** Based on Eqs. (8)–(12), the discrete conservation of

The equation **ΔF=0** describes discrete nonvanishing energy state of spin zero

, ΔF ¼ **0** (29)

arises from the requirement of local gauge invariance. However, based on our theory, this statement is true only partly because the role of local gauge field is reversible and symmetric. The local gauge field is needed for generation of electromagnetic interaction and cancelation it takes place by gravitation for discrete conservation of energy within space–time of baryon frame. The **SO (3)** symmetry group of gauge field translates electromagnetic force to the gravitation force. Therefore, without gravitation force, it is impossible to get invariant performance of strong interactions. The Standard Model, as Kibble showed [29], did not find place for gravity, and that is why it cannot not explain why the elementary particles come in three families with very similar structure but wildly differing masses.

antimatter pair.

local gauge field of strong interactions.

*DOI: http://dx.doi.org/10.5772/intechopen.93087*

of particles, the hypercharge gets formulation **J = 2Q**.

energy at the background gauge field produces condition:

<sup>Δ</sup><sup>F</sup> <sup>¼</sup> **<sup>Δ</sup><sup>S</sup>** *Δt* þ **S1 t1** ; **ΔS** *<sup>Δ</sup><sup>t</sup>* ¼ � **S1 t1**

boson's condensate of the background gauge field.

**229**

Therefore, the ingredients of Eq. (13) generate two equivalent functions:

$$\mathbf{F\_{t}(s\_{1}, \Delta t)} = -\mathbf{F\_{s}(t\_{1}, \Delta s)}\tag{25}$$

The function **Fs** describes displacement in space, while the function **Ft** describes duration of change. When the values of field function are vectors, the plane wave is longitudinal. The space and time portions of energy in the background gauge field form a one-dimensional space–time frame that is why the plane wave in this case is longitudinal. Multiplication of equation **(Es = 1/2Ea)** by scalar **2** leads to the formation of neutral particles **e/e**, **ν/ν**, and **ꝩ/ꝩ** of the background gauge field **2Es = Ea** in the form of spinors, similar to Hamilton quaternions spinors [25]. The ingredients of Hamilton's equation **(I; J; K)** are imaginary quantities, while the products of our model are virtual particles:

$$\mathbf{J}^2 = \mathbf{J}^2 = \mathbf{K}^2 = \mathbf{ij}\mathbf{k} = -\mathbf{1} \, (\mathbf{Hamiltonian}) \tag{26}$$

$$\mathbf{e}/\mathbf{e} = \mathbf{v}/\mathbf{v} = \mathbf{p}/\mathbf{p} = (\mathbf{e}\mathbf{v}\mathbf{p}) = -\mathbf{1} \tag{27}$$

$$(\mathbf{Es} = \mathbf{1}/2\mathbf{Ea}) \left(\mathbf{Space-time phase}\right) \to \text{Inversion to energy phase } (\mathbf{2 Es = Ea}) \tag{28}$$

The equation **(2Es = Ea)** produces a new vector where the scalar is the real number. The inversion transforms integer spin electromagnetic force to the other force being integer **2** spin carrying force. The new force is the gravitation, which with continuous longitudinal wave moves to the background through conjugation of ingredient (**eνꝩ)** of produced neutral spinors.

It is necessary to note that generation of inversion vector space for transformation of energy conservation from space–time phase to energy phase meets all the requirements, required for scalar multiplication procedure, given in the form of axioms [24]. One of the requirements of axioms is the condition **x+(**�**x) = 0**, which is in hold within discrete annihilation of space and time variables (14). Due to the conservation of energy within discrete energy and space–time phases, an event comes to the origin after two full rotations (720°). The generated field has an algebra of zero-dimensional geometric spinor with one-directional helicity to the background origin.

Model (9) describes conservation of energy through non-unitary space–time variables, which unifies fields, particles, and forces within non-unitary energy portions allowing transformation of all the identities to each other. The background plasma-like gauge field can hold the discrete symmetry only through coupling with the local gauge field. The local gauge field of virtual matter, which is the combination of electric and magnetic fields, holds interaction of electrically charged space and handiness carrying time particles within the discrete space–time frame. The driving force for generation of local gauge field's space–time frame of matter is the discrete conservation of energy-momentum pairs.

Due to the absence of space–time frame, the neutral particles of the background gauge field have only color interactions with the feature of "neutral crystals" of time and space portions, bubbling in gauge field condensate with the small wavelength. Recently Wilczek [27] described the similar idea in more details.

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based… DOI: http://dx.doi.org/10.5772/intechopen.93087*

#### **11.4 Invariant translation of symmetries within background/local gauge fields**

Background gauge/local gauge fields exist in the form of field-anti-field pair. Due to the energy-momentum non-commutation, the local gauge field is the non-Abelian, while background gauge is Abelian field. Generation of non-Abelian local gauge field from background Abelian's gauge field is not spontaneously symmetry breaking. The local gauge field at **Ea = 0** of model (9) merges with the background gauge field, as particle-antiparticle pair. In this case, the difference between particle and field disappears. The background and local gauge fields are connected through **SO (3)** rotational matrix which carries invariant translation of particle to field. Quantum mechanics mediates physical quantity by the square of the wave function, but **SO (3)** group of model (9) mediates physical quantity of the background gauge field by coupling of space–time portions of energy, carried in the form of matter– antimatter pair.

Model (9) suggests that CP symmetry of strong interactions is in hold only through cross product of **SU (2) x SO (3)** symmetry groups within two transformations: one is charge cancelation translation, and the second is parity transformation.

Yang-Mills [24, 28] attempted to apply gauge theory to the strong interactions through elevating of global symmetry to local gauge symmetry, but this attempt produced symmetry breaking. Without conjugation of **SU (2)** matrix of space–time frame and energy-momentum exchange interactions **SO (3)** (9), the background gauge field **U (1**) cannot carry invariant translation of mass to the opposite phase of local gauge field of strong interactions.

By Glashow's opinion [2] electromagnetism is mediated not only by photons; it arises from the requirement of local gauge invariance. However, based on our theory, this statement is true only partly because the role of local gauge field is reversible and symmetric. The local gauge field is needed for generation of electromagnetic interaction and cancelation it takes place by gravitation for discrete conservation of energy within space–time of baryon frame. The **SO (3)** symmetry group of gauge field translates electromagnetic force to the gravitation force. Therefore, without gravitation force, it is impossible to get invariant performance of strong interactions. The Standard Model, as Kibble showed [29], did not find place for gravity, and that is why it cannot not explain why the elementary particles come in three families with very similar structure but wildly differing masses.

In our theory electromagnetism and gravitation are unified within **SU (2) x SO (3)** symmetry of local gauge field which involves unification of charges as the internal products of baryon's space–time frame. The genetic code of baryon particles **Es = 1/2 Ea** holds all the internal conservation laws: baryon conservation, isospin conservation, hypercharge conservation, and boson-fermion spin invariant translation. It is known that the hypercharge of **SU (3)** symmetry is one of two quantum numbers of the hadrons and alongside with isospin *I***<sup>3</sup>** follows the formula: **Q=J3 + 1/2Y**. For multiples of particles, the hypercharge gets formulation **J = 2Q**.

According to model (9), at local gauge field, the hypercharge current coupling is the condition **Es = 1/2 Ea** which describes local space–time symmetry at **J3 = 0**. At the background gauge field, the hypercharge conservation **Ea = 2Es** describes multiple bosons, similar to **J = 2Q.** Based on Eqs. (8)–(12), the discrete conservation of energy at the background gauge field produces condition:

$$
\Delta \mathbf{F} = \frac{\Delta \mathbf{S}}{\Delta t} + \frac{\mathbf{S1}}{\mathbf{t\_1}}; \frac{\Delta \mathbf{S}}{\Delta t} = -\frac{\mathbf{S1}}{\mathbf{t\_1}}, \Delta \mathbf{F} = \mathbf{0} \tag{29}
$$

The equation **ΔF=0** describes discrete nonvanishing energy state of spin zero boson's condensate of the background gauge field.

field functions of negative displacement by the conjugation of space and time

Therefore, the ingredients of Eq. (13) generate two equivalent functions:

The function **Fs** describes displacement in space, while the function **Ft** describes duration of change. When the values of field function are vectors, the plane wave is longitudinal. The space and time portions of energy in the background gauge field form a one-dimensional space–time frame that is why the plane wave in this case is longitudinal. Multiplication of equation **(Es = 1/2Ea)** by scalar **2** leads to the formation of neutral particles **e/e**, **ν/ν**, and **ꝩ/ꝩ** of the background gauge field **2Es = Ea** in the form of spinors, similar to Hamilton quaternions spinors [25]. The ingredients of Hamilton's equation **(I; J; K)** are imaginary quantities, while the products of our

ð Þ **Es** ¼ **1***=***2Ea** ð**Space**–**time phase**Þ ! **Inversion to energy phase 2 Es** ð Þ ¼ **Ea**

The equation **(2Es = Ea)** produces a new vector where the scalar is the real number. The inversion transforms integer spin electromagnetic force to the other force being integer **2** spin carrying force. The new force is the gravitation, which with continuous longitudinal wave moves to the background through conjugation

It is necessary to note that generation of inversion vector space for transformation of energy conservation from space–time phase to energy phase meets all the requirements, required for scalar multiplication procedure, given in the form of axioms [24]. One of the requirements of axioms is the condition **x+(**�**x) = 0**, which is in hold within discrete annihilation of space and time variables (14). Due to the conservation of energy within discrete energy and space–time phases, an event comes to the origin after two full rotations (720°). The generated field has an algebra of zero-dimensional geometric spinor with one-directional helicity to the

Model (9) describes conservation of energy through non-unitary space–time variables, which unifies fields, particles, and forces within non-unitary energy portions allowing transformation of all the identities to each other. The background plasma-like gauge field can hold the discrete symmetry only through coupling with the local gauge field. The local gauge field of virtual matter, which is the combination of electric and magnetic fields, holds interaction of electrically charged space and handiness carrying time particles within the discrete space–time frame. The driving force for generation of local gauge field's space–time frame of matter is the

Due to the absence of space–time frame, the neutral particles of the background gauge field have only color interactions with the feature of "neutral crystals" of time and space portions, bubbling in gauge field condensate with the small wavelength.

**s1Δt** ¼ �**t1Δs** (24)

**Ft**ð Þ¼� **s1**,**Δt Fs**ð Þ **t1**, **Δs** (25)

**<sup>2</sup>** <sup>¼</sup> **<sup>K</sup><sup>2</sup>** <sup>¼</sup> **ijk** ¼ �**1 Hamilton** ð Þ (26)

(28)

**e***=***e** ¼ **ν***=***ν** ¼ **ꝩ***=***ꝩ** ¼ ð Þ¼� **eνꝩ 1** (27)

variables:

*Quantum Mechanics*

model are virtual particles:

background origin.

**228**

**I <sup>2</sup>** <sup>¼</sup> **<sup>J</sup>**

of ingredient (**eνꝩ)** of produced neutral spinors.

discrete conservation of energy-momentum pairs.

Recently Wilczek [27] described the similar idea in more details.

#### **12. Principles of isomorphism of SU (2) and SO (3) symmetry groups**

**ΔS S1 Δt t1**

From this formula, we will get:

*DOI: http://dx.doi.org/10.5772/intechopen.93087*

**ΔS S1** <sup>2</sup>

> **ΔS S1** � *Δ***t t1**

vation of energy within invariant translations.

sion **f(x, y) = 0.**

**231**

be described as follows:

**physical laws to frequency**

� **2**

¼

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based…*

**Δt1 t1 Δs1**

**ΔS S1** 

identities, these symmetry groups are not separable from each other.

with matrix multiplication may produce elementary particles.

**<sup>2</sup> <sup>Δ</sup><sup>s</sup> s1**

**s1** 

> **ΔS S1** � *Δ***t t1**

The equations (32) and (33) describe the combination of space–time and energymomentum symmetries in the form **SU (2) x SO (3)** product, which holds conser-

In mathematics isomorphism is a mapping between two structures of the same type that can be reversed. Model (10) describes isomorphism of SU (2) and SO (3) matrixes not only from the point of view of reverse mapping structures; it shows that due to the reciprocal transformation of space–time and energy-momentum

*The rotational symmetry group SO (3) cannot carry translation if the model does not*

The quadratic Eq. (31) with two variables, which is generalized to vector space, is an algebraic expression of quadratic polynomial **P(x, y) = 0** equation. Such a polynomial fundamental equation takes place in conic sectors, having the expres-

At **Ea = 0**, the space and time variables became asymptotically equivalent. The asymptotic limit for these variables having binary relation **f (Δs/S1), f (Δt/t1)** can

> *<sup>f</sup>*ð**<sup>Δ</sup><sup>t</sup> t1** Þ *<sup>f</sup>*ð**<sup>Δ</sup><sup>s</sup> s1** Þ

**14. Translation of space dimensions. Transmutation of dimension-based**

We replaced velocity in linear equation of classic electromagnetic field by the frequency to present electromagnetic field, conserved as the cross product of energy-momentum exchange and local gauge field position of space–time. The electromagnetic field **Ea** of model (9) involves electromagnetic fields, and its relation with the magnetic field **Es** produces the three field symmetry **Es = 1/2Ea.**

Lim **Δs s1** ! 1

*provide the state of origin.* Without initial position of space and time, you cannot build a gauge field theory where the antiparticle cannot find its twin brother in the background gauge field. The energy-momentum exchange eigenvector (12) through angular momentum generates the rotational **SO (3)** symmetry, while the **SU (3)** group of standard model describes only continuous symmetry. The **SO (3)** generates rotation about the origin in Euclidean space. Only this symmetry group

� *<sup>Δ</sup>***<sup>t</sup> t1**

<sup>2</sup>

¼ **0** ð Þa ,

� **<sup>Δ</sup><sup>t</sup> t1 Δt t1**

þ

**Δt t1** <sup>2</sup> (30)

¼ 0 (31)

¼ **0** (32)

¼ **0** ð Þ b (33)

¼ 1 (34)

We developed a new algebra for the isomorphism of **SU (2)** matrix to **SO (3)** group which holds 3D rotation about three-dimensional R<sup>3</sup> Euclidean space, to preserve the origin in discrete mode. The **SO (3)** three-dimensional matrix describes the three-jet performance of elementary particles.

The left side of model (9) is **SU (2)** matrix of space–time, but the right side describes three-dimensional energy-momentum exchange interaction within **SO (3)** group. The inseparable **SU (2)** and **SO (3)** matrixes make inseparable position and momentum. Therefore, the non-separation phenomenon of position and momentum, called uncertainty of quantum mechanics, is the necessary condition to hold discrete invariant translation of symmetries.

The new space and time geometry, which we suggested, is the presentation of new Hilbert space, which is equivalent to Euclidean space where the dimension of a Euclidean-type space may change in accordance with the associated vector space. The **SU (2)** and **U (1)** symmetry groups of standard model do not exist in the same phase, and **U (1)** x **SU (2)** is not the cross products due to the existence of these groups in opposite phases of discrete conservation of energy. Therefore, even the extension of **SU (2) x U (1)** matrixes [2] of standard model to symmetry group **SU (3) x SU (2) x U (1)** [30] cannot describe strong interactions.

For restoration of origin, the eigenvalue of rotation has to have signs 1, and model (9) provides this condition. The eigenvector with eigenvalue **+1** describes extension of baryon space–time frame, while **(1)** in the form of reflection returns the space–time to the origin. Conservation of energy at the origin holds conservation all of the inner products of translation.

The **SU (2)** and **SO (3)** are not subgroups of **U (1)**, as common algebra states; the **SU (2) x SO (3)** and **U (1)** are the products of each other in opposite phases of discrete symmetry. The special orthogonal **SO (3)** rotation symmetry group describes rotation about the origin of the three-dimensional Euclidean space. Orthogonal matrix is the square matrix; a matrix is orthogonal if its transpose is equal to its inverse within equations **Es = 1/2Ea** and **2Es = Ea** which are equal to each other. First is the matrix Q, and second is the inverse matrix.

At **Es = 1/2Ea**, the **SU (2)** symmetry has isomorphic relation to **SO (3)** symmetry, but at **2Es = Ea**, the **SU (2)** symmetry undergoes surjective homomorphism to **SO (3)** symmetry. The surjective homomorphism of the Lie group describes [30] two algebraic structures of the same type, which generates coupling of particle and antiparticle to the same structure. The isomorphism requires symmetry in opposite phases, but Lie algebra does not explain why isomorphic symmetries exist in opposite phases. The surjective homomorphism requires that the ingredients of the homomorphism should have one element, which should be the same for these ingredients. The same ingredient is the mass of particle and antiparticle, which makes them coupling by surjective homomorphism. The particle-antiparticle pair forms a domain-codomain pair where antiparticle codomain is the mathematical image of superposition origin and completely covers the domain function.

At **Ea = 0**, the **SU (2)** matrix gets smooth **2:1** surjective homomorphism to **SO (3)** group matrix which generates **U (1)** symmetry of the background gauge field.

#### **13. Mathematical formulations of SU (2) x SO (3) matrixes**

Multiplication of energy-momentum spin relation **Es = 1/2Ea** to **2Es = Ea** eliminates the difference in energy portions, distributed in space and time phases. Using these equations and model (9), we can get the following equations:

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based… DOI: http://dx.doi.org/10.5772/intechopen.93087*

$$\frac{\frac{\Delta S}{S\_1}}{\frac{\Delta \mathbf{t}}{\mathbf{t}\_1}} = \frac{2\frac{\Delta \mathbf{s}}{\mathbf{s}\_1} - \frac{\Delta \mathbf{t}}{\mathbf{t}\_1}}{\frac{\Delta \mathbf{t}}{\mathbf{t}\_1}}\tag{30}$$

From this formula, we will get:

**12. Principles of isomorphism of SU (2) and SO (3) symmetry groups**

describes the three-jet performance of elementary particles.

*Quantum Mechanics*

tion to hold discrete invariant translation of symmetries.

tion all of the inner products of translation.

**SU (3) x SU (2) x U (1)** [30] cannot describe strong interactions.

each other. First is the matrix Q, and second is the inverse matrix.

We developed a new algebra for the isomorphism of **SU (2)** matrix to **SO (3)** group which holds 3D rotation about three-dimensional R<sup>3</sup> Euclidean space, to preserve the origin in discrete mode. The **SO (3)** three-dimensional matrix

The left side of model (9) is **SU (2)** matrix of space–time, but the right side describes three-dimensional energy-momentum exchange interaction within **SO (3)** group. The inseparable **SU (2)** and **SO (3)** matrixes make inseparable position and momentum. Therefore, the non-separation phenomenon of position and momentum, called uncertainty of quantum mechanics, is the necessary condi-

The new space and time geometry, which we suggested, is the presentation of new Hilbert space, which is equivalent to Euclidean space where the dimension of a Euclidean-type space may change in accordance with the associated vector space. The **SU (2)** and **U (1)** symmetry groups of standard model do not exist in the same phase, and **U (1)** x **SU (2)** is not the cross products due to the existence of these groups in opposite phases of discrete conservation of energy. Therefore, even the extension of **SU (2) x U (1)** matrixes [2] of standard model to symmetry group

For restoration of origin, the eigenvalue of rotation has to have signs 1, and model (9) provides this condition. The eigenvector with eigenvalue **+1** describes extension of baryon space–time frame, while **(1)** in the form of reflection returns the space–time to the origin. Conservation of energy at the origin holds conserva-

The **SU (2)** and **SO (3)** are not subgroups of **U (1)**, as common algebra states; the **SU (2) x SO (3)** and **U (1)** are the products of each other in opposite phases of discrete symmetry. The special orthogonal **SO (3)** rotation symmetry group describes rotation about the origin of the three-dimensional Euclidean space. Orthogonal matrix is the square matrix; a matrix is orthogonal if its transpose is equal to its inverse within equations **Es = 1/2Ea** and **2Es = Ea** which are equal to

At **Es = 1/2Ea**, the **SU (2)** symmetry has isomorphic relation to **SO (3)** symmetry, but at **2Es = Ea**, the **SU (2)** symmetry undergoes surjective homomorphism to **SO (3)** symmetry. The surjective homomorphism of the Lie group describes [30] two algebraic structures of the same type, which generates coupling of particle and antiparticle to the same structure. The isomorphism requires symmetry in opposite phases, but Lie algebra does not explain why isomorphic symmetries exist in opposite phases. The surjective homomorphism requires that the ingredients of the homomorphism should have one element, which should be the same for these ingredients. The same ingredient is the mass of particle and antiparticle, which makes them coupling by surjective homomorphism. The particle-antiparticle pair forms a domain-codomain pair where antiparticle codomain is the mathematical image of superposition origin and completely covers the domain function.

At **Ea = 0**, the **SU (2)** matrix gets smooth **2:1** surjective homomorphism to **SO (3)**

Multiplication of energy-momentum spin relation **Es = 1/2Ea** to **2Es = Ea** eliminates the difference in energy portions, distributed in space and time phases. Using

group matrix which generates **U (1)** symmetry of the background gauge field.

**13. Mathematical formulations of SU (2) x SO (3) matrixes**

these equations and model (9), we can get the following equations:

**230**

$$2\left(\frac{\Delta \mathbf{S}}{\mathbf{S\_1}}\right)^2 - 2\left(\frac{\Delta \mathbf{t\_1}}{\mathbf{t\_1}}\right)\left(\frac{\Delta \mathbf{s\_1}}{\mathbf{s\_1}}\right) + \left(\frac{\Delta \mathbf{t}}{\mathbf{t\_1}}\right)^2 = \mathbf{0} \tag{31}$$

$$
\left[ \left( \frac{\mathbf{A} \mathbf{S}}{\mathbf{S}\_1} \right) - \left( \frac{\mathbf{A} \mathbf{t}}{\mathbf{t}\_1} \right) \right]^2 = \mathbf{0} \tag{32}
$$

$$\frac{\Delta \mathbf{S}}{\mathbf{S\_1}} - \frac{\Delta \mathbf{t}}{\mathbf{t\_1}} = \mathbf{0} \text{ (a)}, \frac{\Delta \mathbf{S}}{\mathbf{S\_1}} - \frac{\Delta \mathbf{t}}{\mathbf{t\_1}} = \mathbf{0} \text{ (b)}\tag{33}$$

The equations (32) and (33) describe the combination of space–time and energymomentum symmetries in the form **SU (2) x SO (3)** product, which holds conservation of energy within invariant translations.

In mathematics isomorphism is a mapping between two structures of the same type that can be reversed. Model (10) describes isomorphism of SU (2) and SO (3) matrixes not only from the point of view of reverse mapping structures; it shows that due to the reciprocal transformation of space–time and energy-momentum identities, these symmetry groups are not separable from each other.

*The rotational symmetry group SO (3) cannot carry translation if the model does not provide the state of origin.* Without initial position of space and time, you cannot build a gauge field theory where the antiparticle cannot find its twin brother in the background gauge field. The energy-momentum exchange eigenvector (12) through angular momentum generates the rotational **SO (3)** symmetry, while the **SU (3)** group of standard model describes only continuous symmetry. The **SO (3)** generates rotation about the origin in Euclidean space. Only this symmetry group with matrix multiplication may produce elementary particles.

The quadratic Eq. (31) with two variables, which is generalized to vector space, is an algebraic expression of quadratic polynomial **P(x, y) = 0** equation. Such a polynomial fundamental equation takes place in conic sectors, having the expression **f(x, y) = 0.**

At **Ea = 0**, the space and time variables became asymptotically equivalent. The asymptotic limit for these variables having binary relation **f (Δs/S1), f (Δt/t1)** can be described as follows:

$$\frac{\text{Lim}}{\frac{\Delta s}{s\_1} \to 1} \frac{f(\frac{\Delta t}{t\_1})}{f(\frac{\Delta s}{s1})} = 1\tag{34}$$

#### **14. Translation of space dimensions. Transmutation of dimension-based physical laws to frequency**

We replaced velocity in linear equation of classic electromagnetic field by the frequency to present electromagnetic field, conserved as the cross product of energy-momentum exchange and local gauge field position of space–time. The electromagnetic field **Ea** of model (9) involves electromagnetic fields, and its relation with the magnetic field **Es** produces the three field symmetry **Es = 1/2Ea.**

Model (10) shows that generation of new space vector takes place when dimension of space–time changes. The energy flux, coupled with the space–time (10), determines the space–time structure and dimension of space and time variables. The third dimension of space and generation of mass takes place in discrete mode at positive value of function **Ea** � **Es > 0**.

**15. The new theory of photon**

*DOI: http://dx.doi.org/10.5772/intechopen.93087*

**15.1 Dual performance of photons**

field.

wave functions.

**233**

Quantum mechanics suggests that photons are electrically neutral and do not couple to other photons. Based on our theory, a photon in the background gauge field behaves as color charge "twin pairs" while in the local gauge field became an electrically charged virtual particle-antiparticle pair. In the local gauge field of matter space–time frame, the interaction of photon with the quarks takes place through generation of fractional charge ingredients of photon and their cross coupling with the quark charges which produces Majorana bosons of the background gauge field. Neutrinos in the local gauge field separate colors of gamma photons for generations of quarks for three fractional protons. You cannot see quarks because photon-photon cross coupling eliminates quarks, translating them to the gauge

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based…*

Photon-antiphoton in the form of Majorana particles do not have independent existence and for conservation of energy have to generate the discrete space–time frame of baryon's matter, holding strong interactions. When an electromagnetic field is on, it keeps conservation laws in baryon structure but, when it is off, trans-

Photon in the gauge field is not a single boson, but it is the composite frame of neutral bosons. Invariant translation of fermions to bosons requires cutoff electromagnetic force (**Ea = 0)** where cross coupling of **Es = 1/2 Ea** shape particles to particles of the background gauge field **2Es = Ea** takes place. The color charge mass of photons of the background gauge field appears in the local gauge field in the form

In accordance with model (9), quanta are the energy-momentum carrying elements, and only the energy-momentum content determines the existence of photons in the form of finite amount of quanta. The portions of energy, carried by

Without clear understanding of half spin phenomenon and the Pauli exclusion principle, we cannot describe photon as a boson. The exclusion principle states that two identical half spin carrying fermions cannot occupy the same quantum state. Pauli's "quantum state" is an abstract point-like state of a particle which does not involve the space–time frame, and his rule does not explain the fact of the existence of two same quarks in baryon's space–time frame. In this sense, Wilczek [16] also raised the question that two identical quark fermions did not appear to obey the normal rules of quantum statistics. It is difficult to understand the pattern of observed baryons using antisymmetric wave functions, as it requires symmetric

The formula **Es = 1/2Ea** explains that space and time portions of energy in the form of particle and antiparticle discretely share the space and only half of the available energy belongs to matter's space portion. On this basis, we modified the exclusion principle to the statement that matter fermion with half spin can present only half portion of the available energy in the form of space. With multiplication of fermion space to scalar, we can produce two fermions that can occupy similar space at different times, holding **2Es = Ea** condition. The vector space of matrix **SO (3**) does multiplication of the code **Es = 1/2Ea** by two to produce two quarks, existing in opposite phases with one force carrier quark **2Es = Ea**: decay of proton's **(+2/3)** quark produces in the opposite phase two other **(1/3)** quarks in the opposite phase. The antiquarks, following the same rule, keep the existence of baryon structure. The Pauli exclusion principle cannot predict such a translation of half spin

forms conservation laws to the background energy phase of gauge field.

of space mass of fractional electric charges of baryon space–time frame.

space–time portions of energy, appear with the integer numbers (10).

Based on model (10), the wave amplitude of space–time is the composite product of instant of time and displacement in space, while the wavelength is the composite product of change of time and local space. The time instant **t1** appears as the genetic code of wave amplitude, while the local space **S1** appears as the genetic code of wavelength in the form of superposition.

Conservation of energy in the form of space and time portions requires transmutation of dimension-based physical laws to unit-less dynamics of frequency, which changes through integer numbers (10). While the energy-momentum content of photon is constant, the phenomenon called mass appears as a unit of change of frequency of energy distribution in space–time frame. Change of the frequency leads to the change of the space length and duration of the interaction, keeping the same physical law regardless of scale. At **ΔS/S1 = 0**, we get **Ea = Es** which shows that when particles move to short or zero distance, the difference between energy and momentum disappears. Thus, mass appears as the space phase equivalent of energy.

To hold conservation of finite amount, energy generates space–time phase through which it moves from one form to another. The **SO (3)** group generates a two-dimensional non-unitary isomorphic space–time symmetry of **SU (2**) matrix which holds the three-dimensional discrete performance of baryon structure through discrete invariant in-out of energy (in the form of so called gluons) to this frame.

Therefore, the discrete in-out external energy (gamma rays, transformed to electromagnetic force) generates additional in-out space dimension in baryon structure of local gauge field. When neutral particles are translated to the background gauge field, the **SO (3)** group eliminates external dimension in baryon structure and returns energy back to vacuum.

Such a performance of **SO (3)** symmetry group is missed in the standard model, and the known symmetry groups of strong interactions do not involve this symmetry. Combination of **SU (2)** non-unitary group with the **SO (3)** matrix generates new principles of fundamental laws which hold invariant translations of all of the natural symmetries through the background gauge field **U (1)**. However, the background gauge field cannot hold its state in continuous symmetry. We may describe the uniform state of a particle in gauge field as **ΔS/Δt=0**, which has isomorphism with the matter–antimatter symmetry at conditions where there is no change in space–time:

$$\left(\frac{\mathbf{E}\mathbf{a} - \mathbf{E}\_{\mathbf{s}}}{\mathbf{E}\_{\mathbf{s}}}\right) = \mathbf{0}, \mathbf{E}\mathbf{a} = \mathbf{E}\mathbf{s} \tag{35}$$

Such a state of a particle generates timeless matter–antimatter annihilation, which violates conservation of energy and leads to the ultraviolence divergences. According to the condition (35), it is very difficult to suggest any valid mechanism without renormalization to eliminate ultraviolence divergences with the equal numbers of matter–antimatter.

Wilzcek [17] suggested that instead of number of virtual particles, we have to speak of the numbers of internal loops in Feynman graphs. However, instead of Feynman diagram, we suggest the energy-momentum loop. Wilczek showed that proton mass in Planck unit arises from the basic unit of color coupling strength, which is of order ½ at the Plank scale. We showed that the color coupling code ½ arises from energy-momentum exchange interaction.

#### **15. The new theory of photon**

Model (10) shows that generation of new space vector takes place when dimension of space–time changes. The energy flux, coupled with the space–time (10), determines the space–time structure and dimension of space and time variables. The third dimension of space and generation of mass takes place in discrete mode at

Based on model (10), the wave amplitude of space–time is the composite prod-

Conservation of energy in the form of space and time portions requires transmutation of dimension-based physical laws to unit-less dynamics of frequency, which changes through integer numbers (10). While the energy-momentum content of photon is constant, the phenomenon called mass appears as a unit of change of frequency of energy distribution in space–time frame. Change of the frequency leads to the change of the space length and duration of the interaction, keeping the same physical law regardless of scale. At **ΔS/S1 = 0**, we get **Ea = Es** which shows that when particles move to short or zero distance, the difference between energy and momentum disappears. Thus, mass appears as the space phase equivalent of energy. To hold conservation of finite amount, energy generates space–time phase through which it moves from one form to another. The **SO (3)** group generates a two-dimensional non-unitary isomorphic space–time symmetry of **SU (2**) matrix which holds the three-dimensional discrete performance of baryon structure through discrete invariant in-out of energy (in the form of so called gluons) to this frame. Therefore, the discrete in-out external energy (gamma rays, transformed to electromagnetic force) generates additional in-out space dimension in baryon structure of local gauge field. When neutral particles are translated to the background gauge field, the **SO (3)** group eliminates external dimension in baryon

Such a performance of **SO (3)** symmetry group is missed in the standard model, and the known symmetry groups of strong interactions do not involve this symmetry. Combination of **SU (2)** non-unitary group with the **SO (3)** matrix generates new principles of fundamental laws which hold invariant translations of all of the natural symmetries through the background gauge field **U (1)**. However, the background gauge field cannot hold its state in continuous symmetry. We may describe the uniform state of a particle in gauge field as **ΔS/Δt=0**, which has isomorphism with the matter–antimatter symmetry at conditions where there is no change in space–time:

¼ **0**, Ea ¼ **Es** (35)

**Ea** � **Es Es** 

Such a state of a particle generates timeless matter–antimatter annihilation, which violates conservation of energy and leads to the ultraviolence divergences. According to the condition (35), it is very difficult to suggest any valid mechanism without renormalization to eliminate ultraviolence divergences with the equal

Wilzcek [17] suggested that instead of number of virtual particles, we have to speak of the numbers of internal loops in Feynman graphs. However, instead of Feynman diagram, we suggest the energy-momentum loop. Wilczek showed that proton mass in Planck unit arises from the basic unit of color coupling strength, which is of order ½ at the Plank scale. We showed that the color coupling code ½

uct of instant of time and displacement in space, while the wavelength is the composite product of change of time and local space. The time instant **t1** appears as the genetic code of wave amplitude, while the local space **S1** appears as the genetic

positive value of function **Ea** � **Es > 0**.

*Quantum Mechanics*

code of wavelength in the form of superposition.

structure and returns energy back to vacuum.

numbers of matter–antimatter.

**232**

arises from energy-momentum exchange interaction.

#### **15.1 Dual performance of photons**

Quantum mechanics suggests that photons are electrically neutral and do not couple to other photons. Based on our theory, a photon in the background gauge field behaves as color charge "twin pairs" while in the local gauge field became an electrically charged virtual particle-antiparticle pair. In the local gauge field of matter space–time frame, the interaction of photon with the quarks takes place through generation of fractional charge ingredients of photon and their cross coupling with the quark charges which produces Majorana bosons of the background gauge field. Neutrinos in the local gauge field separate colors of gamma photons for generations of quarks for three fractional protons. You cannot see quarks because photon-photon cross coupling eliminates quarks, translating them to the gauge field.

Photon-antiphoton in the form of Majorana particles do not have independent existence and for conservation of energy have to generate the discrete space–time frame of baryon's matter, holding strong interactions. When an electromagnetic field is on, it keeps conservation laws in baryon structure but, when it is off, transforms conservation laws to the background energy phase of gauge field.

Photon in the gauge field is not a single boson, but it is the composite frame of neutral bosons. Invariant translation of fermions to bosons requires cutoff electromagnetic force (**Ea = 0)** where cross coupling of **Es = 1/2 Ea** shape particles to particles of the background gauge field **2Es = Ea** takes place. The color charge mass of photons of the background gauge field appears in the local gauge field in the form of space mass of fractional electric charges of baryon space–time frame.

In accordance with model (9), quanta are the energy-momentum carrying elements, and only the energy-momentum content determines the existence of photons in the form of finite amount of quanta. The portions of energy, carried by space–time portions of energy, appear with the integer numbers (10).

Without clear understanding of half spin phenomenon and the Pauli exclusion principle, we cannot describe photon as a boson. The exclusion principle states that two identical half spin carrying fermions cannot occupy the same quantum state. Pauli's "quantum state" is an abstract point-like state of a particle which does not involve the space–time frame, and his rule does not explain the fact of the existence of two same quarks in baryon's space–time frame. In this sense, Wilczek [16] also raised the question that two identical quark fermions did not appear to obey the normal rules of quantum statistics. It is difficult to understand the pattern of observed baryons using antisymmetric wave functions, as it requires symmetric wave functions.

The formula **Es = 1/2Ea** explains that space and time portions of energy in the form of particle and antiparticle discretely share the space and only half of the available energy belongs to matter's space portion. On this basis, we modified the exclusion principle to the statement that matter fermion with half spin can present only half portion of the available energy in the form of space. With multiplication of fermion space to scalar, we can produce two fermions that can occupy similar space at different times, holding **2Es = Ea** condition. The vector space of matrix **SO (3**) does multiplication of the code **Es = 1/2Ea** by two to produce two quarks, existing in opposite phases with one force carrier quark **2Es = Ea**: decay of proton's **(+2/3)** quark produces in the opposite phase two other **(1/3)** quarks in the opposite phase. The antiquarks, following the same rule, keep the existence of baryon structure. The Pauli exclusion principle cannot predict such a translation of half spin

fermions to integer spin particles, while the standard model has no cross **SU (2) x SO (3)** matrixes to carry this translation.

In a similar way we can explain why light cannot be at the same time matter and antimatter, which is the necessary condition to carry a finite amount of energy. Distribution of photon energy within space and time phase colors in the space–time frame generates fractional charges in time phase in the form of positron **(+2/3**, **+2/3**, �**1/3)** and electron **(**�**2/3**, �**2/3**, **+1/3)** of space phase to hold the genetic code **Es = 1/2Ea**. The quarks appear as the ingredients of photon's fractional charges within the space–time frame which carry a virtual baryon structure and the ingredients of nucleons. You cannot cut and separate fractional charges of entangled quarks into two separate species. That is the reason why the ingredients of quark-antiquark pair do not have independent existence, which is specified as *the confinement problem of quantum physics*.

**15.2 The three fractional proton families of baryon frame**

*DOI: http://dx.doi.org/10.5772/intechopen.93087*

flavors of fractional protons, quarks have different colors.

pair is the different state of proton-antiproton pair.

space–time unit.

particles.

**235**

Presently there is no quantum field theory, which may include space–time as the main ingredient of strong interactions. By Weinberg's opinion [1], isospin conservation, which governs strong interactions, has nothing to do with space and time. However, without space–time, it is impossible to produce the theory of strong interactions because space–time is the matrix for flux of energy to baryon frame. Discrete conservation of energy, carried in the space–time by minimum elementary grain of space–time frame of matter (baryon frame), is the same phenomenon called strong nuclear interactions. The strong interactions arise from conservation of energy within the space–time, which has to hold basic elementary baryon's

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based…*

If displacement in the space–time frame of baryon frame has a trend for contraction (**ΔS=0**), the space–time frame of baryon frame disappears, and the ingredients of baryon structure became free particles (**Ea = Es)** which appear as the "asymptotic freedom phenomenon" of gauge field. By quantum physics, energy is borrowed for the generation of particles-antiparticles, but the energy, borrowed from the background gauge field, in reality is required for discrete performance of baryon's frame. According to our theory, the integer proton-neutron pair may exist only within three fractional families, with involvement of other quark flavors, existing through internal color charge interactions between them with untouched spin relations. The condition **2Es = Ea** produces all types of symmetry (**n, ι, mι**) within three fractional proton-neutron families, but the ingredients of this symmetry have a difference only in color mass (**ms**). The proton mass does not come from quarks, but it is comprised of the energy which keeps invariant interactions of three fractional proton-neutron families. To hold color-based interactions between quark

Quantum mechanics suggests that isospin, which identifies proton and neutron

According to the Yang-Mills theory [20], when electromagnetic interaction is neglected, the isotopic spin has no physical significance, and all physical processes would be invariant under isotopic gauge transformations. It was shown that when electromagnetic field is not involved, all interactions are invariances at all space–time points. But these statements could be true only partly because when electromagnetic field is not involved (**Ea = 0**), all transformations move to the background gauge field where space–time forms the frame of integer spin carrying

In accordance with the **SO (3)** symmetry, the local **Es = 1/2Ea** and the background gauge field **2Es = Ea** require existence of **uud –ddu** proton-neutron relation within two rotations**.** Such an existence of quarks determines similar existence of other two fractional proton-neutron families, which occupy top-down location of **uud-udd** in an alternative mode. In this case, the color charges of quarks cancel each other. This principle explains problems, raised by Wilczek [16] who showed that it is difficult to get quark-antiquark color cancelation which needs energy. In accordance with our theory, rotation of fractional proton families' realizes charge

The ingredients of background gauge field appear in the form of dark matter/ dark energy, the composition of which is the same as dark energy/dark matter

cancelation during locating them alternatively at top-down positions.

as the different states of same particle due to the small mass difference, is an approximate symmetry. In accordance with our theory, the extra mass of neutron in comparison with proton arises from coupling of proton-antiproton pairs, which adds mass of color interactions within fractional charges. On this basis, proton and neutron are not the different states of the same particle. The neutron-antineutron

Quarks in the proton-neutron frame exist in the form of fractional charges; that is why we cannot see a fractional proton or fractional neutron, but we can see a pion, which appears from doubling of photon's fractional charges. This mechanism explains the phenomenon that when the quarks of nucleon are poked by highenergy photons, the quarks show behavior as they were free particles [17]. Cross coupling of photon's quark ingredients with the second photon leads to the scaling of the genetic code **Es = 1/2 Ea** to **2Es = Ea** which generates free neutral particles. The cross coupling of fractional charges of a photon in baryon frame through **SO (3)** matrix leads to the formation of a pion—the lightest particle to produce the background gauge field boson which plays a role of a Goldstone boson. In the local gauge field, the photons, as neutrinos, became Dirac particles, while in the background gauge field, they are Majorana pairs.

The condition **Es = 1/2Ea** is the threshold energy to hold a photon within fractional electric charges of baryon frame. Photon seems to be not a fundamental unit and conserved in space–time in the form of fractional quark unit. In such a mechanism, the threshold energy is not the bound energy of electron in metal, but it is the energy required to hold **2Es = Ea** transformation of fractional electric charges which is necessary to produce integer charges**.** The integer electric charge is the combination of fractional charges, produced by coupling of condition **Es = 1/2Ea**:

$$\mathbf{1} = \mathbf{3}/\mathbf{3} = (\mathbf{2}/\mathbf{3}) + (\mathbf{2}/\mathbf{3}) - (\mathbf{1}/\mathbf{3});\ -\mathbf{1} = -\mathbf{3}/\mathbf{3} = (-\mathbf{2}/\mathbf{3}) + (-\mathbf{2}/\mathbf{3}) - (+\mathbf{1}/\mathbf{3})\tag{36}$$

That is why photoemission is not a one-step process, described by Einstein's linear equation, which does not cover these steps. Generation of integer electrons depends on the energy-momentum genetic code (**Ea/Es**�**1**), which determines a threefold frequency.

Model (10) suggests that Planck's emission of photons takes place only through merging of fractional charges. At **ΔS = Δt**, we can get the equation for photon radiation:

$$\frac{\mathbf{E\_s}}{\mathbf{E\_a}} = \frac{\frac{\mathbf{S1}}{\mathbf{t\_1}}}{\frac{\mathbf{s1}}{\mathbf{t\_1}} - \mathbf{1}} \tag{37}$$

Radiation takes place uniformly through the reduction of frequency by integer numbers, which describe numbers of energy portions in relation to total energy. The **Ea** in Eq. (37) presents the total numbers of elementary quanta. At Planck scale with the uniform distribution of energy in space and time phase, using condition (37), we can get the Planck formulation **Es = hν** where **h** presents the vacuum expectation value of background energy **(Ea).**

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based… DOI: http://dx.doi.org/10.5772/intechopen.93087*

#### **15.2 The three fractional proton families of baryon frame**

fermions to integer spin particles, while the standard model has no cross

ingredients of nucleons. You cannot cut and separate fractional charges of

In a similar way we can explain why light cannot be at the same time matter and antimatter, which is the necessary condition to carry a finite amount of energy. Distribution of photon energy within space and time phase colors in the

space–time frame generates fractional charges in time phase in the form of positron **(+2/3**, **+2/3**, �**1/3)** and electron **(**�**2/3**, �**2/3**, **+1/3)** of space phase to hold the genetic code **Es = 1/2Ea**. The quarks appear as the ingredients of photon's fractional charges within the space–time frame which carry a virtual baryon structure and the

entangled quarks into two separate species. That is the reason why the ingredients of quark-antiquark pair do not have independent existence, which is specified as *the*

Quarks in the proton-neutron frame exist in the form of fractional charges; that is why we cannot see a fractional proton or fractional neutron, but we can see a pion, which appears from doubling of photon's fractional charges. This mechanism explains the phenomenon that when the quarks of nucleon are poked by highenergy photons, the quarks show behavior as they were free particles [17]. Cross coupling of photon's quark ingredients with the second photon leads to the scaling of the genetic code **Es = 1/2 Ea** to **2Es = Ea** which generates free neutral particles. The cross coupling of fractional charges of a photon in baryon frame through **SO (3)** matrix leads to the formation of a pion—the lightest particle to produce the background gauge field boson which plays a role of a Goldstone boson. In the local gauge field, the photons, as neutrinos, became Dirac particles, while in the back-

The condition **Es = 1/2Ea** is the threshold energy to hold a photon within fractional electric charges of baryon frame. Photon seems to be not a fundamental unit and conserved in space–time in the form of fractional quark unit. In such a mechanism, the threshold energy is not the bound energy of electron in metal, but it is the energy required to hold **2Es = Ea** transformation of fractional electric charges which is necessary to produce integer charges**.** The integer electric charge is the combina-

**1** ¼ **3***=***3** ¼ ð Þþ **2***=***3** ð Þ� **2***=***3** ð Þ **1***=***3** ; � **1** ¼ �**3***=***3** ¼ �ð Þþ � **2***=***3** ð Þ� þ **2***=***3** ð Þ **1***=***3**

That is why photoemission is not a one-step process, described by Einstein's linear equation, which does not cover these steps. Generation of integer electrons depends on the energy-momentum genetic code (**Ea/Es**�**1**), which determines a

Model (10) suggests that Planck's emission of photons takes place only through

**S1 t1 s1**

Radiation takes place uniformly through the reduction of frequency by integer numbers, which describe numbers of energy portions in relation to total energy. The **Ea** in Eq. (37) presents the total numbers of elementary quanta. At Planck scale with the uniform distribution of energy in space and time phase, using condition (37), we can get the Planck formulation **Es = hν** where **h** presents the vacuum

**t1** � **<sup>1</sup>** (37)

merging of fractional charges. At **ΔS = Δt**, we can get the equation for photon

**Es Ea** ¼ (36)

tion of fractional charges, produced by coupling of condition **Es = 1/2Ea**:

**SU (2) x SO (3)** matrixes to carry this translation.

*Quantum Mechanics*

*confinement problem of quantum physics*.

ground gauge field, they are Majorana pairs.

expectation value of background energy **(Ea).**

threefold frequency.

radiation:

**234**

Presently there is no quantum field theory, which may include space–time as the main ingredient of strong interactions. By Weinberg's opinion [1], isospin conservation, which governs strong interactions, has nothing to do with space and time. However, without space–time, it is impossible to produce the theory of strong interactions because space–time is the matrix for flux of energy to baryon frame. Discrete conservation of energy, carried in the space–time by minimum elementary grain of space–time frame of matter (baryon frame), is the same phenomenon called strong nuclear interactions. The strong interactions arise from conservation of energy within the space–time, which has to hold basic elementary baryon's space–time unit.

If displacement in the space–time frame of baryon frame has a trend for contraction (**ΔS=0**), the space–time frame of baryon frame disappears, and the ingredients of baryon structure became free particles (**Ea = Es)** which appear as the "asymptotic freedom phenomenon" of gauge field. By quantum physics, energy is borrowed for the generation of particles-antiparticles, but the energy, borrowed from the background gauge field, in reality is required for discrete performance of baryon's frame. According to our theory, the integer proton-neutron pair may exist only within three fractional families, with involvement of other quark flavors, existing through internal color charge interactions between them with untouched spin relations. The condition **2Es = Ea** produces all types of symmetry (**n, ι, mι**) within three fractional proton-neutron families, but the ingredients of this symmetry have a difference only in color mass (**ms**). The proton mass does not come from quarks, but it is comprised of the energy which keeps invariant interactions of three fractional proton-neutron families. To hold color-based interactions between quark flavors of fractional protons, quarks have different colors.

Quantum mechanics suggests that isospin, which identifies proton and neutron as the different states of same particle due to the small mass difference, is an approximate symmetry. In accordance with our theory, the extra mass of neutron in comparison with proton arises from coupling of proton-antiproton pairs, which adds mass of color interactions within fractional charges. On this basis, proton and neutron are not the different states of the same particle. The neutron-antineutron pair is the different state of proton-antiproton pair.

According to the Yang-Mills theory [20], when electromagnetic interaction is neglected, the isotopic spin has no physical significance, and all physical processes would be invariant under isotopic gauge transformations. It was shown that when electromagnetic field is not involved, all interactions are invariances at all space–time points. But these statements could be true only partly because when electromagnetic field is not involved (**Ea = 0**), all transformations move to the background gauge field where space–time forms the frame of integer spin carrying particles.

In accordance with the **SO (3)** symmetry, the local **Es = 1/2Ea** and the background gauge field **2Es = Ea** require existence of **uud –ddu** proton-neutron relation within two rotations**.** Such an existence of quarks determines similar existence of other two fractional proton-neutron families, which occupy top-down location of **uud-udd** in an alternative mode. In this case, the color charges of quarks cancel each other. This principle explains problems, raised by Wilczek [16] who showed that it is difficult to get quark-antiquark color cancelation which needs energy. In accordance with our theory, rotation of fractional proton families' realizes charge cancelation during locating them alternatively at top-down positions.

The ingredients of background gauge field appear in the form of dark matter/ dark energy, the composition of which is the same as dark energy/dark matter

composition of universe. The portion of every boson in gauge field is 33%, which explains the predicted dark matter composition.

during decay of the space–time phase of baryon frame and within 2**Es = Ea** invariance translation became vector boson of the background gauge field. The **e/e** and **ν/ν** pairs, produced simultaneously, do not have independent existence, and with gamma rays they form three-jet particles to hold **2Es = Ea** symmetry of the

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based…*

When the flux of electromagnetic interactions to baryon structure is neglected **(Ea = 0**), electrically charged interactions disappear, but color interactions without change are translated to the background gauge field. The color interactions in baryonic frame do not touch spin interactions of baryonic quarks but hold interactions within three fractional proton-neutron families. Therefore, the color of light photon as a variable is needed to generate translations between electrically charged

The color charge of quarks is required to carry interaction between fractional protons. The correct mass of proton can be calculated only from color-based interactions within fractional protons, and the theories based on a common protonelectron structure cannot produce correct proton mass. The interaction between fractional protons is spin invariant and determined by the color interactions. The six of eight gluons participate within the three fractional proton-neutron families, two

The ingredients of exchange interaction in baryon's space–time frame carry three symmetric interactions. The first is the symmetric energy-momentum interaction, regulated by conservation of spin numbers **(Es = 1/2Ea)**, which takes place between quarks. The second symmetric interactions take place internally within ingredients of baryon frame, (a) the internal color-based symmetric flavor interactions within quarks (called gluons color charge interactions), which combine two symmetric internal interactions **a = b1+ b2**: (a1) the internal mass-based symmetric interactions within neutrinos and (b2) the internal mass-based symmetric interac-

At **Ea = 0** takes place invariant translation of color- and mass-based internal symmetric interactions to the background gauge field to hold the symmetric internal interactions within neutral electron and neutrino families. In reverse translation of energy conservation from background energy phase to local space–time phase, the color charge transforms to electric charges of quark-antiquark families. The energy inserted to quark families of baryon frame is the gluon of gamma photons from the gauge energy phase. Due to the discrete insertion of gamma photons to quark frame of baryon structure, the

The QCD theory is a non-Abelian gauge theory (Yang-Mills theory) and based on approximate **SU (3)** symmetry. Gell-Mann suggested [31] that quarks do not have space–time frame. Such an approach was the main reason for the appearance of approximate SU (3) symmetry because point-like behavior of quarks cannot carry conservation of energy. The other problem of the Gell-Mann approach was due to the application of Lagrangian continuous field, which produces approximation for perturbative theories. In addition, the theory used nonsymmetric four-momentum frame of special relativity. In accordance with our theory, without the discrete space–

Our theory shows that quarks are not Gell-Mann's mathematical construct; they are ingredients of photon's fractional charges, distributed within the space–time frame of baryon frame. Han [32] desired to construct models in which the quarks

background gauge field.

tions within electron families.

**237**

spin **Es = 1/2 Ea** and color **2Es = Ea** interactions.

*DOI: http://dx.doi.org/10.5772/intechopen.93087*

between proton-antiproton-neutron-anti-neutron interactions.

mass of individual quarks is very less than the proton-neutron masses.

**17. New principles of quantum chromodynamics theory**

time symmetry, all field theories will produce approximate symmetry.

had integer value electric charges but was not able to deliver a theory.

#### **15.3 What is the Planck scale? Where did it come from?**

It is well known that the Plank scale is the magnitude of space, time, and energy below which the prediction of quantum theories is no longer valid and quantum effects of gravity are expected to dominate. Planck units are derived by normalization of the numerical values of certain fundamental constants to 1: **c = ℏ = ℎ = ℇo=k=1**.

Planck did normalization of different constants regardless of their dimensions. However, as we showed through the example of energy-mass transformation of SR, such a normalization can be done if physical quantities were expressed with dimensionless units, which give numbers. The relation of changes to their initial value, which we applied, gives proper normalization, which is a dimensionless nonunitary operator. Model (9) describes the Planck scale as the boundary position of a particle in space–time where the change of space presents wavelength, while amplitude is the initial space locality. The conditions of model (9) **ΔS = S1, Δt=t1** are the boundary condition for existence of the space–time which may present the Planck space and time. Model (9) describes normalization of all the dimensionless parameters to 1. The space–time triangle wave with the equal wavelength and amplitude is the Planck scale of space–time**.** When **ΔS < S1**, there is the no space– time frame and strong interactions of baryon frame. This is the phenomenon called vanishing of the effective coupling at short distances.

At high-energy region, close to **ΔS=0**, there is no consumption of energy for displacement of space which presents quarks as a point-like particle. The point-like interaction out of space–time "is equivalent to no interaction," because at point-like particles, there is no conservation of energy and there is no particle. In the similar way light cannot be identified as a point-like particle because light without emission from the space–time frame cannot exist. Without space–time with local position, energy is not conserved, and baryon structure does not exist. Therefore, without mathematics of space–time frame, we cannot explain strong interactions and "asymptotic freedom of baryon quarks at short distances."

When the local momentum merges with the initial momentum, the local position also merges with the initial position. Therefore, due to the non-separable conservation of energy-momentum in the space–time frame, momentum and position are not separable identities. The time-energy relation of the uncertainty principle involves interval of external time, which flow independently of measurement. However, in the concept of production of space–time position from energy conservation, the outcomes of the uncertainty principle probably will be different. The condition **Es** ≥ **1/2Ea** of the model describes the limit above which **Ea** may present the Planck scale, where the space–time and local position do not exist.

Following to the genetic code **Es = 1/2Ea**, existence of position and momentum in different phases generates non-commutation of these identities.

#### **16. Gluons**

The standard model does not provide any information on gluon's origin. Based on our theory, only the origin of a particle can give information on how it will behave. This is the requirement of the causality that "past determines future."

Invariant translation from the local gauge field to the background field shows that gamma rays are the products of transformation of electromagnetic energy

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based… DOI: http://dx.doi.org/10.5772/intechopen.93087*

during decay of the space–time phase of baryon frame and within 2**Es = Ea** invariance translation became vector boson of the background gauge field. The **e/e** and **ν/ν** pairs, produced simultaneously, do not have independent existence, and with gamma rays they form three-jet particles to hold **2Es = Ea** symmetry of the background gauge field.

When the flux of electromagnetic interactions to baryon structure is neglected **(Ea = 0**), electrically charged interactions disappear, but color interactions without change are translated to the background gauge field. The color interactions in baryonic frame do not touch spin interactions of baryonic quarks but hold interactions within three fractional proton-neutron families. Therefore, the color of light photon as a variable is needed to generate translations between electrically charged spin **Es = 1/2 Ea** and color **2Es = Ea** interactions.

The color charge of quarks is required to carry interaction between fractional protons. The correct mass of proton can be calculated only from color-based interactions within fractional protons, and the theories based on a common protonelectron structure cannot produce correct proton mass. The interaction between fractional protons is spin invariant and determined by the color interactions. The six of eight gluons participate within the three fractional proton-neutron families, two between proton-antiproton-neutron-anti-neutron interactions.

The ingredients of exchange interaction in baryon's space–time frame carry three symmetric interactions. The first is the symmetric energy-momentum interaction, regulated by conservation of spin numbers **(Es = 1/2Ea)**, which takes place between quarks. The second symmetric interactions take place internally within ingredients of baryon frame, (a) the internal color-based symmetric flavor interactions within quarks (called gluons color charge interactions), which combine two symmetric internal interactions **a = b1+ b2**: (a1) the internal mass-based symmetric interactions within neutrinos and (b2) the internal mass-based symmetric interactions within electron families.

At **Ea = 0** takes place invariant translation of color- and mass-based internal symmetric interactions to the background gauge field to hold the symmetric internal interactions within neutral electron and neutrino families. In reverse translation of energy conservation from background energy phase to local space–time phase, the color charge transforms to electric charges of quark-antiquark families. The energy inserted to quark families of baryon frame is the gluon of gamma photons from the gauge energy phase. Due to the discrete insertion of gamma photons to quark frame of baryon structure, the mass of individual quarks is very less than the proton-neutron masses.

#### **17. New principles of quantum chromodynamics theory**

The QCD theory is a non-Abelian gauge theory (Yang-Mills theory) and based on approximate **SU (3)** symmetry. Gell-Mann suggested [31] that quarks do not have space–time frame. Such an approach was the main reason for the appearance of approximate SU (3) symmetry because point-like behavior of quarks cannot carry conservation of energy. The other problem of the Gell-Mann approach was due to the application of Lagrangian continuous field, which produces approximation for perturbative theories. In addition, the theory used nonsymmetric four-momentum frame of special relativity. In accordance with our theory, without the discrete space– time symmetry, all field theories will produce approximate symmetry.

Our theory shows that quarks are not Gell-Mann's mathematical construct; they are ingredients of photon's fractional charges, distributed within the space–time frame of baryon frame. Han [32] desired to construct models in which the quarks had integer value electric charges but was not able to deliver a theory.

composition of universe. The portion of every boson in gauge field is 33%, which

It is well known that the Plank scale is the magnitude of space, time, and energy below which the prediction of quantum theories is no longer valid and quantum

Planck did normalization of different constants regardless of their dimensions. However, as we showed through the example of energy-mass transformation of SR,

At high-energy region, close to **ΔS=0**, there is no consumption of energy for displacement of space which presents quarks as a point-like particle. The point-like interaction out of space–time "is equivalent to no interaction," because at point-like particles, there is no conservation of energy and there is no particle. In the similar way light cannot be identified as a point-like particle because light without emission from the space–time frame cannot exist. Without space–time with local position, energy is not conserved, and baryon structure does not exist. Therefore, without mathematics of space–time frame, we cannot explain strong interactions and

When the local momentum merges with the initial momentum, the local position also merges with the initial position. Therefore, due to the non-separable conservation of energy-momentum in the space–time frame, momentum and position are not separable identities. The time-energy relation of the uncertainty principle involves interval of external time, which flow independently of measurement. However, in the concept of production of space–time position from energy conservation, the outcomes of the uncertainty principle probably will be different. The condition **Es** ≥ **1/2Ea** of the model describes the limit above which **Ea** may present

Following to the genetic code **Es = 1/2Ea**, existence of position and momentum

The standard model does not provide any information on gluon's origin. Based on our theory, only the origin of a particle can give information on how it will behave. This is the requirement of the causality that "past determines future." Invariant translation from the local gauge field to the background field shows that gamma rays are the products of transformation of electromagnetic energy

explains the predicted dark matter composition.

**c = ℏ = ℎ = ℇo=k=1**.

*Quantum Mechanics*

**16. Gluons**

**236**

**15.3 What is the Planck scale? Where did it come from?**

vanishing of the effective coupling at short distances.

"asymptotic freedom of baryon quarks at short distances."

the Planck scale, where the space–time and local position do not exist.

in different phases generates non-commutation of these identities.

effects of gravity are expected to dominate. Planck units are derived by normalization of the numerical values of certain fundamental constants to 1:

such a normalization can be done if physical quantities were expressed with dimensionless units, which give numbers. The relation of changes to their initial value, which we applied, gives proper normalization, which is a dimensionless nonunitary operator. Model (9) describes the Planck scale as the boundary position of a particle in space–time where the change of space presents wavelength, while amplitude is the initial space locality. The conditions of model (9) **ΔS = S1, Δt=t1** are the boundary condition for existence of the space–time which may present the Planck space and time. Model (9) describes normalization of all the dimensionless parameters to 1. The space–time triangle wave with the equal wavelength and amplitude is the Planck scale of space–time**.** When **ΔS < S1**, there is the no space– time frame and strong interactions of baryon frame. This is the phenomenon called

The other problem of the field theories is that, as Gross [28] perfectly suggested, quantum field theories do not know which field to use and cannot explain why all the hadrons, baryons, and mesons appeared to be equally fundamental. The field theories do not clarify properly the nature of gauge field and unify the four forces for description of fundamental interactions.

Heisenberg [35] suggested that the proton and neutron are different states of the same particle, which should produce integer spin for the nucleon because of the addition of the angular momentum of the constituents. He called this rule addition law and suggested that full spin of the nucleon is always integer if the mass number

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based…*

Sakata suggested [36] that the even-odd rule and addition law can be applied for other particles as well. He suggested the model of hadrons, which comprised triplet of proton, neutron, and lambda, but later the quark model was suggested where triplet of **uds** quarks replaced **pnλ**. Sakata's model could not explain why hadrons should follow triplet performance of particles, and Sakata suggested that three **pnλ** particles are composite states of some hypothetical object called B

The mystery of triplet particles generated significant concern for particle physics theories when in 1964 unusual decay spectrum of kaon was reported [37]. Decay of neutral kaon produced mixture of **π π + ꝩ**, which by the author's opinion "no physical process would accomplish this decay and any alternative explanation of the effect requires highly nonphysical behavior of three body decay of neutral kaon." The author suggested that the presence of two-pion mode implies that the neutral kaon meson is not pure eigenstate. Such a decay process leads to the new direction

The eigenvalue (12) of model (9) shows that the triplet performance of hadron holds condition 2**Es = Ea** and has pure eigenstate to hold symmetry. This eigenstate requires existence of symmetry of integer-half-integer particles with the condition **π π + ꝩ**, which meets the requirement of eigenstate (12). Therefore, there is no symmetry breaking in kaon decay to **π π + ꝩ**, and the force called weak interaction is the gravitation force which holds the existence of the nucleon in discrete symmetry within **Es = 1/2Ea** and **2Es = Ea** invariant energy translations.

Translation of local gauge symmetry **Es = 1/2Ea** to background symmetry **2Es = Ea**, due to the existence of quark flavors in three families of fractional protons, requires counterpart mixing of quark flavors with generation of kaons. Flavor mixing appears through mixing of **SU (2)** and **SO (3)** matrixes. *Due to the existence of three fractional proton-neutron pairs, the formation of three kaons is the necessary*

Kobayashi [38] showed that CP violation would occur if irreducible complex number appears in the element of mixing. By terminology, the irreducible polynomial has a meaning that it cannot be factored into the product of two nonconstant polynomials. The symmetric reduction of condition 2**Es = Ea** to **Es = 1/2Ea** meets this requirement. The other condition for CP violation, as Kobayashi mentioned, is that the complex number remains in the polynomial equation, which cannot be removed by the phase adjoint of the particle state. The polynomial Eq. (32), describing mixing of **SU (2) x SO (3)** matrixes meets this requirement as well. By Kobayashi's opinion, flavor mixing arises between gauge symmetry and particle states. Kobayashi's statement is partly equivalent to our approach only while flavor mixing is the requirement of invariant translation within the background local

The standard model suggests that the CP violation is due to the essential difference between particles and antiparticles. Based on our theory, particles and antiparticles exist in different phases and are connected through symmetry mediator electromagnetic energy; when it is off, the antiparticle as the displacement merges

Formation of integer spin within proton-neutron pair in the nucleon through the addition law of proton and neutron is not possible because the proton and neutron do not exist in the same phase. The integer spin at an even mass number is described

is even; the full spin is half-integer if the mass number is odd.

*DOI: http://dx.doi.org/10.5772/intechopen.93087*

of studies of particle physics, called spontaneous symmetry breaking.

matter.

*condition to hold discrete symmetry.*

with its superposition twin particle.

gauge fields.

**239**

Model (10) combines all fundamental interactions within only electromagnetic and gravitation forces, strength of which changes with the integer numbers. The background gauge symmetry is the vacuum, which has no independent existence; that is why it mediates local gauge field. Such features of the background gauge field eliminate renormalization procedure, which is widely applied in quantum field theories. At maximum boundary energy (vacuum expectation value **Ea**), energy does not runaway to ultraviolent divergence due to translation of energy for separation of space **e/e** and time **ν/ν** spin one neutral pairs. In this case, separation of **U (1)** matrix into two symmetries takes place with the generation of space–time **SU (2**) and energy-momentum **SO (3)** matrixes. It is reduction from **2Es = Ea** background symmetry to the local gauge field symmetry **Es = 1/2 Ea** with generations of ½ spin carrying fermions and integer spin carrying photons of electromagnetic force. This invariance translation generates electromagnetic force with the positive sign **(EaEs)/Es.**

QCD is based partly on Poincare symmetry [33] that involves: (a) Abelian Lie group, (b) rotation in space to the non-Abelian Lie group, and (c) transformations connecting two uniformly moving bodies. However, having the excellent statements of (a) and (b), Poincare symmetry is not free from the problems due to the application of Minkowski's four-momentum space–time isometries that produces a semi-direct product of the translations. However, the statement (c) does not hold conservation of energy because it ignores boundary of motion.

The Wikipedia discussion [33] on Poincare symmetry shows that it might be possible to extend the Poincare algebra to produce super-Poincare algebra that may lead to the supersymmetry between spatial and fermionic directions. However, Poincare symmetry due to the absence of initial position cannot deliver conservation of energy at origin.

Nambu [34] suggested that the nucleon mass arises largely as self-energy of some primary fermion field, similar to the appearance of energy gap in the theory of superconductivity. According to his opinion, the nucleon mass is a manifestation of some unknown primary interaction between originally massless fermions. In addition, the pion is not the primary agent of strong interactions, and the nature of primary interaction is not clear.

In accordance with our theory, Nambu's coupling is the discrete cutoff electromagnetic energy **(Ea = 0)** to baryon space–time frame, turning fermions to bosons of the background gauge field, which performs as the superconductive medium due to the absence of fermionic space–time frame of "free" boson particles of condensate.

Our theory explains the ratio of spin constituents on the basis of ratio of transference **σ<sup>T</sup>** and longitudinal waves **σ<sup>l</sup>** of virtual photon (**R** = **σT/σl**) discussed by Gross [28]. At **Es = 1/2Ea** we get transference waves **σ<sup>l</sup> = 0**, while at **2Es = Ea** it transforms to longitudinal wave of virtual bosons **σ<sup>T</sup> = 0**. If the constituent has spin zero, the **σ<sup>T</sup>** became zero **σ<sup>T</sup> = 0**, but if spin is ½ the **σ<sup>l</sup>** became zero.

#### **18. The triplet model of hadron particles and problems of quantum mechanics**

It is necessary to note that the three particles performance of nucleons was the mystery of strong interactions.

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based… DOI: http://dx.doi.org/10.5772/intechopen.93087*

Heisenberg [35] suggested that the proton and neutron are different states of the same particle, which should produce integer spin for the nucleon because of the addition of the angular momentum of the constituents. He called this rule addition law and suggested that full spin of the nucleon is always integer if the mass number is even; the full spin is half-integer if the mass number is odd.

Sakata suggested [36] that the even-odd rule and addition law can be applied for other particles as well. He suggested the model of hadrons, which comprised triplet of proton, neutron, and lambda, but later the quark model was suggested where triplet of **uds** quarks replaced **pnλ**. Sakata's model could not explain why hadrons should follow triplet performance of particles, and Sakata suggested that three **pnλ** particles are composite states of some hypothetical object called B matter.

The mystery of triplet particles generated significant concern for particle physics theories when in 1964 unusual decay spectrum of kaon was reported [37]. Decay of neutral kaon produced mixture of **π π + ꝩ**, which by the author's opinion "no physical process would accomplish this decay and any alternative explanation of the effect requires highly nonphysical behavior of three body decay of neutral kaon." The author suggested that the presence of two-pion mode implies that the neutral kaon meson is not pure eigenstate. Such a decay process leads to the new direction of studies of particle physics, called spontaneous symmetry breaking.

The eigenvalue (12) of model (9) shows that the triplet performance of hadron holds condition 2**Es = Ea** and has pure eigenstate to hold symmetry. This eigenstate requires existence of symmetry of integer-half-integer particles with the condition **π π + ꝩ**, which meets the requirement of eigenstate (12). Therefore, there is no symmetry breaking in kaon decay to **π π + ꝩ**, and the force called weak interaction is the gravitation force which holds the existence of the nucleon in discrete symmetry within **Es = 1/2Ea** and **2Es = Ea** invariant energy translations.

Translation of local gauge symmetry **Es = 1/2Ea** to background symmetry **2Es = Ea**, due to the existence of quark flavors in three families of fractional protons, requires counterpart mixing of quark flavors with generation of kaons. Flavor mixing appears through mixing of **SU (2)** and **SO (3)** matrixes. *Due to the existence of three fractional proton-neutron pairs, the formation of three kaons is the necessary condition to hold discrete symmetry.*

Kobayashi [38] showed that CP violation would occur if irreducible complex number appears in the element of mixing. By terminology, the irreducible polynomial has a meaning that it cannot be factored into the product of two nonconstant polynomials. The symmetric reduction of condition 2**Es = Ea** to **Es = 1/2Ea** meets this requirement. The other condition for CP violation, as Kobayashi mentioned, is that the complex number remains in the polynomial equation, which cannot be removed by the phase adjoint of the particle state. The polynomial Eq. (32), describing mixing of **SU (2) x SO (3)** matrixes meets this requirement as well. By Kobayashi's opinion, flavor mixing arises between gauge symmetry and particle states. Kobayashi's statement is partly equivalent to our approach only while flavor mixing is the requirement of invariant translation within the background local gauge fields.

The standard model suggests that the CP violation is due to the essential difference between particles and antiparticles. Based on our theory, particles and antiparticles exist in different phases and are connected through symmetry mediator electromagnetic energy; when it is off, the antiparticle as the displacement merges with its superposition twin particle.

Formation of integer spin within proton-neutron pair in the nucleon through the addition law of proton and neutron is not possible because the proton and neutron do not exist in the same phase. The integer spin at an even mass number is described

The other problem of the field theories is that, as Gross [28] perfectly suggested, quantum field theories do not know which field to use and cannot explain why all the hadrons, baryons, and mesons appeared to be equally fundamental. The field theories do not clarify properly the nature of gauge field and unify the four forces

Model (10) combines all fundamental interactions within only electromagnetic and gravitation forces, strength of which changes with the integer numbers. The background gauge symmetry is the vacuum, which has no independent existence; that is why it mediates local gauge field. Such features of the background gauge field eliminate renormalization procedure, which is widely applied in quantum field theories. At maximum boundary energy (vacuum expectation value **Ea**), energy does not runaway to ultraviolent divergence due to translation of energy for separation of space **e/e** and time **ν/ν** spin one neutral pairs. In this case, separation of **U (1)** matrix into two symmetries takes place with the generation of space–time **SU (2**) and energy-momentum **SO (3)** matrixes. It is reduction from **2Es = Ea** background symmetry to the local gauge field symmetry **Es = 1/2 Ea** with generations of ½ spin carrying fermions and integer spin carrying photons of electromagnetic force. This invariance translation generates electromagnetic force with the

QCD is based partly on Poincare symmetry [33] that involves: (a) Abelian Lie group, (b) rotation in space to the non-Abelian Lie group, and (c) transformations connecting two uniformly moving bodies. However, having the excellent statements of (a) and (b), Poincare symmetry is not free from the problems due to the application of Minkowski's four-momentum space–time isometries that produces a semi-direct product of the translations. However, the statement (c) does not hold

The Wikipedia discussion [33] on Poincare symmetry shows that it might be possible to extend the Poincare algebra to produce super-Poincare algebra that may lead to the supersymmetry between spatial and fermionic directions. However, Poincare symmetry due to the absence of initial position cannot deliver conserva-

Nambu [34] suggested that the nucleon mass arises largely as self-energy of some

In accordance with our theory, Nambu's coupling is the discrete cutoff electromagnetic energy **(Ea = 0)** to baryon space–time frame, turning fermions to bosons of the background gauge field, which performs as the superconductive medium due to the absence of fermionic space–time frame of "free" boson particles of condensate. Our theory explains the ratio of spin constituents on the basis of ratio of transference **σ<sup>T</sup>** and longitudinal waves **σ<sup>l</sup>** of virtual photon (**R** = **σT/σl**) discussed by Gross [28]. At **Es = 1/2Ea** we get transference waves **σ<sup>l</sup> = 0**, while at **2Es = Ea** it transforms to longitudinal wave of virtual bosons **σ<sup>T</sup> = 0**. If the constituent has spin

primary fermion field, similar to the appearance of energy gap in the theory of superconductivity. According to his opinion, the nucleon mass is a manifestation of some unknown primary interaction between originally massless fermions. In addition, the pion is not the primary agent of strong interactions, and the nature of primary

zero, the **σ<sup>T</sup>** became zero **σ<sup>T</sup> = 0**, but if spin is ½ the **σ<sup>l</sup>** became zero.

**18. The triplet model of hadron particles and problems of quantum**

It is necessary to note that the three particles performance of nucleons was the

conservation of energy because it ignores boundary of motion.

for description of fundamental interactions.

positive sign **(EaEs)/Es.**

*Quantum Mechanics*

tion of energy at origin.

interaction is not clear.

**mechanics**

**238**

mystery of strong interactions.

by symmetry **2Es = Ea**, while half spin at an odd mass number is expressed by the condition **Es = 1/2Ea.**

symmetries of nature. Simply, we developed a new gauge theory of photon, which describes all the fundamental laws through conjugation of the discrete space–time **SU (2)** frame and energy-momentum **SO (3)** symmetry group. At background gauge supersymmetry **Ea = 2Es**, all the forces and interactions are symmetrically entangled. Based on the theory, *gravitation appears as the short-range force, which holds discrete performance of electromagnetic field for the existence of the nucleon in discrete mode.* Nature outlined this rule to avoid approximate symmetry in its

*Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based…*

fundamental laws.

*DOI: http://dx.doi.org/10.5772/intechopen.93087*

**Author details**

**241**

Aghaddin Mamedov

SABIC Technology Center, Sugar Land, TX, USA

provided the original work is properly cited.

\*Address all correspondence to: amamedov@sabic.com; aghaddinm@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Transformation from kaons to bosons of gauge field involves the following steps: coupling of fractional proton-antiproton pairs to kaon ! coupling to neutral kaon ! decay to pions ! coupling to neutral pions ! decay of neutral pions to quark ingredients with participation of intermediate W bosons through discrete translation of **Es = 1/2Ea** and **2Es = Ea** symmetries to each other**.**

#### **19. The principles of isospin symmetry**

Kibble [29] showed that the proton and neutron are not identical which the reason for generation of approximate symmetry. The proton has an electric charge, but the neutron does not. On this basis, the isospin symmetry, which describes proton-neutron symmetry by **SU (2)** group, was accepted as an approximate symmetry.

The standard model suggests that while isospin is an approximate symmetry, it must be broken in some way [29]. However, the addition of symmetry breaking terms generates non-renormalizable theories, producing infinite results. Therefore, the reason why the symmetry must be broken remained a mystery of particle physics. On this basis, the standard model, avoiding the need to add explicit symmetry breaking terms, suggested spontaneous symmetry breaking [29]. However, the spontaneous symmetry breaking theory of the standard model, producing massive bosons, did not explain the main problem of isospin that generates asymmetry: why the neutron has more mass than the proton or the proton has less mass than the neutron.

In accordance with quantum mechanics, the physical situation is unchanged if the electron wave function is multiplied by a phase factor [29]. This transformation involves a constant (α) and an imaginary number. The problem of such a transformation is that the constant (α) describes space–time in exponential function without involvement of space–time variables and their boundary. The other problem of this transformation is that if the electron wave function is multiplied by the phase factor, the physical situation changes and produces different phase symmetries.

In Kibble's analysis there is one excellent statement that spontaneous symmetry breaking occurs when ground state or vacuum does not share the underlying symmetry of the theory. As we showed, the background gauge symmetry does not exist independently and exists only in conjugation with the local gauge field, which appears as invariant translation of energy from the background vacuum.

Therefore, the isospin symmetry is not related to proton-neutron translation. As we showed in this chapter, isospin between the neutron and proton exist within the translation of proton-antiproton pair to neutron-antineutron pair with rotation of local gauge field to the background gauge field. The **ΔS/S1** and **Δt/t1** operators of model (9) within **2x2** non-unitary matrix have the exact **SU (2)** symmetry and carry this translation. This symmetry within the **Es = 1/2Ea** genetic code of the energy-momentum isospin symmetry generates a supersymmetry within the three particles' performance of baryon frame which exist in discrete mode in conjugation with the background vacuum.

#### **20. Conclusion**

We developed a new supersymmetric gauge field theory of photon, which describes fundamental laws of physics through invariant translation of discrete *Development of Supersymmetric Background/Local Gauge Field Theory of Nucleon Based… DOI: http://dx.doi.org/10.5772/intechopen.93087*

symmetries of nature. Simply, we developed a new gauge theory of photon, which describes all the fundamental laws through conjugation of the discrete space–time **SU (2)** frame and energy-momentum **SO (3)** symmetry group. At background gauge supersymmetry **Ea = 2Es**, all the forces and interactions are symmetrically entangled. Based on the theory, *gravitation appears as the short-range force, which holds discrete performance of electromagnetic field for the existence of the nucleon in discrete mode.* Nature outlined this rule to avoid approximate symmetry in its fundamental laws.

### **Author details**

by symmetry **2Es = Ea**, while half spin at an odd mass number is expressed by the

Kibble [29] showed that the proton and neutron are not identical which the reason for generation of approximate symmetry. The proton has an electric charge, but the neutron does not. On this basis, the isospin symmetry, which describes proton-neutron symmetry by **SU (2)** group, was accepted as an approximate sym-

The standard model suggests that while isospin is an approximate symmetry, it must be broken in some way [29]. However, the addition of symmetry breaking terms generates non-renormalizable theories, producing infinite results. Therefore, the reason why the symmetry must be broken remained a mystery of particle physics. On this basis, the standard model, avoiding the need to add explicit symmetry breaking terms, suggested spontaneous symmetry breaking [29]. However, the spontaneous symmetry breaking theory of the standard model, producing massive bosons, did not explain the main problem of isospin that generates asymmetry: why the neutron has more mass than the proton or the proton has less mass than the

In accordance with quantum mechanics, the physical situation is unchanged if the electron wave function is multiplied by a phase factor [29]. This transformation involves a constant (α) and an imaginary number. The problem of such a transformation is that the constant (α) describes space–time in exponential function without involvement of space–time variables and their boundary. The other problem of this transformation is that if the electron wave function is multiplied by the phase factor, the physical situation changes and produces different phase symmetries. In Kibble's analysis there is one excellent statement that spontaneous symmetry breaking occurs when ground state or vacuum does not share the underlying symmetry of the theory. As we showed, the background gauge symmetry does not exist independently and exists only in conjugation with the local gauge field, which appears as invariant translation of energy from the background vacuum.

Therefore, the isospin symmetry is not related to proton-neutron translation. As we showed in this chapter, isospin between the neutron and proton exist within the translation of proton-antiproton pair to neutron-antineutron pair with rotation of local gauge field to the background gauge field. The **ΔS/S1** and **Δt/t1** operators of model (9) within **2x2** non-unitary matrix have the exact **SU (2)** symmetry and carry this translation. This symmetry within the **Es = 1/2Ea** genetic code of the energy-momentum isospin symmetry generates a supersymmetry within the three particles' performance of baryon frame which exist in discrete mode in conjugation

We developed a new supersymmetric gauge field theory of photon, which describes fundamental laws of physics through invariant translation of discrete

tion of **Es = 1/2Ea** and **2Es = Ea** symmetries to each other**.**

**19. The principles of isospin symmetry**

Transformation from kaons to bosons of gauge field involves the following steps: coupling of fractional proton-antiproton pairs to kaon ! coupling to neutral kaon ! decay to pions ! coupling to neutral pions ! decay of neutral pions to quark ingredients with participation of intermediate W bosons through discrete transla-

condition **Es = 1/2Ea.**

*Quantum Mechanics*

metry.

neutron.

with the background vacuum.

**20. Conclusion**

**240**

Aghaddin Mamedov SABIC Technology Center, Sugar Land, TX, USA

\*Address all correspondence to: amamedov@sabic.com; aghaddinm@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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*Quantum Mechanics*

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**Chapter 12**

**Abstract**

**1. Introduction**

**245**

from the course of high school physics.

Confinement

*Eugen M. Sheregii*

Realization of the Quantum

splitting into symmetrical and anti-symmetrical states.

Shubnikov-de Haas effect, high electron mobility transistors

In this chapter the three main technologies are described, which allows for the implementation of quantum structures (QS)—quantum wells (QWs) and heterostructures. These are liquid phase epitaxy (LPE), molecular beam epitaxy (MBE), and metal-organic chemical vapor deposition (MOCVD). The most important properties, including the quantum Hall effect (QHE), of two-dimensional electron gas (2DEG) arising in a heterojunction on the boundary of two phases—the socalled interface—are also presented. The 2DEG properties in different kinds of QW are described. Double quantum wells as interesting example of quantum structure is considered also including such a spectacular quantum-mechanical phenomenon as

**Keywords:** hetero-structure, liquid phase epitaxy, molecular beam epitaxy, interface, single quantum well, quantum well, electron transport in quantum structures, two-dimensional electron gas (2DEG), quantum Hall effect,

The entrapping of electrons in an infinite quantum well (QW) is one of the basic issues of quantum mechanics showing its difference from classical mechanics. In fact, nature has given us a natural quantum well—atom. Coulomb's potential of the atomic nucleus creates the edges of this well (see **Figure 1**)—a very narrow well, about 1 Å (10<sup>10</sup> m) width. As it was shown in previous chapters, in such a narrow well the electron can occupy only certain energy states—discrete and not continuous energy values—as it is in the macro-world. This was indicated by the linear emission spectra of atoms discovered at the end of the nineteenth century. Their interpretation forced Niels Bohr to introduce discrete electronic states, so alien to classical physics, into the historically first quantum atom theory, which is familiar

However, this rectangular quantum well, which is the subject of students' exercises in quantum mechanics course, until the early 1980s was a theoretical issue. As will be shown in the next paragraphs, the progress of semiconductor technology, particularly the development of the *molecular beam epitaxy* (MBE), allowed the production of hetero-structures with *very sharp interface* (transition between two material phases) and later also quantum wells with width less than 100 angstroms and with finite potential edges, which changed the situation cardinally: the issue of two-dimensional electron gas (2DEG) appeared and the quantum Hall effect (QHE)

#### **Chapter 12**

## Realization of the Quantum Confinement

*Eugen M. Sheregii*

#### **Abstract**

In this chapter the three main technologies are described, which allows for the implementation of quantum structures (QS)—quantum wells (QWs) and heterostructures. These are liquid phase epitaxy (LPE), molecular beam epitaxy (MBE), and metal-organic chemical vapor deposition (MOCVD). The most important properties, including the quantum Hall effect (QHE), of two-dimensional electron gas (2DEG) arising in a heterojunction on the boundary of two phases—the socalled interface—are also presented. The 2DEG properties in different kinds of QW are described. Double quantum wells as interesting example of quantum structure is considered also including such a spectacular quantum-mechanical phenomenon as splitting into symmetrical and anti-symmetrical states.

**Keywords:** hetero-structure, liquid phase epitaxy, molecular beam epitaxy, interface, single quantum well, quantum well, electron transport in quantum structures, two-dimensional electron gas (2DEG), quantum Hall effect, Shubnikov-de Haas effect, high electron mobility transistors

#### **1. Introduction**

The entrapping of electrons in an infinite quantum well (QW) is one of the basic issues of quantum mechanics showing its difference from classical mechanics. In fact, nature has given us a natural quantum well—atom. Coulomb's potential of the atomic nucleus creates the edges of this well (see **Figure 1**)—a very narrow well, about 1 Å (10<sup>10</sup> m) width. As it was shown in previous chapters, in such a narrow well the electron can occupy only certain energy states—discrete and not continuous energy values—as it is in the macro-world. This was indicated by the linear emission spectra of atoms discovered at the end of the nineteenth century. Their interpretation forced Niels Bohr to introduce discrete electronic states, so alien to classical physics, into the historically first quantum atom theory, which is familiar from the course of high school physics.

However, this rectangular quantum well, which is the subject of students' exercises in quantum mechanics course, until the early 1980s was a theoretical issue. As will be shown in the next paragraphs, the progress of semiconductor technology, particularly the development of the *molecular beam epitaxy* (MBE), allowed the production of hetero-structures with *very sharp interface* (transition between two material phases) and later also quantum wells with width less than 100 angstroms and with finite potential edges, which changed the situation cardinally: the issue of two-dimensional electron gas (2DEG) appeared and the quantum Hall effect (QHE)

was discovered—a qualitatively new phenomenon. In this manner, quantum wells are soluble models and have provided tests for quantum theory. Also, the applications appeared very quickly—already in the 80-ch formed hetero-lasers—the first solid-state lasers, and transistors on hetero-structures sizes that do not exceed of 100 nm, which led to the production of modern micro-processors with a packing density of 20,000 transistors in a spatial centimeter.

In this way quantum mechanics contributed to the emergence of the third industrial revolution—electronics and radio-communications—as well as the promised fourth one, computerization (without microprocessors would be impossible) and global communication network, the Internet, which without semiconductor lasers would not have been created either. To achieve this duty, it was necessary to develop appropriate technologies. The first was *liquid phase epitaxy* (LPE).

offset. The QW exists only for electrons, and if the electron concentration increases, the Fermi level (FL) moves upwardly, and the conduction band will intersect at the QW, shown enlarged separately in **Figure 3**. You can see that it is a QW with a triangular shape. It should be noted that the shape of QW takes place also in the metal-oxide semiconductor (MOS) structures. Energy states for electrons in the real QW for the Al0.3Ga0.7As/GaAs junction were calculated by Zawadzki and Pfeffer [4]: the well depth is about 500 meV, and the resonance states occur at 200 meV and 360 meV from the bottom of the well. According to these calculations, the width of the well is about 200 Å or 0.02 μm. This fact explains why heterostructures with QW visible in the experiment could not have been obtained earlier using known crystal growth technologies and obtaining the p-n junctions by diffusion method. These methods did not allow for such a required change in the composition over several crystal lattice parameters. LPE methods have the advantage that with relative simplicity, a liquid AlGaAs solution with the necessary composition is poured onto a previously prepared (well-polished and heated to a temperature of about 600°C) GaAs substrate and the substrate will not melt during crystallization. Also, the diffusion of atoms is too slow for them to penetrate into the solid phase. Thanks to this, the required sharpness of the transition (junction) is preserved. However, the thickness of the AlGaAs layer should not exceed 1–2 μm. The last limitation is related to the upper layer stresses, resulting from incompatibility of the crystal lattice parameters of the substrate and the applied layer—socalled the lattice mismatch—minimal in the case of the GaAs and the AlGaAs solid

*Triangular QW formed in the conduction band at the heterojunction GaAs/AlGaAs shown in Figure 2.*

*Energy band diagram for the GaAs/AlGaAs heterojunction point QW, which would mean that the well is filled*

**Figure 2.**

**Figure 3.**

**247**

*with electrons.*

*Realization of the Quantum Confinement DOI: http://dx.doi.org/10.5772/intechopen.93112*

#### **2. LPE technology and the hetero-structure production**

LPE is the deposition from a liquid phase (a solution or melt) of a thin monocrystalline layer which is isostructural to the crystal of the substrate [1]. For the production of hetero-structures, the LPE was first used by Zhores Alferov with colleagues at the Ioffe Physical and Technical Institute in St. Petersburg [2]. They produced the hetero-structures based on the GaAs/AlGaAs n-p heterojunctions (unlike the usual p-n junction, which can be called a homo-junction) [3]. The zone scheme of such p-n heterojunction is presented in **Figure 2**. This diagram clearly shows that a quantum well forms at the interface in the conduction band from the side of GaAs, i.e., a semiconductor with a smaller energy gap (about 1.4 eV). In the case of the solid solution Al0.3Ga0.7As that is 1.9 eV, QW is created by the discontinuity in the conduction band profile as a function of distance *x.* The discontinuity takes place in the case of the valence band too, and it manifests itself as a leap called *Realization of the Quantum Confinement DOI: http://dx.doi.org/10.5772/intechopen.93112*

#### **Figure 2.**

was discovered—a qualitatively new phenomenon. In this manner, quantum wells are soluble models and have provided tests for quantum theory. Also, the applications appeared very quickly—already in the 80-ch formed hetero-lasers—the first solid-state lasers, and transistors on hetero-structures sizes that do not exceed of 100 nm, which led to the production of modern micro-processors with a packing

*Natural quantum well of the hydrogen atom created by the potential of the atomic nucleus* E = e/r.

In this way quantum mechanics contributed to the emergence of the third industrial revolution—electronics and radio-communications—as well as the promised fourth one, computerization (without microprocessors would be impossible) and global communication network, the Internet, which without semiconductor lasers would not have been created either. To achieve this duty, it was necessary to

develop appropriate technologies. The first was *liquid phase epitaxy* (LPE).

LPE is the deposition from a liquid phase (a solution or melt) of a thin monocrystalline layer which is isostructural to the crystal of the substrate [1]. For the production of hetero-structures, the LPE was first used by Zhores Alferov with colleagues at the Ioffe Physical and Technical Institute in St. Petersburg [2]. They produced the hetero-structures based on the GaAs/AlGaAs n-p heterojunctions (unlike the usual p-n junction, which can be called a homo-junction) [3]. The zone scheme of such p-n heterojunction is presented in **Figure 2**. This diagram clearly shows that a quantum well forms at the interface in the conduction band from the side of GaAs, i.e., a semiconductor with a smaller energy gap (about 1.4 eV). In the case of the solid solution Al0.3Ga0.7As that is 1.9 eV, QW is created by the discontinuity in the conduction band profile as a function of distance *x.* The discontinuity takes place in the case of the valence band too, and it manifests itself as a leap called

**2. LPE technology and the hetero-structure production**

density of 20,000 transistors in a spatial centimeter.

**Figure 1.**

*Quantum Mechanics*

**246**

*Energy band diagram for the GaAs/AlGaAs heterojunction point QW, which would mean that the well is filled with electrons.*

**Figure 3.** *Triangular QW formed in the conduction band at the heterojunction GaAs/AlGaAs shown in Figure 2.*

offset. The QW exists only for electrons, and if the electron concentration increases, the Fermi level (FL) moves upwardly, and the conduction band will intersect at the QW, shown enlarged separately in **Figure 3**. You can see that it is a QW with a triangular shape. It should be noted that the shape of QW takes place also in the metal-oxide semiconductor (MOS) structures. Energy states for electrons in the real QW for the Al0.3Ga0.7As/GaAs junction were calculated by Zawadzki and Pfeffer [4]: the well depth is about 500 meV, and the resonance states occur at 200 meV and 360 meV from the bottom of the well. According to these calculations, the width of the well is about 200 Å or 0.02 μm. This fact explains why heterostructures with QW visible in the experiment could not have been obtained earlier using known crystal growth technologies and obtaining the p-n junctions by diffusion method. These methods did not allow for such a required change in the composition over several crystal lattice parameters. LPE methods have the advantage that with relative simplicity, a liquid AlGaAs solution with the necessary composition is poured onto a previously prepared (well-polished and heated to a temperature of about 600°C) GaAs substrate and the substrate will not melt during crystallization. Also, the diffusion of atoms is too slow for them to penetrate into the solid phase. Thanks to this, the required sharpness of the transition (junction) is preserved. However, the thickness of the AlGaAs layer should not exceed 1–2 μm. The last limitation is related to the upper layer stresses, resulting from incompatibility of the crystal lattice parameters of the substrate and the applied layer—socalled the lattice mismatch—minimal in the case of the GaAs and the AlGaAs solid

solution with 30% AlAs as it is only 0.01 Å (4.65 Å for GaAs and 4.66 Å for Al0.3Ga0.7As) [4]. The first semiconductor lasers were produced thanks to LPE technology developed for the GaAs/AlGaAs heterojunction in the early 1970s [3]. However, more advanced technology was needed to improve the production of the heterojunction lasers.

reactor. MOCVD technology is much cheaper to operate (no vacuum) and is used as

The improved interface quality of hetero-structures through the use of MBE technology has led to the discovery of the unique properties of two-dimensional electron gas. The point is that the quantum well, which is located at the interface in the conduction band, naturally fills with electrons. The electrons are located in a layer with a thickness less than 200 Å. This means that they are actually in a plane that adheres to the interface (parallel to the interface) with a negligible thickness compared to two other dimensions. That is, a two-dimensional electron gas is

One of the basic properties of 2DEG is that electrons occupy one of the energy states of the quantum well at the interface. We will call this state as the energy sub-band and the dispersion law—energy dependence from quasi-momentum ℇð Þ*k* for 2DEG can be written in the case where the interface plane is the plane (yz) and

x—the direction of growth of the hetero-structure layers (as it is shown in

ℇ*<sup>i</sup>* is the energy value of the sub-band *i, m\** effective mass of electrons.

*<sup>π</sup>*ℏ2, where m\* is the effective mass of electrons.

*E k*ð Þ¼ *Ei* <sup>þ</sup> <sup>ℏ</sup><sup>2</sup> *<sup>k</sup>*<sup>2</sup>

where *i* is 1, 2, 3, … number of the sub-band, *ħ* = *h*/2*π* the Planck constant, and

Graphically the expression (1) is presented in **Figure 4**. The parabolic sub-band

The density of states function shown in **Figure 5** corresponds to such a law of dispersion. In contrast to the bulk material where the function of density of states is proportional to √*E*, in the 2D case we have steps corresponding to each value of *E*i:

It is obvious that the dispersion law (1) is a consequence of the restriction of movement in the x direction, in other words, by the quantum confinement, which

*<sup>z</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> *y*

<sup>2</sup>*m*<sup>∗</sup> (1)

industrial manufacturing technology for QS and devices on their base.

**3.3 Two-dimensional electron gas (2DEG)**

*Realization of the Quantum Confinement DOI: http://dx.doi.org/10.5772/intechopen.93112*

**Figure 2**), in the following way:

causes the energy quantization.

corresponds to each value of *i.*

*D E*ð Þ¼ *<sup>i</sup> <sup>m</sup>*<sup>∗</sup>

**Figure 4.**

**249**

*The energy sub-bands corresponding to Eq. (1) where ky = 0.*

created at the interface, which we will denote as 2DEG.

#### **3. MBE technology and the production of the solid-state QW**

#### **3.1 Description of the MBE technology**

The width of the quantum well in the case of GaAs/Al0.3Ga0.7As heterojunction is on the order of 150–200 Å, and the technology capabilities of LPE technology are on the border of these requirements to keep the production of devices based on them. For this reason, in the 1970s, a fundamentally new technology was developed that allowed a significant leap in the development of the semiconductor devices as well as the solid-state physics, generally. The MBE technology is based on the method of the crystal growth from the gas phase, but the use of computers made it possible to achieve precision previously unattainable [5]. First of all, it concerns the composition control (the composition control is so closely that practically every atom deposited on the substrate is calculated) but it is also the substrate temperature is much lower (450°C) than in the LPE method what is important because it reduces the diffusion intensity of atoms and has significantly improved the quality of the interface. But on the other hand, it is an expensive technology because it requires a high vacuum—10<sup>11</sup> Torr—which must be sustained continuously over several years. On the other hand, this extraordinary high vacuum allows the use of mass spectroscopy in the reactor for precise control of the composition and existing impurities. It should be recalled that the intrinsic properties of semiconductor materials were achieved only after chemists learned to clean the input materials from impurities at a concentration level of 10<sup>12</sup> cm<sup>3</sup> , which in turn means chemical purity 99.9999999%. Achieving such chemical purity of input materials requires huge amounts of labor and energy. The use of a high vacuum of the order of 10<sup>11</sup> Torr means additional "dilution" in the dopant concentration in reactor, which allowed the use of input materials in the effusors—sources of elements in the MBE machine—with a chemical purity lower by one row: 99,999999%.

The MBE process was noticed in the late 1970s at Bell Telephone Laboratories by Arthur and LePore [6]. But, the main role of this method has become the production of quantum structures (QS) from the 1980s [7] and above all—heterostructures and quantum wells.

Another technology that also relates to high tech is the metal-organic chemical vapor deposition (MOCVD) in some ways competitive to MBE because it allows obtaining high-quality quantum structures also.

#### **3.2 MOCVD technology**

The MOCVD involves the use of gases—carriers of elements used in QS built from GaAs, AlGaAs, InGaAs, and others. We call these gases metal-organic, for example, three-methyl-gal (Ga(CH3)3), three-methyl-aluminum (Al(CH3)3), or three-hydrogen of arsenic (AsH3). These substances are contained in bottles in a liquid state at about – 60°C, and are admitted as gases (still cool) to a reactor where the touching surface of the substrate at 500°C to immediately distributed to the constituent elements and relatively heavy metals as Al, As, Ga deposited on the substrate surface at this time how much lighter C and H are pumped out of the

solution with 30% AlAs as it is only 0.01 Å (4.65 Å for GaAs and 4.66 Å for Al0.3Ga0.7As) [4]. The first semiconductor lasers were produced thanks to LPE technology developed for the GaAs/AlGaAs heterojunction in the early 1970s [3]. However, more advanced technology was needed to improve the production of the

**3. MBE technology and the production of the solid-state QW**

The width of the quantum well in the case of GaAs/Al0.3Ga0.7As heterojunction is on the order of 150–200 Å, and the technology capabilities of LPE technology are on the border of these requirements to keep the production of devices based on them. For this reason, in the 1970s, a fundamentally new technology was developed that allowed a significant leap in the development of the semiconductor devices as well as the solid-state physics, generally. The MBE technology is based on the method of the crystal growth from the gas phase, but the use of computers made it possible to achieve precision previously unattainable [5]. First of all, it concerns the composition control (the composition control is so closely that practically every atom deposited on the substrate is calculated) but it is also the substrate temperature is much lower (450°C) than in the LPE method what is important because it reduces the diffusion intensity of atoms and has significantly improved the quality of the interface. But on the other hand, it is an expensive technology because it requires a high vacuum—10<sup>11</sup> Torr—which must be sustained continuously over several years. On the other hand, this extraordinary high vacuum allows the use of mass spectroscopy in the reactor for precise control of the composition and existing impurities. It should be recalled that the intrinsic properties of semiconductor materials were achieved only after chemists learned to clean the input materials

ical purity 99.9999999%. Achieving such chemical purity of input materials requires huge amounts of labor and energy. The use of a high vacuum of the order of 10<sup>11</sup> Torr means additional "dilution" in the dopant concentration in reactor, which allowed the use of input materials in the effusors—sources of elements in the

The MBE process was noticed in the late 1970s at Bell Telephone Laboratories by Arthur and LePore [6]. But, the main role of this method has become the production of quantum structures (QS) from the 1980s [7] and above all—hetero-

Another technology that also relates to high tech is the metal-organic chemical vapor deposition (MOCVD) in some ways competitive to MBE because it allows

The MOCVD involves the use of gases—carriers of elements used in QS built from GaAs, AlGaAs, InGaAs, and others. We call these gases metal-organic, for example, three-methyl-gal (Ga(CH3)3), three-methyl-aluminum (Al(CH3)3), or three-hydrogen of arsenic (AsH3). These substances are contained in bottles in a liquid state at about – 60°C, and are admitted as gases (still cool) to a reactor where the touching surface of the substrate at 500°C to immediately distributed to the constituent elements and relatively heavy metals as Al, As, Ga deposited on the substrate surface at this time how much lighter C and H are pumped out of the

MBE machine—with a chemical purity lower by one row: 99,999999%.

, which in turn means chem-

heterojunction lasers.

*Quantum Mechanics*

**3.1 Description of the MBE technology**

from impurities at a concentration level of 10<sup>12</sup> cm<sup>3</sup>

obtaining high-quality quantum structures also.

structures and quantum wells.

**3.2 MOCVD technology**

**248**

reactor. MOCVD technology is much cheaper to operate (no vacuum) and is used as industrial manufacturing technology for QS and devices on their base.

#### **3.3 Two-dimensional electron gas (2DEG)**

The improved interface quality of hetero-structures through the use of MBE technology has led to the discovery of the unique properties of two-dimensional electron gas. The point is that the quantum well, which is located at the interface in the conduction band, naturally fills with electrons. The electrons are located in a layer with a thickness less than 200 Å. This means that they are actually in a plane that adheres to the interface (parallel to the interface) with a negligible thickness compared to two other dimensions. That is, a two-dimensional electron gas is created at the interface, which we will denote as 2DEG.

One of the basic properties of 2DEG is that electrons occupy one of the energy states of the quantum well at the interface. We will call this state as the energy sub-band and the dispersion law—energy dependence from quasi-momentum ℇð Þ*k* for 2DEG can be written in the case where the interface plane is the plane (yz) and x—the direction of growth of the hetero-structure layers (as it is shown in **Figure 2**), in the following way:

$$E(k) = E\_i + \hbar^2 \frac{k\_x^2 + k\_y^2}{2m^\*} \tag{1}$$

where *i* is 1, 2, 3, … number of the sub-band, *ħ* = *h*/2*π* the Planck constant, and ℇ*<sup>i</sup>* is the energy value of the sub-band *i, m\** effective mass of electrons.

It is obvious that the dispersion law (1) is a consequence of the restriction of movement in the x direction, in other words, by the quantum confinement, which causes the energy quantization.

Graphically the expression (1) is presented in **Figure 4**. The parabolic sub-band corresponds to each value of *i.*

The density of states function shown in **Figure 5** corresponds to such a law of dispersion. In contrast to the bulk material where the function of density of states is proportional to √*E*, in the 2D case we have steps corresponding to each value of *E*i: *D E*ð Þ¼ *<sup>i</sup> <sup>m</sup>*<sup>∗</sup> *<sup>π</sup>*ℏ2, where m\* is the effective mass of electrons.

**Figure 4.** *The energy sub-bands corresponding to Eq. (1) where ky = 0.*

**Figure 5.** *The function of the state density for 2DEG in a quantum well.*

This stepped nature of the function of the state density for 2DEG in a quantum well is manifested in a multitude of phenomena including the dependence of the current through the heterojunction on the gate voltage, on which the transistor operated on the GaAs/AlGaAs hetero-structure is based, the so-called high electron mobility transistor (HEMT) [8].

#### **3.4 Quantum Hall effect**

The most spectacular expression of 2DEG in QS is quantum Hall effect discovered by von Klitzing [9] in 1980. We can say that QHE is a manifestation of quantum mechanics on macroscopic scales [10].

Experimentally, QHE shows the remarkable transport data as it is shown in **Figure 6** for a real device in the quantum Hall regime which is the same as in classical Hall effect when magnetic field *B* is perpendicular to the plane of the sample *xy* and to the current *I* directed along the *x*-axis. Then, in the direction perpendicular to the movement of the charges (electrons), an additional transverse voltage is created, called the Hall voltage UH. In classical Hall effect, the Hall resistance RH is simply a linear function of magnetic field and resistivity also *ρxy* � B. In QHE we see a series of the so-called Hall plateaus in which *ρxy* is a universal constant

$$
\rho\_{\infty} = \frac{1}{\nu} \frac{h}{\mathfrak{e}^2} \tag{2}
$$

dissipate, and it cannot move in the electric field—no electric current exists and ρxx is zero. In this situation the Hall voltage is constant until the Fermi level does not reach the next Landau level, and the next step of the Hall voltage takes place (in the case of bulk material, there are always scattering channels causing the presence of electric current in a strong magnetic field, therefore the dependence of the Hall voltage on the magnetic field is continuous and reflects the continuity of the density

*(a) QHE (the Hall resistance* RH *as function of magnetic field* B*); (b) the Shubnikov-de Haas oscillations*

As a result, the QHE is now used to maintain the standard of electrical resistance

The magnitude h/e2 = 25,812,80 Ω is so important as constant of the fine struc-

To obtain a quantum well with a rectangular shape, it should be placed close enough to two heterojunctions as shown in **Figure 7**. How close? The experiment shows that at a 200 nm distance, two GaAs/AlGaAs heterojunctions exhibit the properties of a rectangular QW [11]. It can be seen from **Figure 7** that these two

function of states (see **Figure 5**)).

*Realization of the Quantum Confinement DOI: http://dx.doi.org/10.5772/intechopen.93112*

*(magnetoresistance* ρxx(B)*) for hetero-structure GaAs/AlGaAs [11].*

**Figure 6.**

**251**

by metrology laboratories globally.

ture in the quantum electrodynamics.

**3.5 Quasi-rectangular quantum well**

(where *e* is the electron charge and *ν* = 1,2, … an integer which means the number of the states occupied by electrons under the Fermi level and is called as *filling factor*) independent of all microscopic details (including the precise value of the magnetic field). Associated with each of these plateaus is a dramatic decrease in the dissipative resistivity *ρxx* ! 0 which drops as much as 13 orders of magnitude in the plateau regions.

QHE is a two-dimensional phenomenon because when the magnetic field B is perpendicular to the plane of the hetero-structure, the movement of the electrons in the plane of the quantum well is completely quantized. This quantization is universal and independent of all microscopic details such as the type of semiconductor material, the purity of the sample, the precise value of the magnetic field, and so forth. The growth of the magnetic field causes an increase of the distance between Landau levels, ℏ*ω<sup>c</sup>* = *eB/m\* ,* and when the Fermi level is located between Landau levels then, for the electrons occupying the Fermi level, there are no states to

**Figure 6.**

This stepped nature of the function of the state density for 2DEG in a quantum well is manifested in a multitude of phenomena including the dependence of the current through the heterojunction on the gate voltage, on which the transistor operated on the GaAs/AlGaAs hetero-structure is based, the so-called high electron

The most spectacular expression of 2DEG in QS is quantum Hall effect discov-

Experimentally, QHE shows the remarkable transport data as it is shown in **Figure 6** for a real device in the quantum Hall regime which is the same as in classical Hall effect when magnetic field *B* is perpendicular to the plane of the sample *xy* and to the current *I* directed along the *x*-axis. Then, in the direction perpendicular to the movement of the charges (electrons), an additional transverse voltage is created, called the Hall voltage UH. In classical Hall effect, the Hall resistance RH is simply a linear function of magnetic field and resistivity also *ρxy* � B. In QHE we see a series of the so-called Hall plateaus in which *ρxy* is a

> *<sup>ρ</sup>xy* <sup>¼</sup> <sup>1</sup> *ν h*

(where *e* is the electron charge and *ν* = 1,2, … an integer which means the number of the states occupied by electrons under the Fermi level and is called as *filling factor*) independent of all microscopic details (including the precise value of the magnetic field). Associated with each of these plateaus is a dramatic decrease in the dissipative resistivity *ρxx* ! 0 which drops as much as 13 orders of magnitude in

QHE is a two-dimensional phenomenon because when the magnetic field B is perpendicular to the plane of the hetero-structure, the movement of the electrons in the plane of the quantum well is completely quantized. This quantization is universal and independent of all microscopic details such as the type of semiconductor material, the purity of the sample, the precise value of the magnetic field, and so forth. The growth of the magnetic field causes an increase of the distance between

levels then, for the electrons occupying the Fermi level, there are no states to

*,* and when the Fermi level is located between Landau

*<sup>e</sup>*<sup>2</sup> (2)

ered by von Klitzing [9] in 1980. We can say that QHE is a manifestation of

mobility transistor (HEMT) [8].

quantum mechanics on macroscopic scales [10].

*The function of the state density for 2DEG in a quantum well.*

**3.4 Quantum Hall effect**

**Figure 5.**

*Quantum Mechanics*

universal constant

the plateau regions.

Landau levels, ℏ*ω<sup>c</sup>* = *eB/m\**

**250**

*(a) QHE (the Hall resistance* RH *as function of magnetic field* B*); (b) the Shubnikov-de Haas oscillations (magnetoresistance* ρxx(B)*) for hetero-structure GaAs/AlGaAs [11].*

dissipate, and it cannot move in the electric field—no electric current exists and ρxx is zero. In this situation the Hall voltage is constant until the Fermi level does not reach the next Landau level, and the next step of the Hall voltage takes place (in the case of bulk material, there are always scattering channels causing the presence of electric current in a strong magnetic field, therefore the dependence of the Hall voltage on the magnetic field is continuous and reflects the continuity of the density function of states (see **Figure 5**)).

As a result, the QHE is now used to maintain the standard of electrical resistance by metrology laboratories globally.

The magnitude h/e2 = 25,812,80 Ω is so important as constant of the fine structure in the quantum electrodynamics.

#### **3.5 Quasi-rectangular quantum well**

To obtain a quantum well with a rectangular shape, it should be placed close enough to two heterojunctions as shown in **Figure 7**. How close? The experiment shows that at a 200 nm distance, two GaAs/AlGaAs heterojunctions exhibit the properties of a rectangular QW [11]. It can be seen from **Figure 7** that these two

**Figure 7.** *QW formed from two heterojunctions.*

heterojunctions must be a mirror image of each other: first is the growth of the GaAs layer—the QW—and next is of the AlGaAs layer, the *barrier* for QW. It is clear that such a well has a form still far from a rectangular well, but it is already known how to achieve the form of a rectangular potential: stretch the middle GaAs layer as much as possible. In this case we would have a very wide quantum well.

But there is another way of special engineering allowing to obtain a real rectangular QW considered in Section 3.7. Modeling of quasi-rectangular QW using heterojunctions InGaAs/InAlGaAs will be shown in the next paragraph.

#### **3.6 Modeling of quasi-rectangular QW based on the InGaAs/InAlGaAs heterojunctions**

Heterojunctions InGaAs/InAlGaAs are important advantage in comparison with GaAs/AlGaAs because low effective mass of electrons—adding the InAs to the QW material, i.e., to GaAs—allows effective mass to be significantly reduced from 0.65 m0 for GaAs to even 0.4m0 for InGaAs with 65% of the InAs. This means that the electron mobility is almost doubled, which is the main goal of the HEMT modeling. On the other hand, the composition of two solid solutions—InxGa1-xAs for QW and InyAl1-yAs for a barrier—can be selected so as to minimize mismatch of the lattice parameters. Such hetero-structures are isomorphic. In this way, the issue was the production of QW for HEMT based on isomorphic hetero-structures. That could be used in industry, so the production technology also had to be industrial. To implement this duty, the MOCVD technology was developed at the Institute of Electronic Materials Technology (ITME) in Warsaw for the production of isomorphic hetero-structures based on InGaAs/InAlAs heterojunctions. The structures are consisted from single InxGa1-xAs QW and from the two InyAl1-yAs layers—barriers. Four types (see **Table 1**) of different forms of structures with a single QW (SQW) were produced by MOCVD on semi-insulating GaAs at ITME by W. Strupiński group and tested at the Center for Microelectronics and Nanotechnology at the University of Rzeszów during the years 2005–2014 [12, 13]. After that, the program of producing double quantum wells (DQW) and multiple quantum wells (MQW) was developed in years 2015–2018 [14–16].

The magneto-transport measurements, i.e., Hall's resistance curves RH (B) or Rxy (B)

In order to interpret the curves Rxy (B) and Rxx (B) for the QW 1088 which is practically a triangle QW (the electrons are located in the bottom left triangle), the

1*=*2

 

<sup>¼</sup> *<sup>E</sup>*<sup>∗</sup> *g* 2*m*<sup>∗</sup> *c*

<sup>1</sup>*=*<sup>2</sup>

� 4*eF*ℏ*π*ð Þ *i* þ 3*=*4 (3)

**concentration (10<sup>12</sup> cm**�**<sup>2</sup>**

5.0

**)**

and longitudinal magnetoresistance *ρ*xx (B), were performed for all the presented SQWs. It is clearly seen in**Figure 9** the plateau on the curve Rxy (B) corresponding to the filling factors ν = 3, 6, 8, etc. Explanation of this values of filling factor is presented in

**Sample Channel parameters δ-doping donor**

1093 75 20 Sharp interface 2.5

1607 65 23.5 Sharp interface 3.5 1088 53 20 Sharp interface 0.7

**QW profile**

channel

**Thickness (nm)**

1098 65 20 Changing composition in a

*Parameters of the channels and descriptions of the interfaces for the SQWs.*

**Figure 10** where the curve of Rxy (B) as well as Rxx (B) is interpreted.

 

<sup>2</sup> ln *<sup>b</sup>*<sup>1</sup>*=*<sup>2</sup> � *<sup>a</sup>*<sup>1</sup>*=*<sup>2</sup> ð Þ *b* � *a*

theory developed by W. Zawadzki [17] was used

*<sup>b</sup>*<sup>1</sup>*=*<sup>2</sup> <sup>þ</sup> ð Þ *<sup>b</sup>* � *<sup>a</sup>*

*The cross section of the SQW grown by the MOCVD.*

**Composition of In (%)**

*Realization of the Quantum Confinement DOI: http://dx.doi.org/10.5772/intechopen.93112*

ð Þ *<sup>a</sup>* <sup>þ</sup> *<sup>b</sup> <sup>a</sup>*<sup>1</sup>*=*<sup>2</sup>

**253**

**Figure 8.**

**Table 1.**

#### *3.6.1 SQWs*

In **Figure 8**, cross section of SQWs obtained by MOCVD on semi-insulated GaAs substrates is shown. If the δ-doping layer with Si is at the top above QW and below at QW then, the shape of the QW is symmetrical as in **Figure 9**, if and only at the top, a QW is asymmetric as in **Figure 10**.

#### *Realization of the Quantum Confinement DOI: http://dx.doi.org/10.5772/intechopen.93112*


#### **Table 1.**

heterojunctions must be a mirror image of each other: first is the growth of the GaAs layer—the QW—and next is of the AlGaAs layer, the *barrier* for QW. It is clear that such a well has a form still far from a rectangular well, but it is already known how to achieve the form of a rectangular potential: stretch the middle GaAs layer as much as possible. In this case we would have a very wide quantum well.

But there is another way of special engineering allowing to obtain a real rectan-

Heterojunctions InGaAs/InAlGaAs are important advantage in comparison with GaAs/AlGaAs because low effective mass of electrons—adding the InAs to the QW material, i.e., to GaAs—allows effective mass to be significantly reduced from 0.65 m0 for GaAs to even 0.4m0 for InGaAs with 65% of the InAs. This means that the electron mobility is almost doubled, which is the main goal of the HEMT modeling. On the other hand, the composition of two solid solutions—InxGa1-xAs for QW and InyAl1-yAs for a barrier—can be selected so as to minimize mismatch of the lattice parameters. Such hetero-structures are isomorphic. In this way, the issue was the production of QW for HEMT based on isomorphic hetero-structures. That could be used in industry, so the production technology also had to be industrial. To implement this duty, the MOCVD technology was developed at the Institute of Electronic Materials Technology (ITME) in Warsaw for the production of isomorphic hetero-structures based on InGaAs/InAlAs heterojunctions. The structures are consisted from single InxGa1-xAs QW and from the two InyAl1-yAs layers—barriers. Four types (see **Table 1**) of different forms of structures with a single QW (SQW) were produced by MOCVD on semi-insulating GaAs at ITME by W. Strupiński group and tested at the Center for Microelectronics and Nanotechnology at the University of Rzeszów during the years 2005–2014 [12, 13]. After that, the program of producing double quantum wells (DQW) and multiple quantum wells (MQW)

In **Figure 8**, cross section of SQWs obtained by MOCVD on semi-insulated GaAs substrates is shown. If the δ-doping layer with Si is at the top above QW and below at QW then, the shape of the QW is symmetrical as in **Figure 9**, if and only at the

gular QW considered in Section 3.7. Modeling of quasi-rectangular QW using heterojunctions InGaAs/InAlGaAs will be shown in the next paragraph.

**3.6 Modeling of quasi-rectangular QW based on the InGaAs/InAlGaAs**

**heterojunctions**

*QW formed from two heterojunctions.*

**Figure 7.**

*Quantum Mechanics*

was developed in years 2015–2018 [14–16].

top, a QW is asymmetric as in **Figure 10**.

*3.6.1 SQWs*

**252**

*Parameters of the channels and descriptions of the interfaces for the SQWs.*

#### **Figure 8.**

*The cross section of the SQW grown by the MOCVD.*

The magneto-transport measurements, i.e., Hall's resistance curves RH (B) or Rxy (B) and longitudinal magnetoresistance *ρ*xx (B), were performed for all the presented SQWs. It is clearly seen in**Figure 9** the plateau on the curve Rxy (B) corresponding to the filling factors ν = 3, 6, 8, etc. Explanation of this values of filling factor is presented in **Figure 10** where the curve of Rxy (B) as well as Rxx (B) is interpreted.

In order to interpret the curves Rxy (B) and Rxx (B) for the QW 1088 which is practically a triangle QW (the electrons are located in the bottom left triangle), the theory developed by W. Zawadzki [17] was used

$$a(a+b)a^{1/2}b^{1/2} + (b-a)^2 \ln \left| \frac{b^{1/2} - a^{1/2}}{(b-a)^{1/2}} \right| = \left[ \frac{E\_g^\*}{2m\_c^\*} \right]^{1/2} \times 4eF\hbar\pi(i+3/4) \tag{3}$$

#### **Figure 9.**

*The Hall resistivity curve Rxy (B) and longitudinal magnetoresistivity curve Rxx (B) for the SQW 1088 (see Table 1) with asymmetric shape of QW [12, 13].*

where *a=E – <sup>E</sup>┴;b=Eg +E+E┴; E* is the energy of sub-band in QW*, <sup>E</sup>┴* is the Landau level energy sought, *Eg* is the energy gap, and *<sup>F</sup>* is the electrical field strength caused by interface and is determined by linear potential *U = eFz*.

Results of calculations presented in **Figure 10** show that the intersection of Landau levels of two energy sub-subbands takes place; hence the picture of QHE and SdH oscillations is more complicated but is perfectly explained by the theory for the triangle QW.

#### **3.7 Special engineering of a rectangular QW**

The special engineering of QW involves changing the composition of the solid solution in the well to compensate for the reduction in potential at the left and right corner of the bottom of the well. The schema of such compensation is shown in **Figure 11**: there is a change in the composition in the quantum well from the left side of the interface and the right side too. This mild change from x = 0.53 to 0.65 (on the left and vice versa from 0.65 to 0.53 on the right) accurately compensates for the value of the energy gap, as well as the decrease in the bottom of the well the conduction band—so that it becomes almost flat.

The theoretical interpretation of experimental curves presented in **Figure 12** was

¼ ℏ2

*<sup>π</sup>*<sup>2</sup>ð Þ *<sup>i</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> 2*m*<sup>∗</sup>

<sup>0</sup> *<sup>a</sup>*<sup>2</sup>*<sup>k</sup>* (4)

performed by curves of the Landau level (LL) (presented above) calculated

*Interpretation of the QHE curve and magnetoresistance curve for three-angle SQW 1088 [12, 13].*

ð Þ *E* � *E*<sup>⊥</sup> *Eg* þ *E* þ *E*<sup>⊥</sup>

*Eg*

according the theory of Zawadzki [17]:

*Schema of the rectangular QW (1098) engineering [13].*

**Figure 10.**

*Realization of the Quantum Confinement DOI: http://dx.doi.org/10.5772/intechopen.93112*

**Figure 11.**

**255**

This fact that we are dealing with an excellent rectangular quantum well confirms the experimental magneto-transport curves obtained for QW 1098. *Realization of the Quantum Confinement DOI: http://dx.doi.org/10.5772/intechopen.93112*

**Figure 10.** *Interpretation of the QHE curve and magnetoresistance curve for three-angle SQW 1088 [12, 13].*

**Figure 11.**

where *a=E – <sup>E</sup>┴;b=Eg +E+E┴; E* is the energy of sub-band in QW*, <sup>E</sup>┴* is the Landau level energy sought, *Eg* is the energy gap, and *<sup>F</sup>* is the

*The Hall resistivity curve Rxy (B) and longitudinal magnetoresistivity curve Rxx (B) for the SQW 1088*

*U = eFz*.

**254**

**Figure 9.**

*Quantum Mechanics*

for the triangle QW.

**3.7 Special engineering of a rectangular QW**

*(see Table 1) with asymmetric shape of QW [12, 13].*

the conduction band—so that it becomes almost flat.

electrical field strength caused by interface and is determined by linear potential

Results of calculations presented in **Figure 10** show that the intersection of Landau levels of two energy sub-subbands takes place; hence the picture of QHE and SdH oscillations is more complicated but is perfectly explained by the theory

The special engineering of QW involves changing the composition of the solid solution in the well to compensate for the reduction in potential at the left and right corner of the bottom of the well. The schema of such compensation is shown in **Figure 11**: there is a change in the composition in the quantum well from the left side of the interface and the right side too. This mild change from x = 0.53 to 0.65 (on the left and vice versa from 0.65 to 0.53 on the right) accurately compensates for the value of the energy gap, as well as the decrease in the bottom of the well—

This fact that we are dealing with an excellent rectangular quantum well confirms the experimental magneto-transport curves obtained for QW 1098.

*Schema of the rectangular QW (1098) engineering [13].*

The theoretical interpretation of experimental curves presented in **Figure 12** was performed by curves of the Landau level (LL) (presented above) calculated according the theory of Zawadzki [17]:

$$\frac{\left((E - E\_{\perp})\left(E\_{\rm g} + E + E\_{\perp}\right)\right)}{E\_{\rm g}} = \frac{\hbar^2 \pi^2 (i + 1)^2}{2m\_0^\* a^2 k} \tag{4}$$

Pauli's principle was known in relation to atoms, molecules, and crystal theory, while for the first time an artificial object was generated in which this principle was spectacularly confirmed—in an electron system consisting of two closely spaced QWs. In the inset of **Figure 13**, the potential profile of two QWs with narrow barrier between wells is shown. Due to narrow barrier, the tunneling among QWs is facilitated and electrons in these two QWs constitute the common electron system. This system is subject to Pauli's principle, as a result of which there are electron states in which the spin part of the wave function has the opposite sign—

symmetrical and anti-symmetrical functions and correspondently symmetrical and anti-symmetrical states—separated by the energy gap, the so-called SAS gap.

For first time, this effect was considered in the work of G. S. Boebinger et al. [18] where the GaAs/AlGaAs DQWs produced by MBE technology were investigated. This fact was observed experimentally on the QHE curves: where quantum Hall states at odd integer ν (filling factor) were missing, the ν = odd quantum Hall states

Magneto-transport phenomena were studied also for the InGaAs/InAlAs DQWs.

� �ℏ*ωcj* <sup>þ</sup> *Ej* <sup>þ</sup> *<sup>V</sup><sup>F</sup>*

exp *En*<sup>0</sup> *<sup>j</sup>*

1

<sup>0</sup> � *μ<sup>c</sup>* h i*=kBT* n o <sup>þ</sup> <sup>1</sup>

*j j*<sup>0</sup> *qxy* � � (8)

*nj* (7)

9 = ;

�

*μBB, and ΔSAS—splitting on the*

*jj*<sup>0</sup> *qxy* � � is factor of screening [19].

� � � �

� � � 2 is

In addition to QHE and SdH oscillation, magneto-phonon resonance was also studied and interpreted using the LL energy theory for the DQW [19]:

2

8 < :

where *j* is number of the energy sub-band, *n* is number of the LL, *An*0*<sup>n</sup> qxy*

*Enj* <sup>¼</sup> *<sup>n</sup>* � <sup>1</sup>

*dqxyqxy* �

þ ð∞

0

� � � 2 *VF*

*Three kinds of splitting of energy states in DQW: cyclotron ħɷc, spin splitting g\**

*<sup>n</sup> qxy* � � � �

matrix element for two Landau levels n and n', *V<sup>F</sup>*

� *An*<sup>0</sup>

�

*3.8.2 Magneto-transport phenomena*

*Realization of the Quantum Confinement DOI: http://dx.doi.org/10.5772/intechopen.93112*

originate from the SAS gap [18].

*VF nj* ¼ � <sup>1</sup> 2*π* X *n*0 *j* 0

**Figure 13.**

**257**

*symmetric and antisymmetric states.*

**Figure 12.** *The Hall resistivity curve Rxy (B) and longitudinal magnetoresistivity curve Rxx (B) for the SQW 1098 [13].*

$$E\_{\perp} = -\frac{E\_{\rm g}}{2} + \frac{E\_{\rm g}}{2} \sqrt{\mathbf{1} + \frac{4\mu\_{B}B}{E\_{\rm g}} \left[ f\_{1} \frac{m\_{0}}{m\_{\rm c}^{\*}} \left( n + \frac{1}{2} \right) \pm \frac{1}{2} \mathbf{g}\_{0}^{\*} f\_{2} \right]} \tag{5}$$

$$f\_1 = \frac{\left(E\_\text{g} + \Delta\right)\left(E\_\perp + E\_\text{g}\frac{2}{3}\Delta\right)}{\left(E\_\text{g} + \frac{2}{3}\Delta\right)\left(E\_\perp + E\_\text{g} + \Delta\right)}\\f\_2 = \frac{E\_\text{g} + \frac{2}{3}\Delta}{E\_\perp + E\_\text{g} + \Delta} \tag{6}$$

where *<sup>E</sup>* is the energy of sub-band in QW, *<sup>E</sup>┴* is the Landau level energy sought, and *Eg* is the energy gap, while Δ is the value of the spin-orbit splitting, i is the number of sub-band, n is the number of LL, μ<sup>B</sup> is the Bohr magneton, and *m*<sup>∗</sup> *<sup>c</sup>* is the effective mass of electrons on the bottom of the conduction band. As you can see in the right side of Eq. (4), the energies of states in a rectangular well with a correction for the finite potential through coefficient *k* are described.

It is seen that theoretical curve of the Fermi level in the course of the magnetic field reflected both the plateau of the Rxr(B) and the maxima of the Rxx(B) experimental curves: the QHE plateau positions correspond to the FL positions between LL that simultaneously correspond to the minima of the SdH oscillations.

In this way, it can be said that thanks to special engineering, it has been possible to make a *real rectangular potential of QW described by quantum-mechanical theory*.

#### **3.8 Double quantum well**

#### *3.8.1 The SAS-splitting*

Technology successes have allowed us to experimentally confirm one interesting quantum-mechanical phenomenon—it concerns the splitting into symmetrical and anti-symmetrical states thanks to the *Pauli exclusion principle*, in other words, *exchange interaction*.

#### *Realization of the Quantum Confinement DOI: http://dx.doi.org/10.5772/intechopen.93112*

Pauli's principle was known in relation to atoms, molecules, and crystal theory, while for the first time an artificial object was generated in which this principle was spectacularly confirmed—in an electron system consisting of two closely spaced QWs. In the inset of **Figure 13**, the potential profile of two QWs with narrow barrier between wells is shown. Due to narrow barrier, the tunneling among QWs is facilitated and electrons in these two QWs constitute the common electron system. This system is subject to Pauli's principle, as a result of which there are electron states in which the spin part of the wave function has the opposite sign symmetrical and anti-symmetrical functions and correspondently symmetrical and anti-symmetrical states—separated by the energy gap, the so-called SAS gap.

#### *3.8.2 Magneto-transport phenomena*

For first time, this effect was considered in the work of G. S. Boebinger et al. [18] where the GaAs/AlGaAs DQWs produced by MBE technology were investigated. This fact was observed experimentally on the QHE curves: where quantum Hall states at odd integer ν (filling factor) were missing, the ν = odd quantum Hall states originate from the SAS gap [18].

Magneto-transport phenomena were studied also for the InGaAs/InAlAs DQWs. In addition to QHE and SdH oscillation, magneto-phonon resonance was also studied and interpreted using the LL energy theory for the DQW [19]:

$$E\_{\eta j} = \left(n - \frac{1}{2}\right) \hbar a\_{\circ j} + E\_j + V\_{\eta j}^F \tag{7}$$

$$\begin{split} V\_{\eta j}^F &= -\frac{1}{2\pi} \sum\_{\pi' j'} \int\_0^{+\infty} dq\_{\mathrm{xy}} q\_{\mathrm{xy}} \times \left\{ \frac{1}{\exp\left\{ \left[ E\_{\pi' j'} - \mu\_\epsilon \right] / k\_B T \right\} + 1} \right\} \\ &\times \left| A\_{\mathrm{\pi' n}} \left( q\_{\mathrm{xy}} \right) \right|^2 V\_{\mathrm{jj}}^F \left( q\_{\mathrm{xy}} \right) \end{split} \tag{8}$$

where *j* is number of the energy sub-band, *n* is number of the LL, *An*0*<sup>n</sup> qxy* � � � � � � � � 2 is matrix element for two Landau levels n and n', *V<sup>F</sup> jj*<sup>0</sup> *qxy* � � is factor of screening [19].

#### **Figure 13.**

*Three kinds of splitting of energy states in DQW: cyclotron ħɷc, spin splitting g\* μBB, and ΔSAS—splitting on the symmetric and antisymmetric states.*

*<sup>E</sup>*<sup>⊥</sup> ¼ � *Eg*

**Figure 12.**

*Quantum Mechanics*

**3.8 Double quantum well**

*3.8.1 The SAS-splitting*

*exchange interaction*.

**256**

2 þ *Eg* 2

1 þ

*<sup>f</sup>* <sup>1</sup> <sup>¼</sup> *Eg* <sup>þ</sup> <sup>Δ</sup> � � *<sup>E</sup>*<sup>⊥</sup> <sup>þ</sup> *Eg*

for the finite potential through coefficient *k* are described.

*Eg* <sup>þ</sup> <sup>2</sup>

4*μBB Eg*

*The Hall resistivity curve Rxy (B) and longitudinal magnetoresistivity curve Rxx (B) for the SQW 1098 [13].*

*f* 1 *m*<sup>0</sup> *m*<sup>∗</sup> *c*

2 <sup>3</sup> <sup>Δ</sup> � �

where *<sup>E</sup>* is the energy of sub-band in QW, *<sup>E</sup>┴* is the Landau level energy sought, and *Eg* is the energy gap, while Δ is the value of the spin-orbit splitting, i is the number of sub-band, n is the number of LL, μ<sup>B</sup> is the Bohr magneton, and *m*<sup>∗</sup>

effective mass of electrons on the bottom of the conduction band. As you can see in the right side of Eq. (4), the energies of states in a rectangular well with a correction

It is seen that theoretical curve of the Fermi level in the course of the magnetic field reflected both the plateau of the Rxr(B) and the maxima of the Rxx(B) experimental curves: the QHE plateau positions correspond to the FL positions between

In this way, it can be said that thanks to special engineering, it has been possible to make a *real rectangular potential of QW described by quantum-mechanical theory*.

Technology successes have allowed us to experimentally confirm one interesting quantum-mechanical phenomenon—it concerns the splitting into symmetrical and anti-symmetrical states thanks to the *Pauli exclusion principle*, in other words,

LL that simultaneously correspond to the minima of the SdH oscillations.

<sup>3</sup> <sup>Δ</sup> � � *<sup>E</sup>*<sup>⊥</sup> <sup>þ</sup> *Eg* <sup>þ</sup> <sup>Δ</sup> � � *<sup>f</sup>* <sup>2</sup> <sup>¼</sup> *Eg* <sup>þ</sup> <sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s � �

*n* þ 1 2 � �

� 1 2 *g* ∗ <sup>0</sup> *f* <sup>2</sup>

<sup>3</sup> Δ

*<sup>E</sup>*<sup>⊥</sup> <sup>þ</sup> *Eg* <sup>þ</sup> <sup>Δ</sup> (6)

(5)

*<sup>c</sup>* is the

The combination of this DQW theory with the Landau level theory presented above for SQW (Eq. (4)) gives us the following equation:

$$\frac{\left(E - E\_{\perp}\right)\left(E\_{\rm g} + E + E\_{\perp}\right)}{E\_{\rm g}} = \frac{\hbar^2 \pi^2 (i + 1)^2}{2m\_0^\* a^2 k} \pm \left(\Delta\_0 + 0.38 E\_{\perp}\right) \tag{9}$$

Adding Eqs. (5) and (6) to this (9) allows the calculation of the LL energy in DQW. The value of ΔSAS in Eq. (9) depends from magnetic field *B* as function of energy E┴:

$$
\Delta\_{\text{SAS}} = \Delta\_0 + \mathbf{0.38 E}\_{\perp}.\tag{10}
$$

**Figure 15.**

**Figure 16.**

**259**

*2* ! *CVE – band 8; 1* ! *CVE – band 9 [16].*

*CO2 in the atmosphere) [16].*

*Realization of the Quantum Confinement DOI: http://dx.doi.org/10.5772/intechopen.93112*

*Optical absorption curve obtained by averaging the optical reflection R0 when scanning the sample surface 2506. The recorded absorption bands are renumbered (the double minimum 5 is due to the strong absorption of*

*The electron states in the DQW and optical transitions responsible for absorption bands shown in Figure 15: 1* ! *2 – band 1; 2* ! *3 – band 2; 1* ! *3 – band 3; 3* ! *4 – band 4; 2* ! *4 – band 6; 1* ! *4 – band 7;*

where Δ<sup>0</sup> is the ΔSAS value without magnetic field.

In **Figure 14**, the LL energies for DQW 2506 (see **Table 1**) and interpretation of the Rxy curve obtained for this DQW are presented. The splits caused by the SAS gap are clearly seen on the Rxx (B) curve. These are experimental data that indirectly indicate the SAS-splitting in DQW. But on the same DQW it was possible to make optical measurements from which the energy states were directly determined.

These optical measurements that concern the optical reflection in the infrared region were made using infrared microscope.

#### *3.8.3 Direct determination of the energy states in the DQW*

The experiment on the infrared reflection was performed at the Frascati National Laboratory in Italy. Synchrotron radiation served the brilliant infrared radiation source.

**Figure 14.** *Interpretation of Rxy (B) curve for DQW 2506 [15] .*

**Figure 15.**

The combination of this DQW theory with the Landau level theory presented

*<sup>π</sup>*2ð Þ *<sup>i</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> 2*m*<sup>∗</sup>

<sup>0</sup> *<sup>a</sup>*2*<sup>k</sup>* � ð Þ <sup>Δ</sup><sup>0</sup> <sup>þ</sup> <sup>0</sup>*:*38*E*<sup>⊥</sup> (9)

ΔSAS ¼ Δ<sup>0</sup> þ 0*:*38 E⊥*:* (10)

¼ ℏ2

Adding Eqs. (5) and (6) to this (9) allows the calculation of the LL energy in DQW. The value of ΔSAS in Eq. (9) depends from magnetic field *B* as function of

In **Figure 14**, the LL energies for DQW 2506 (see **Table 1**) and interpretation of the Rxy curve obtained for this DQW are presented. The splits caused by the SAS gap are clearly seen on the Rxx (B) curve. These are experimental data that indirectly indicate the SAS-splitting in DQW. But on the same DQW it was possible to make optical measurements from which the energy states were directly determined. These optical measurements that concern the optical reflection in the infrared

The experiment on the infrared reflection was performed at the Frascati National Laboratory in Italy. Synchrotron radiation served the brilliant infrared

above for SQW (Eq. (4)) gives us the following equation:

where Δ<sup>0</sup> is the ΔSAS value without magnetic field.

*3.8.3 Direct determination of the energy states in the DQW*

region were made using infrared microscope.

ð Þ *E* � *E*<sup>⊥</sup> *Eg* þ *E* þ *E*<sup>⊥</sup>

*Eg*

energy E┴:

*Quantum Mechanics*

radiation source.

**Figure 14.**

**258**

*Interpretation of Rxy (B) curve for DQW 2506 [15] .*

*Optical absorption curve obtained by averaging the optical reflection R0 when scanning the sample surface 2506. The recorded absorption bands are renumbered (the double minimum 5 is due to the strong absorption of CO2 in the atmosphere) [16].*

#### **Figure 16.**

*The electron states in the DQW and optical transitions responsible for absorption bands shown in Figure 15: 1* ! *2 – band 1; 2* ! *3 – band 2; 1* ! *3 – band 3; 3* ! *4 – band 4; 2* ! *4 – band 6; 1* ! *4 – band 7; 2* ! *CVE – band 8; 1* ! *CVE – band 9 [16].*

Supplying the infrared microscope with such a brilliant radiation source allowed for unique results [16]. For the first time, the energies of electron states were determined directly in DQW (see **Figures 15** and **16**) analogically as it was once done for the natural H2 molecule [20]. From **Figure 16** it can be seen that the delta-SAS varies depending on the *j* number of the energy sub-band from 3.1 meV for *j* = 1 to 9.4 meV for *j* = 4.

**References**

2004

[1] Herman MA, Richter W, Sitter H. Liquid phase epitaxy. In: Epitaxy. Springer Series in Materials Science. Vol. 62. Berlin, Heidelberg: Springer;

*Realization of the Quantum Confinement DOI: http://dx.doi.org/10.5772/intechopen.93112*

> determination of the fine-structure constant based on quantized hall resistance. Physical Review Letters.

[10] Girvin SM. The Quantum Hall Effect: Novel Excitations and Broken Symmetries. New York: Springer-

[11] Weisbuch C, Vinter B. Quantum Semiconductor Structures. San Diego:

[12] Tomaka G, Sheregii EM, Kąkol T, Strupiński W, Jasik A, Jakiela R. Charge carriers parameters in the conductive channels of HEMTs. Physica Status

Marchewka M, Tomaka G, Kolek A, Stadle A, et al. Parallel magnetotransport in multiple quantum well structures. Low Temperature Physics.

Marchewka M, Woźny M, Tomaka G. Magnetophonon resonance in double quantum Wells. Physical Review B.

Tralle I, Ploch D, Tomaka G, Furdak M, et al. Magnetospectroscopy of double quantum wells. Physica E. 2008;**40**:

symmetric and ant-symmetric states in

temperature. Physical Review B. 2009;

[17] Zawadzki W. Theory of optical transitions in inversion layers of narrowgap semiconductors. Journal of Physics C: Solid State Physics. 1983;**16**:229

[15] Marchewka M, Sheregii EM,

[16] Marchewka M, Sheregii EM, Tralle I, Marcelli A, Piccinini M, Cebulski J. Optically detected

double quantum Wells at room

1980;**45**:494

Verlag; 1999

Academic Press; 2007

Solidi (A). 2003;**195**(127)

[13] Sheregii EM, Ploch D,

[14] Płoch D, Sheregii EM,

2004;**30**:1146

2009;**79**:195434

894-904

**80**:125316

[2] Alferov ZI, Andreev VM, Korol'kov VI, Portnoy EL, Yakovenko AA. AlAs-GaAs Heterojunction Injection Lasers with a Low Room-temperature

[3] Alferov ZI. Nobel lecture: The double

applications in physics, electronics, and technology. Reviews of Modern Physics.

[4] Zawadzki W, Pfeffer P. Average forces in bound and resonant quantum states. Physical Review B: Condensed Matter and Materials Physics. 2001;**64**:

[5] Herman MA, Sitter H. Molecular beam epitaxy. In: Fundamentals and Current Status. Springer Series in Materials Science. Vol. 7. Heidelberg:

[6] Arthur JR, LePore JJ. GaAs, GaP, and GaAsxP1–<sup>x</sup> Epitaxial films grown by molecular beam deposition. Journal of Vacuum Science and Technology. 1969;

[7] Sakaki H. Prospects of advanced quantum nano-structures and roles of molecular beam epitaxy. In: International Conference on Molecular Bean Epitaxy.

[8] Mimura T. The early history of the high electron mobility transistor (HEMT). IEEE Transactions on Microwave Theory and Techniques.

[9] von Klitzing K, Dorda G, Pepper M.

New method for high-accuracy

2002. p. 5. DOI: 10.1109/MBE

2002;**50**(3):780-782

Springer Verlag; 1996

hetero-structure concept and its

Threshold Soviet Physics Semiconductors. 1969;**3**:460

2001;**73**:767

235313

**6**:545

**261**

#### **3.9 Conclusion**

The implementation of the quantum-mechanical problem of electron entrapment in a quantum well has been described. Various shapes of quantum wells produced by advanced technologies as MBE and MOCVD, as well as based on different materials—are considered. Quantum wells based on GaAs/AlGaAs heterojunctions are especially important for the production of the semiconductor lasers, while the ones based on InGaAs/InAlAs heterojunctions are for the production of the HEMT transistors. Thanks to special engineering, it has been possible to make a *real rectangular potential of QW described by quantum-mechanical theory.*

Research into *double quantum wells* is a significant cognitive interest as an analogue of a two-atom hydrogen molecule in solid-state physics where essential role plays such quantum-mechanical phenomenon as *exchange interaction*. It can be predicted that their applications in electronics will also not make us wait long.

### **Author details**

Eugen M. Sheregii University of Rzeszow, Rzeszow, Poland

\*Address all correspondence to: sheregii@ur.edu.pl

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Realization of the Quantum Confinement DOI: http://dx.doi.org/10.5772/intechopen.93112*

#### **References**

Supplying the infrared microscope with such a brilliant radiation source allowed

The implementation of the quantum-mechanical problem of electron entrapment in a quantum well has been described. Various shapes of quantum wells produced by advanced technologies as MBE and MOCVD, as well as based on different materials—are considered. Quantum wells based on GaAs/AlGaAs heterojunctions are especially important for the production of the semiconductor lasers, while the ones based on InGaAs/InAlAs heterojunctions are for the production of the HEMT transistors. Thanks to special engineering, it has been possible to make a *real rectangular potential of QW described by quantum-mechanical theory.* Research into *double quantum wells* is a significant cognitive interest as an analogue of a two-atom hydrogen molecule in solid-state physics where essential role plays such quantum-mechanical phenomenon as *exchange interaction*. It can be predicted that their applications in electronics will also not make us wait long.

for unique results [16]. For the first time, the energies of electron states were determined directly in DQW (see **Figures 15** and **16**) analogically as it was once done for the natural H2 molecule [20]. From **Figure 16** it can be seen that the delta-SAS varies depending on the *j* number of the energy sub-band from 3.1 meV for *j* = 1

to 9.4 meV for *j* = 4.

*Quantum Mechanics*

**3.9 Conclusion**

**Author details**

Eugen M. Sheregii

**260**

University of Rzeszow, Rzeszow, Poland

provided the original work is properly cited.

\*Address all correspondence to: sheregii@ur.edu.pl

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

[1] Herman MA, Richter W, Sitter H. Liquid phase epitaxy. In: Epitaxy. Springer Series in Materials Science. Vol. 62. Berlin, Heidelberg: Springer; 2004

[2] Alferov ZI, Andreev VM, Korol'kov VI, Portnoy EL, Yakovenko AA. AlAs-GaAs Heterojunction Injection Lasers with a Low Room-temperature Threshold Soviet Physics Semiconductors. 1969;**3**:460

[3] Alferov ZI. Nobel lecture: The double hetero-structure concept and its applications in physics, electronics, and technology. Reviews of Modern Physics. 2001;**73**:767

[4] Zawadzki W, Pfeffer P. Average forces in bound and resonant quantum states. Physical Review B: Condensed Matter and Materials Physics. 2001;**64**: 235313

[5] Herman MA, Sitter H. Molecular beam epitaxy. In: Fundamentals and Current Status. Springer Series in Materials Science. Vol. 7. Heidelberg: Springer Verlag; 1996

[6] Arthur JR, LePore JJ. GaAs, GaP, and GaAsxP1–<sup>x</sup> Epitaxial films grown by molecular beam deposition. Journal of Vacuum Science and Technology. 1969; **6**:545

[7] Sakaki H. Prospects of advanced quantum nano-structures and roles of molecular beam epitaxy. In: International Conference on Molecular Bean Epitaxy. 2002. p. 5. DOI: 10.1109/MBE

[8] Mimura T. The early history of the high electron mobility transistor (HEMT). IEEE Transactions on Microwave Theory and Techniques. 2002;**50**(3):780-782

[9] von Klitzing K, Dorda G, Pepper M. New method for high-accuracy

determination of the fine-structure constant based on quantized hall resistance. Physical Review Letters. 1980;**45**:494

[10] Girvin SM. The Quantum Hall Effect: Novel Excitations and Broken Symmetries. New York: Springer-Verlag; 1999

[11] Weisbuch C, Vinter B. Quantum Semiconductor Structures. San Diego: Academic Press; 2007

[12] Tomaka G, Sheregii EM, Kąkol T, Strupiński W, Jasik A, Jakiela R. Charge carriers parameters in the conductive channels of HEMTs. Physica Status Solidi (A). 2003;**195**(127)

[13] Sheregii EM, Ploch D, Marchewka M, Tomaka G, Kolek A, Stadle A, et al. Parallel magnetotransport in multiple quantum well structures. Low Temperature Physics. 2004;**30**:1146

[14] Płoch D, Sheregii EM, Marchewka M, Woźny M, Tomaka G. Magnetophonon resonance in double quantum Wells. Physical Review B. 2009;**79**:195434

[15] Marchewka M, Sheregii EM, Tralle I, Ploch D, Tomaka G, Furdak M, et al. Magnetospectroscopy of double quantum wells. Physica E. 2008;**40**: 894-904

[16] Marchewka M, Sheregii EM, Tralle I, Marcelli A, Piccinini M, Cebulski J. Optically detected symmetric and ant-symmetric states in double quantum Wells at room temperature. Physical Review B. 2009; **80**:125316

[17] Zawadzki W. Theory of optical transitions in inversion layers of narrowgap semiconductors. Journal of Physics C: Solid State Physics. 1983;**16**:229

#### *Quantum Mechanics*

[18] Boebinger GS, Jiang HW, Pfeiffer LN, West KW. Magnetic-fielddriven destruction of quantum hall states in a double quantum well. Physical Review Letters. 1990;**64**:1793

[19] Huang D, Manasreh MO. Effects of the screened exchange interaction on the tunneling and Landau gaps in double quantum wells. Physical Review B. 1996;**54**:2044

[20] Woodgate GK. Elementary Atomic Structure. Oxford: Clarendon Press; 1980

[18] Boebinger GS, Jiang HW,

*Quantum Mechanics*

Review B. 1996;**54**:2044

1980

**262**

Pfeiffer LN, West KW. Magnetic-fielddriven destruction of quantum hall states in a double quantum well. Physical Review Letters. 1990;**64**:1793

[19] Huang D, Manasreh MO. Effects of the screened exchange interaction on the tunneling and Landau gaps in double quantum wells. Physical

[20] Woodgate GK. Elementary Atomic Structure. Oxford: Clarendon Press;

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Quantum Mechanics

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