Questions regarding the use of the book should be directed to the Rights and Permissions Department of INTECHOPEN LIMITED (permissions@intechopen.com). Sivasankaran works as an associate professor in the Department of Mechanical Engineering, College of Engineering, Qassim University.

## 95 Application of Spin-Orbit Coupling in Exotic Graphene Structures

### Introduction

The main assumption of the tight-binding approach to band spectra is that the atoms in the crystal interact very weakly. Again, our view of the crystal is tightly bound (atoms do not lose their identity).

### Band spectra in the tight-binding approach: effects of the overlaps between neighboring orbitals

*Quantum mechanics in a nutshell**Naive tight-binding approach to band theory**Tight-binding band calculation: properly done Recognizing that H crystal ¼ ∑ i � H atomi þ W i �**A simple model that yields an exact tight-binding band spectrum*

Band spectra in the close-binding approach: effects of neighbor-orbital overlaps between neighbor-orbitals. Moreover, the multicenter integrals in the cosine spectrum are unevenly neglected at the top and bottom of the exact spectrum.

### Quantum Monte Carlo method for systems with strongly correlated fermions

*Quantum statistical mechanics in a nutshell**Monte Carlo pursuit of the ground state**The case of fermions*

Therefore, the approximation performs worse for the top than for the bottom of the band. Quantum Monte Carlo method for systems with strongly correlated fermions. of the remaining atoms in the solid.

### Non-equilibrium routes to soft solids

*The non-equilibrium potential (NEP)**The FitzHugh-Nagumo model and its NEP**Arrays of excitable elements**Spatiotemporal pattern formation in arrays of FHN neurons*

For the state vector of the entire set of electrons in the crystal to be completely antisymmetric in exchange, these operators must anticommute with each other, unless they refer to the same location and spin projection. As a consequence of PBC, the NEP landscape along slow manifolds is symmetric with respect to the u1¼u2 line.

### Conclusions

The derivation of quantum theory with respect to fields requires new interpretations of the uncertainty principle, the correspondence principle, complementarity and force. If the atom is in an excited state, the energy lies at the location of the electron.

Excitation

### Decay

The behavior of the radiation field can also be described microscopically by a Lagrangian density. Then the action integral of the Lagrangian density over a certain region of space-time must stand still for all small variations of the continuous coordinates in the region, provided that the discrete coordinates on the boundary remain invariant.

### Interpretation of mathematical models 1 Fully relativistic quantum mechanics

*The path integral formulation**Matrix mechanics**Wave mechanics*

Wave properties, on the other hand, occur in free space when field boundaries have no reference point, so they cannot be observed at all. Wave properties, on the other hand, occur in free space when field boundaries have no reference point, so they cannot be observed at all.

### Discussion

While in classical theory forces are three-dimensional vectors with direction and magnitude, in a fully relativistic theory they are four-dimensional and symmetrical in the coordinates. They have orientation in space-time, but not direction, with magnitude determined by the instantaneous separation of field boundaries according to (1).

### Conclusion

In nuclear magnetic resonance (NMR) experiments [12], a major role of the interface in the enhancement of quantum diffusion was found. The present work is devoted to the development of a self-consistent description of the quantum behavior of 4He atoms in the double boundary proposed in [14].

### Model of the twin boundary

We apply this treatment to the quantum and thermal description of twin boundaries in some metals.

### Atomic potential in continual description

For the homogeneous part of the free energy Eq. 1), the maximum and minimum positions are. A is shown in solid lines and layer B is shown in dotted lines. a) View perpendicular to the layers.

### The atomic potential and hard sphere model in hcp phase

The first term Uan1ð Þx is an anisotropic and non-linear part of the potential in the displacement direction Ox. The analysis (see [14]) of the term Upnðx;ξÞ makes it possible to determine the anisotropic atomic potential Eq. 10) in the following simple form: .

### Quantum atomic spheres and ellipsoids in hcp phase and in the twin boundary

Quantum atomic spheres and ellipsoids in hcp phase and in the twin boundary and in the twin boundary. The probability density at a distance of R0 equal to the radius of the atom in the hcp phase (half the distance between the centers of neighboring atoms in the crystal) is.

Classic atomic thermal spheres and ellipsoids in hcp phase and the twin boundary

The self-consistent description of the twin boundary

### Atom as anisotropic harmonic oscillator in the boundary, one axis In continual description inside the boundary, we have found a change of the

Quantum and thermal description of TB is self-consistent, i.e. the parameters Eq. 31) is varied as a function of some parameter q, which in turn is a function of these parameters:. The hard ellipsoidal model Eq. 35) is used to obtain a new local atomic potential and a new ellipsoid shape. i) The third and additional step qualitatively replicates the previous steps in the same way.

### The self-consistent correspondence of the potential and the uniaxial hard ellipsoid model

The self-consistent agreement between the potential and the uniaxial hard ellipsoidal model hard ellipsoidal model. So for the classical model of the twin boundary (stacking fault) it is possible to estimate the bulk density of the barrier height h and the surface energy density WT according to Eq.

### Discussion and conclusion

*Physical interpretation**Band gap**Mobility**Carrier diffusion**Thermal expansion coefficient*

It is shown that in the twin boundary, the potential of the atom is softer in the direction of the rearrangement of the atomic planes. On average - while at q¼2=3 the instability of the structure occurs in the system of atomic ellipsoids.

### Experimental

The diffusion of charge carriers is fundamental to tune the thermoelectric properties of oxide semiconductors. Oxide semiconductors have a high density of intrinsic defects, which basically affect the diffusion of charge carriers.

### Results and discussion

Seebeck coefficient for Cu2InO4, CuAlO2 and Zn2GeO4 thin films grown by thermal evaporation technique, respectively. All graphs showed that the value of the Seebeck coefficient increases as the annealing temperature increases.

### Conclusion

The power factor increases significantly with increasing annealing and measurement temperature because both the Seebeck coefficient and the electrical conductivity increase. The power factor increases significantly with increasing annealing and measurement temperature because both the Seebeck coefficient and the electrical conductivity increase.

### Methodology

*Data extraction procedure**Detail of data analysis**Amorphous alloy development: evolution of patent application**Amorphous alloy development: analysis by country and assignee nationality Figure 5 shows the number of patent application in various countries and their**Analysis by top ten patent assignees**Technological development strategy analysis: analysis of the top five patent families**Technological exploitation analysis: five most-cited patents*

In this research, patent data was analyzed to explore the technological development of metallic glass materials. The top five families and the five most cited patents are also explored in this study. From subsequent analysis, it was found that the first noticeable increase in the number of patents.

Furthermore, the ratio of copper to nickel and/or cobalt is in the range of 1:2 to 2:1.

### Discharge modeling

*Boundary and initial conditions**Numerical method*

In chapter 5 the influence of the BOLSIG+ input data on the argon discharge is shown. In chapter 6 the characteristics of the Ne discharge are presented with input data from BOLSIG+. The transport equations of the electron energy, electron and ion are also spatially discretized with the finite difference technique.

The discretization of the time terms at the real position of the finite difference technique has been used.

### Results and discussion of argon discharge

*Influence of the voltage and gas pressure*

In this section, we will analyze the spatiotemporal evolution of the anomalous glow discharge in the existence of the metastable atom density. In the second zone, we observe a pseudo exit of the cathodic region, which is characterized by an importance of the ion concentration and the insignificance of the electron. In this part we will study the effect of voltage and gas pressure on argon discharge.

These circumstances of the charged particle manipulate the behavior of the metastable atoms in the study indicated, ie. the cathodic region overflows with electron and metastable atomic concentrations, which go faster the ion species in the existence of the electric field.

### Validity of the model

The results reached the beginning of the BMA database evaluated with those obtained by the experimental method [43, 44]. We find that the initial results of the BMA database are in excellent judgment of conformity against those experimental results [43, 44].

### Influence for input data of argon abnormal glow discharge

*Effect of the metastable lifetime on the characteristics of argon abnormal glow discharge*

In addition to our model identification, properties of the discharge, the average electron energy and the metastable atomic concentrations. We recall that the preceding results were obtained with input data calculated by multiterm estimation of the Boltzmann equation. We remind again that the previous results are identical when calculated excluding the tariff coefficients Komand Kiom.

Consequently, we can calculate the properties of the argon abnormal glow discharge exclusively from Komand Kiom coefficients.

### Characteristics of the neon discharge through entering data of the BOLSIG+

As a consequence, the utilization of the experimental or artificial value of the metastable lifetime has an inconsistency of the abnormal glow discharge characteristics. In this part, we will investigate the characteristics of the anomalous neon glow discharge by inputting data from the BOLSIG+ code. We note that the largest of the metastable atomic concentration is equal to 1.6151011cm3.

Effect of metastable lifetime on the argon spatial distribution of the density of metastable atoms in the steady state.

Conclusion

### Appendix A

*Graphitic wormhole**Electronic structure**Case of massive fermions**Spin-orbit coupling in the wormhole connecting nanotube**Graphene black hole**Spinor fields in biological systems**Circular Artin braid group representation for spinor field in genetic code**Classification of loop braid group in genetic code**Conclusions**Application of Ising model**Phase separation and wetting/dewetting**Lattice-based liquid-gas model**Spin glasses**Mathematical formulation in one dimension**Case A: free boundary with zero field**Critical phenomena**Scaling hypothesis and renormalization group theory**Physical realization: simulation results based on Ising model We now discuss some of the simulation results obtained using Ising model**Experimental details**Porphyrins and hemeproteins**General and basic concepts about nanotechnology, nano-/microrobots (motors), and spintronics**Applications of porphyrins and hemeproteins in spintronics 1 Porphyrins and derivatives**Hemeproteins**Application of porphyrins and hemeproteins in the construction and working of micro-/nanorobots**Porphyrins and derivatives**Hemeproteins**Conclusions and perspectives**Problems statement**Equilibrium equations and Hooke’s law in parabolic coordinates**Boundary conditions**Solution of stated boundary value problems**Interior boundary value problems**Exterior boundary value problems**Test problems**Internal problem**External problem**Conclusion*

Two-dimensional numerical simulation of the DC glow discharge in the normal mode and with. An important measurable quantity in the carbon nanostructures, including the nanotubular part of the graphitic wormhole, is the spin-orbit coupling. In the work, the finite temperature local density of states is predicted which is a realization of the Hawking-Unruh effect.

In [26–28] we introduced a new representation of the genetic code in the time series using a modeling by strings and D-branes. In the following we set h¼0 in our definition of the Chern-Simons current for biology for the simpler derivation of formulas. Interaction and reaction of the antioxidant MnIII [Meso-Tetrakis(4-NMethyI Pyridinium)Porphyrin] with the apoptosis reporter lipid Phosphatidylserine.

### Some basic formulas in parabolic coordinates

Whenξ1!∞, then displacements and stresses tend to zero, that is, the boundary conditions (10) are satisfied. Whenη1!∞, then displacements and stresses tend to zero, that is, the boundary conditions (100) are satisfied. The solution of the equilibrium equations is obtained by the method of separation of variables.

Bodies bounded by a parabola are common in practice, for example in construction, mechanical engineering, biology, medicine, etc., the study of the deformed state of such bodies is topical, therefore, in my opinion, the setting of the discussed problems in the chapter and the way they are solved are interesting from a practical point of view.

### Solution of system of partial differential equations We solve the system of partial differential equations (2)

*Formulation of the problem**A new mathematical modelling of nonlinear generalized micropolar thermoviscoelasticity problem**BEM simulation for temperature field**BEM simulation for micropolar thermoviscoelastic fields**A new boundary element technique for simulation and optimization of solid deformable bodies under different loads**Numerical examples and discussion of results*

F1k=F10< ε is valid; to check the fairness of the selection it is necessary. Representation of solutions of some boundary value problems of elasticity by a sum of the solutions of other boundary value problems. To find the optimal boundary conditions for temperature, the following function can be used.

To find the optimal boundary conditions for temperature, the following function can be used.