_{1}

^{*}

The Bloch NMR as an analytical tool was able to address the fundamental features in the learning of spintronics. Beside confirming past assertions on the Rashba spin-orbit interaction, thermal motion of hole and electron spin and features of the quantum well, it was also able to explain the condition necessary for Rashba splitting within the quantum well. When the Rashba energy is 43 meV, it modified the Ehrenfest’s theorem to hold for an external magnetic field. The confinement potential which is the strength of the Rashba spin-orbit interaction was shown to be controlled magnetically.

The Bloch NMR model has over the years proven to be a good diagnostic tool for investigating properties of known quantities. For example, the Bloch NMR equations has shown good success in the study biological and physiological properties of living tissue [1,2]. Besides medical applications, the Bloch NMR equations were used to investigate the thermodynamic properties of system e.g. the Wegner distribution function [

The technological and computational prospects of semiconductor spintronics have been explored in various researches [5-10]. One of the notable works in the semiconductor spintronics is the Rashba term [11-13]. One of the major successes of the Rashba term is the measurement of the strength of spin splitting using the Rashba energy of split state and coupling parameter. The Rashba energies can be tuned by changing the composition parameters of the surface alloy [14,15] while the Rashba coupling parameter (α_{R}) was calculated as 10^{−1} eVÅ for conventional narrow-gap semiconductor structures. Unfortunately, low Rashba coupling parameter do not supports the 2 d spintronics device. Early this year, a layered semiconductor (bismuth tellurohalides) was found to exhibit a large spin splitting [

In this paper, the Bloch NMR equations were used as a diagnostic tool to analyze the Rashba energies state in a rotating magnetic frame. Derivations of operational equations were used to solve the Rashba energy with the mind set of investigating the conditions necessary for the presence of Rashba splitting in quantum wells and the possibility of controlling the confinement potential magnetically.

The following theoretical assumptions were made in order to incorporate the Bloch NMR into the spintronics device 1) The Rashba parameter was substituted for the spin coupling in the Bloch NMR because they both are synonymous in operation.

2) Let the excitation frequencies in each concept be the same

3) The macroscopic magnetization of the localized was equated with the equilibrium magnetization of the Bloch model.

4) Semiconductors have various electron-hole recombination rates [25,26]. The speed of the holes and electrons is assumed.

5) Both the g tends to 1, likewise the frequency difference between Larmor frequencies.

The Rashba energy and Rashba parameter are related in the equations shown below

The magnetic potential introduced by Zeeman term

The resulting spin-splitting of s-type electron states is given by

From Equations (1)-(5), Equation (6) was derived

where E_{R} = Rashba energy, α_{B} = Rashba parameter, m^{*} = effective mass, u = magnetic potential, σ = spin index, Pauli spin matrices, for electric spin, it is given as

where a = lattice constant, g = g-factor. μ_{B} = mobility.

In this section, a mathematical algorithm to describe in detail the translational mechanical properties of the Bloch NMR equation was developed. A sample of atomic crystal structure is analyzed by the x, y, z component (in the rotating frame) of magnetization given by the Bloch equations which may be written as follows

where the frequency difference between Larmor frequency and frame of reference, is the Rabi frequency, is the Larmor frequency, M_{x}, M_{y} are the transverse magnetization, M_{z} is the longitudinal magnetization, M_{o} is the equilibrium magnetization. The movement of the holes and electrons is assumed to move at a velocity V_{e} and V_{h} and at distances L_{h} & L_{e} from the conduction and valence bands.

where and

The solution of the above equations can be arranged in matrix form as shown below

The matrix multiplication as shown below leads to the steady solutions of the Bloch equations in the rotating frame of reference are shown below

These solutions directly give the frequency response of the magnetization. This idea gives the possibilities of quantitatively calculating the measured signal if a spin system is characterized by relaxation times T_{1} and T_{2}. The term is proportional to the radio frequency power P. At the state of no saturation i.e. for low power P, this term is small .

The operational equations used for the simulations were worked out and systems of equations were generated. Equations (6), (20) and (21) yield

where

where

Equations (17)-(19) yield when

Applying Equations (23)-(27)

Ploting the first and second term of Equations (28) and (29) where

The Rashba energy along the vertical and horizontal axes are represented diagrammatically (shown in

In this section, the simulations of Equations (22)-(24) under various conditions e.g. the time relaxations and magnetic moments. Time relaxations of 9 ns was applied (Laura Fanea et al., 2011) to investigate the behavior of the Rashba energy with respect to its longitudinal and transverse component.

Thermal motion of the hole was investigated in ^{−9} s. Its results was in accordance with past papers [27,28] which was interpreted that the stochastic modulation of the interaction between the heavy and light hole sub-bands may induce a nonadiabatic transitions between them, which may eventually lead to J-relaxation and dephasing. In

Unlike the findings of Eerdunchaolu et al. [_{1} & χ_{2} (which is described as the height). More importantly, it reveals that Rashba split of orbit interaction effect is not dependent on the height in the y direction. This is because the effective magnetic field which is produced by spin-orbit coupling (due to the lack of structure inversion symmetry) is approximately perpendicular to the electron momentum in the quantum ring. _{x}, M_{y} or M_{z}. Very much like the reaction of the electric

fields in the quantum well, the behavioral features of the effective magnetic field in the conduction band (upper part of

The Bloch NMR model has been proven to be efficient in analyzing the fundamentals of spintronics. It was discovered that an increase in the Rashba energy makes no sig-

nificant changes in χ_{1} & χ_{2} (which was described as height). This result, makes the Rashba split of orbit interaction effect independent on the height along the y direction as shown in Figures 9 and 10. Furthermore, this led to the alteration of Ehrenfest’s theorem (which originally applies to the external electric fields), but now applies to the external magnetic field. Therefore, the confinement potential which is the strength of the Rashba spin-orbit interaction can also be controlled magnetically. This idea may be expatiated upon based on further research. Also the Bloch NMR was also effective to analyze the quantum well i.e. even though it is a standing wave, it could still experience Rashba splitting when the quantized electron energy E_{e} ≥ 1 meV and the length of the quantum well is 6 nm.

This work is self-funded. I appreciate Mrs. Jennifer Emetere for editing the script. I appreciate the Head of Physics Department of the above named institution.